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1
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Abstract
I Introduction
I-A Models
I-A1 Correlated Gaussian Mixture Models
I-A2 Correlated Contextual Stochastic Block Models
I-B Prior Works
I-B 1 Graph Matching
Matching correlated random graphs
Database alignment
Attributed graph matching
I-B2 Community Recovery in Correlated Random Graphs
I-C Our Contributions
I-D Notation
II Correlated Gaussian Mixture Models
II-A Exact Matching on Correlated Gaussian Mixture Models
II-B Exact Community Recovery in Correlated Gaussian Mixture Models
III Correlated Contextual Stochastic Block Models
III-A Exact Matching
III-B Exact Community Recovery
IV Outline of the Proof
IV-A Proof Sketch of Theorem 1
IV-B Proof Sketch of Theorem 5
V Discussion and Open Problems
V-1 Closing the information-theoretic gap for exact matching
V-2 Closing the information-theoretic gap for exact community recovery
V-3 Generalizing to more communities
V-4 Multiple correlated graphs
V-5 Efficient algorithms
VI Proof of Theorem 1: Achievability of Exact Matching in Correlated Gaussian Mixture Models
VII Proof of Theorem 2: Impossibility of Exact Matching in Correlated Gaussian Mixture Models
VIII Proof of Theorem 3: Achievability of Exact Community Recovery in Correlated Gaussian Mixture Models
IX Proof of Theorem 5 : Achievability of Exact Matching in Correlated Contextual Stochastic Block Models
IX-A The k𝑘kitalic_k-core matching and the proof of Theorem 5
IX-B Lemmas for the analysis of k𝑘kitalic_k-core matching
IX-C Proofs of Theorems 12 and 13 and Lemmas 5 and 6
X Proof of Theorem 6 : Impossibility of Exact Matching in Correlated Contextual Stochastic Block Models
X-A Proof of Lemma 9
XI Proof of Theorem 7: Achievability of Exact Community Recovery in Correlated Contextual Stochastic Block Models
XII Proof of Theorem 8 : Impossiblity of Exact Community Recovery in Correlated Contextual Stochastic Block Models
XII-A Proof of Lemma 11
XIII Proof of Lemma 1
XIV Technical Tools
References
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Table of Contents
Abstract
I Introduction
I-A Models
I-A1 Correlated Gaussian Mixture Models
I-A2 Correlated Contextual Stochastic Block Models
I-B Prior Works
I-B 1 Graph Matching
Matching correlated random graphs
Database alignment
Attributed graph matching
I-B2 Community Recovery in Correlated Random Graphs
I-C Our Contributions
I-D Notation
II Correlated Gaussian Mixture Models
II-A Exact Matching on Correlated Gaussian Mixture Models
II-B Exact Community Recovery in Correlated Gaussian Mixture Models
III Correlated Contextual Stochastic Block Models
III-A Exact Matching
III-B Exact Community Recovery
IV Outline of the Proof
IV-A Proof Sketch of Theorem 1
IV-B Proof Sketch of Theorem 5
V Discussion and Open Problems
V-1 Closing the information-theoretic gap for exact matching
V-2 Closing the information-theoretic gap for exact community recovery
V-3 Generalizing to more communities
V-4 Multiple correlated graphs
V-5 Efficient algorithms
VI Proof of Theorem 1: Achievability of Exact Matching in Correlated Gaussian Mixture Models
VII Proof of Theorem 2: Impossibility of Exact Matching in Correlated Gaussian Mixture Models
VIII Proof of Theorem 3: Achievability of Exact Community Recovery in Correlated Gaussian Mixture Models
IX Proof of Theorem 5 : Achievability of Exact Matching in Correlated Contextual Stochastic Block Models
IX-A The k𝑘kitalic_k-core matching and the proof of Theorem 5
IX-B Lemmas for the analysis of k𝑘kitalic_k-core matching
IX-C Proofs of Theorems 12 and 13 and Lemmas 5 and 6
X Proof of Theorem 6 : Impossibility of Exact Matching in Correlated Contextual Stochastic Block Models
X-A Proof of Lemma 9
XI Proof of Theorem 7: Achievability of Exact Community Recovery in Correlated Contextual Stochastic Block Models
XII Proof of Theorem 8 : Impossiblity of Exact Community Recovery in Correlated Contextual Stochastic Block Models
XII-A Proof of Lemma 11
XIII Proof of Lemma 1
XIV Technical Tools
References
| 484
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2
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We study community detection in multiple networks whose nodes and edges are jointly correlated. This setting arises naturally in applications such as social platforms, where a shared set of users may exhibit both correlated friendship patterns and correlated attributes across different platforms. Extending the classical Stochastic Block Model (SBM) and its contextual counterpart (CSBM), we introduce the correlated CSBM, which incorporates structural and attribute correlations across graphs. To build intuition, we first analyze correlated Gaussian Mixture Models, wherein only correlated node attributes are available without edges, and identify the conditions under which an estimator minimizing the distance between attributes achieves exact matching of nodes across the two databases. For correlated CSBMs, we develop a two-step procedure that first applies k𝑘kitalic_k-core matching to most nodes using edge information, then refines the matching for the remaining unmatched nodes by leveraging their attributes with a distance-based estimator. We identify the conditions under which the algorithm recovers the exact node correspondence, enabling us to merge the correlated edges and average the correlated attributes for enhanced community detection. Crucially, by aligning and combining graphs, we identify regimes in which community detection is impossible in a single graph but becomes feasible when side information from correlated graphs is incorporated. Our results illustrate how the interplay between graph matching and community recovery can boost performance, broadening the scope of multi-graph, attribute-based community detection.
| 275
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
Abstract
We study community detection in multiple networks whose nodes and edges are jointly correlated. This setting arises naturally in applications such as social platforms, where a shared set of users may exhibit both correlated friendship patterns and correlated attributes across different platforms. Extending the classical Stochastic Block Model (SBM) and its contextual counterpart (CSBM), we introduce the correlated CSBM, which incorporates structural and attribute correlations across graphs. To build intuition, we first analyze correlated Gaussian Mixture Models, wherein only correlated node attributes are available without edges, and identify the conditions under which an estimator minimizing the distance between attributes achieves exact matching of nodes across the two databases. For correlated CSBMs, we develop a two-step procedure that first applies k𝑘kitalic_k-core matching to most nodes using edge information, then refines the matching for the remaining unmatched nodes by leveraging their attributes with a distance-based estimator. We identify the conditions under which the algorithm recovers the exact node correspondence, enabling us to merge the correlated edges and average the correlated attributes for enhanced community detection. Crucially, by aligning and combining graphs, we identify regimes in which community detection is impossible in a single graph but becomes feasible when side information from correlated graphs is incorporated. Our results illustrate how the interplay between graph matching and community recovery can boost performance, broadening the scope of multi-graph, attribute-based community detection.
| 291
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3
|
Identifying community labels of nodes from a given graph or database–often referred to as community recovery or community detection–is a fundamental problem in network analysis, with wide-ranging applications in machine learning, social network analysis, and biology. The principal insight behind many community detection approaches is that nodes within the same community are typically more strongly connected or share similar attributes compared to nodes in different communities. However, real-world networks frequently deviate from such idealized patterns due to noise or the presence of anomalous nodes whose behaviors do not conform to typical intra-community connectivity or attribute similarities, thereby complicating the community recovery process.
A variety of probabilistic models has been developed to provide rigorous frameworks for community detection. Among these, the Stochastic Block Model (SBM) introduced by Holland, Laskey, and Leinhardt [1] remains one of the most widely studied. In the classical SBM, n𝑛nitalic_n nodes are partitioned into r𝑟ritalic_r communities, and edges form with probability p∈[0,1]𝑝01p\in[0,1]italic_p ∈ [ 0 , 1 ] between nodes in the same community, versus q∈[0,p)𝑞0𝑝q\in[0,p)italic_q ∈ [ 0 , italic_p ) between nodes in different communities. Such models capture community structures effectively in various settings–for example, social networks where edges represent friendships or interactions. In SBMs with p=alognn𝑝𝑎𝑛𝑛p=\frac{a\log n}{n}italic_p = divide start_ARG italic_a roman_log italic_n end_ARG start_ARG italic_n end_ARG, q=blognn𝑞𝑏𝑛𝑛q=\frac{b\log n}{n}italic_q = divide start_ARG italic_b roman_log italic_n end_ARG start_ARG italic_n end_ARG for constants a,b>0𝑎𝑏0a,b>0italic_a , italic_b > 0 and r=Θ(1)𝑟Θ1r=\Theta(1)italic_r = roman_Θ ( 1 ), it has been established in [2, 3, 4] that exact community recovery is information-theoretically achievable if and only if a−b>r𝑎𝑏𝑟\sqrt{a}-\sqrt{b}>\sqrt{r}square-root start_ARG italic_a end_ARG - square-root start_ARG italic_b end_ARG > square-root start_ARG italic_r end_ARG. Although SBMs incorporate only network structure, node attributes can play an equally important role in determining community memberships.
Since the standard SBM focuses on graph connectivity alone, it neglects potentially informative node attributes. To remedy this, the Contextual Stochastic Block Model (CSBM) includes node attributes alongside structural connectivity. For instance, in [5], the authors consider a two-community CSBM in which each node has a Gaussian-distributed attribute vector of dimension d𝑑ditalic_d, with mean either 𝝁𝝁{\boldsymbol{\mu}}bold_italic_μ or −𝝁𝝁-{\boldsymbol{\mu}}- bold_italic_μ (depending on the node’s community) and covariance 𝑰dsubscript𝑰𝑑{\boldsymbol{I}}_{d}bold_italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT. This augmented framework leverages both graph structure and node attributes, leading to improved community recovery. Indeed, it is shown in [5] that the Signal-to-Noise Ratio (SNR), derived from both edges and attributes, governs the feasibility of exact community recovery, thereby demonstrating that combining these two sources of information can outperform methods relying on only one.
While CSBMs incorporate attributes in a single-network context, many real-world settings naturally feature multiple, correlated networks. For example, users often participate in more than one social platform (e.g., Facebook and LinkedIn), giving rise to correlated friendship relationships. However, due to privacy and security concerns, user identities are anonymized across different platforms, making it nontrivial to match corresponding users. This task, known as graph matching, is critical for leveraging correlated graphs for downstream tasks. Indeed, finding the exact correspondence between all nodes (i.e., exact matching) enables the construction of a combined graph by merging edges from both platforms, thereby facilitating more accurate community detection. Past research [6, 7] has shown that once exact matching is attainable in correlated SBMs, community recovery becomes easier than if only one graph were available. Moreover, Gaudio et al. [8] established precise information-theoretic limits for exact community recovery under correlated SBMs.
Building on these insights, this work addresses scenarios in which both edges and node attributes are correlated across multiple networks. For example, a user on platforms like Facebook and LinkedIn may exhibit similar friendship connections as well as comparable personal attributes. We posit that these correlated sources of information can significantly enhance community recovery. Concretely, we propose extending Contextual Stochastic Block Models to the correlated setting, resulting in correlated CSBMs.
As a preliminary step, we first investigate correlated Gaussian Mixture Models, which capture correlated attributes alone (i.e., without edges). By generalizing the database alignment techniques from [9, 10], we identify the conditions under which an estimator minimizing the distance between node attributes can reliably recover the underlying permutation π∗:[n]→[n]:subscript𝜋→delimited-[]𝑛delimited-[]𝑛\pi_{*}:[n]\to[n]italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT : [ italic_n ] → [ italic_n ], even when community memberships are initially unknown. For correlated CSBMs, we introduce a two-step algorithm for exact matching: in the first step, we apply the k𝑘kitalic_k-core matching approach [11], which leverages edge information to recover matching for n−o(n)𝑛𝑜𝑛n-o(n)italic_n - italic_o ( italic_n ) nodes; in the second step, we employ the minimum-distance estimator using the node attributes of the remaining unmatched nodes, thereby finalizing the alignment. We derive conditions where this two-step algorithm achieves exact matching.
Once the node alignment is established, merging the correlated edges yields a denser graph structure, while averaging correlated node attributes increases the effective SNR. Crucially, by aligning and combining graphs, we identify regimes in which community detection is impossible in a single graph but becomes feasible when side information from correlated graphs is incorporated. This strategy offers deeper insights into how the interplay between graph matching and community detection can improve overall performance. To the best of our knowledge, we are the first to investigate community recovery in correlated graphs that incorporate correlated node attributes, thereby broadening the scope of existing research on multi-graph and attribute-based community detection.
| 1,455
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
I Introduction
Identifying community labels of nodes from a given graph or database–often referred to as community recovery or community detection–is a fundamental problem in network analysis, with wide-ranging applications in machine learning, social network analysis, and biology. The principal insight behind many community detection approaches is that nodes within the same community are typically more strongly connected or share similar attributes compared to nodes in different communities. However, real-world networks frequently deviate from such idealized patterns due to noise or the presence of anomalous nodes whose behaviors do not conform to typical intra-community connectivity or attribute similarities, thereby complicating the community recovery process.
A variety of probabilistic models has been developed to provide rigorous frameworks for community detection. Among these, the Stochastic Block Model (SBM) introduced by Holland, Laskey, and Leinhardt [1] remains one of the most widely studied. In the classical SBM, n𝑛nitalic_n nodes are partitioned into r𝑟ritalic_r communities, and edges form with probability p∈[0,1]𝑝01p\in[0,1]italic_p ∈ [ 0 , 1 ] between nodes in the same community, versus q∈[0,p)𝑞0𝑝q\in[0,p)italic_q ∈ [ 0 , italic_p ) between nodes in different communities. Such models capture community structures effectively in various settings–for example, social networks where edges represent friendships or interactions. In SBMs with p=alognn𝑝𝑎𝑛𝑛p=\frac{a\log n}{n}italic_p = divide start_ARG italic_a roman_log italic_n end_ARG start_ARG italic_n end_ARG, q=blognn𝑞𝑏𝑛𝑛q=\frac{b\log n}{n}italic_q = divide start_ARG italic_b roman_log italic_n end_ARG start_ARG italic_n end_ARG for constants a,b>0𝑎𝑏0a,b>0italic_a , italic_b > 0 and r=Θ(1)𝑟Θ1r=\Theta(1)italic_r = roman_Θ ( 1 ), it has been established in [2, 3, 4] that exact community recovery is information-theoretically achievable if and only if a−b>r𝑎𝑏𝑟\sqrt{a}-\sqrt{b}>\sqrt{r}square-root start_ARG italic_a end_ARG - square-root start_ARG italic_b end_ARG > square-root start_ARG italic_r end_ARG. Although SBMs incorporate only network structure, node attributes can play an equally important role in determining community memberships.
Since the standard SBM focuses on graph connectivity alone, it neglects potentially informative node attributes. To remedy this, the Contextual Stochastic Block Model (CSBM) includes node attributes alongside structural connectivity. For instance, in [5], the authors consider a two-community CSBM in which each node has a Gaussian-distributed attribute vector of dimension d𝑑ditalic_d, with mean either 𝝁𝝁{\boldsymbol{\mu}}bold_italic_μ or −𝝁𝝁-{\boldsymbol{\mu}}- bold_italic_μ (depending on the node’s community) and covariance 𝑰dsubscript𝑰𝑑{\boldsymbol{I}}_{d}bold_italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT. This augmented framework leverages both graph structure and node attributes, leading to improved community recovery. Indeed, it is shown in [5] that the Signal-to-Noise Ratio (SNR), derived from both edges and attributes, governs the feasibility of exact community recovery, thereby demonstrating that combining these two sources of information can outperform methods relying on only one.
While CSBMs incorporate attributes in a single-network context, many real-world settings naturally feature multiple, correlated networks. For example, users often participate in more than one social platform (e.g., Facebook and LinkedIn), giving rise to correlated friendship relationships. However, due to privacy and security concerns, user identities are anonymized across different platforms, making it nontrivial to match corresponding users. This task, known as graph matching, is critical for leveraging correlated graphs for downstream tasks. Indeed, finding the exact correspondence between all nodes (i.e., exact matching) enables the construction of a combined graph by merging edges from both platforms, thereby facilitating more accurate community detection. Past research [6, 7] has shown that once exact matching is attainable in correlated SBMs, community recovery becomes easier than if only one graph were available. Moreover, Gaudio et al. [8] established precise information-theoretic limits for exact community recovery under correlated SBMs.
Building on these insights, this work addresses scenarios in which both edges and node attributes are correlated across multiple networks. For example, a user on platforms like Facebook and LinkedIn may exhibit similar friendship connections as well as comparable personal attributes. We posit that these correlated sources of information can significantly enhance community recovery. Concretely, we propose extending Contextual Stochastic Block Models to the correlated setting, resulting in correlated CSBMs.
As a preliminary step, we first investigate correlated Gaussian Mixture Models, which capture correlated attributes alone (i.e., without edges). By generalizing the database alignment techniques from [9, 10], we identify the conditions under which an estimator minimizing the distance between node attributes can reliably recover the underlying permutation π∗:[n]→[n]:subscript𝜋→delimited-[]𝑛delimited-[]𝑛\pi_{*}:[n]\to[n]italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT : [ italic_n ] → [ italic_n ], even when community memberships are initially unknown. For correlated CSBMs, we introduce a two-step algorithm for exact matching: in the first step, we apply the k𝑘kitalic_k-core matching approach [11], which leverages edge information to recover matching for n−o(n)𝑛𝑜𝑛n-o(n)italic_n - italic_o ( italic_n ) nodes; in the second step, we employ the minimum-distance estimator using the node attributes of the remaining unmatched nodes, thereby finalizing the alignment. We derive conditions where this two-step algorithm achieves exact matching.
Once the node alignment is established, merging the correlated edges yields a denser graph structure, while averaging correlated node attributes increases the effective SNR. Crucially, by aligning and combining graphs, we identify regimes in which community detection is impossible in a single graph but becomes feasible when side information from correlated graphs is incorporated. This strategy offers deeper insights into how the interplay between graph matching and community detection can improve overall performance. To the best of our knowledge, we are the first to investigate community recovery in correlated graphs that incorporate correlated node attributes, thereby broadening the scope of existing research on multi-graph and attribute-based community detection.
| 1,472
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4
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We introduce two new models to capture correlations: correlated Gaussian Mixture Models, which focus on node attributes alone, and correlated Contextual Stochastic Block Models, which integrate both node attributes and graph structure.
| 40
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
I Introduction
I-A Models
We introduce two new models to capture correlations: correlated Gaussian Mixture Models, which focus on node attributes alone, and correlated Contextual Stochastic Block Models, which integrate both node attributes and graph structure.
| 61
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5
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First, we assign d𝑑ditalic_d dimensional features (or attributes) to n𝑛nitalic_n nodes. Let V1:=[n]assignsubscript𝑉1delimited-[]𝑛V_{1}:=[n]italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT := [ italic_n ] denote the set of nodes in the first database, and for each node i∈V1𝑖subscript𝑉1i\in V_{1}italic_i ∈ italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, the node attribute is given by
, 1 = 𝒙i=𝝁σi+𝒛i,subscript𝒙𝑖𝝁subscript𝜎𝑖subscript𝒛𝑖{\boldsymbol{x}}_{i}=\boldsymbol{\mu}\sigma_{i}+{\boldsymbol{z}}_{i},bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = bold_italic_μ italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + bold_italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ,. , 2 = . , 3 = (1)
where 𝝁∈ℝd𝝁superscriptℝ𝑑\boldsymbol{\mu}\in\mathbb{R}^{d}bold_italic_μ ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT, σi∈{−1,+1}subscript𝜎𝑖11\sigma_{i}\in\{-1,+1\}italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ { - 1 , + 1 } and 𝒛i∼𝒩(0,𝑰d)similar-tosubscript𝒛𝑖𝒩0subscript𝑰𝑑{\boldsymbol{z}}_{i}\sim\mathcal{N}(0,{\boldsymbol{I}}_{d})bold_italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∼ caligraphic_N ( 0 , bold_italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) for all i∈[n]𝑖delimited-[]𝑛i\in[n]italic_i ∈ [ italic_n ].
Let 𝝈:={σi}i=1nassign𝝈superscriptsubscriptsubscript𝜎𝑖𝑖1𝑛{\boldsymbol{\sigma}}:=\{\sigma_{i}\}_{i=1}^{n}bold_italic_σ := { italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT be the vector of community labels associated with the nodes in V1subscript𝑉1V_{1}italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT.
Next, for the node set V2:=[n]assignsubscript𝑉2delimited-[]𝑛V_{2}:=[n]italic_V start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT := [ italic_n ], we assign the attribute
, 1 = 𝒚i=𝝁σi+ρ𝒛i+1−ρ2𝒘i,subscript𝒚𝑖𝝁subscript𝜎𝑖𝜌subscript𝒛𝑖1superscript𝜌2subscript𝒘𝑖{\boldsymbol{y}}_{i}=\boldsymbol{\mu}\sigma_{i}+\rho{\boldsymbol{z}}_{i}+\sqrt%
{1-\rho^{2}}{\boldsymbol{w}}_{i},bold_italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = bold_italic_μ italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + italic_ρ bold_italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + square-root start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG bold_italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ,. , 2 = . , 3 = (2)
where ρ∈[0,1]𝜌01\rho\in[0,1]italic_ρ ∈ [ 0 , 1 ] and 𝒘i∼𝒩(0,𝑰d)similar-tosubscript𝒘𝑖𝒩0subscript𝑰𝑑{\boldsymbol{w}}_{i}\sim\mathcal{N}(0,{\boldsymbol{I}}_{d})bold_italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∼ caligraphic_N ( 0 , bold_italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) for all i∈[n]𝑖delimited-[]𝑛i\in[n]italic_i ∈ [ italic_n ].
Equivalently, for each i∈[n]𝑖delimited-[]𝑛i\in[n]italic_i ∈ [ italic_n ], the pair (𝒙i,𝒚i)subscript𝒙𝑖subscript𝒚𝑖({\boldsymbol{x}}_{i},{\boldsymbol{y}}_{i})( bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) can be represented as
, 1 = (𝒙i,𝒚i)∼𝒩([𝝁σi𝝁σi],Σd),similar-tosubscript𝒙𝑖subscript𝒚𝑖𝒩delimited-[]matrix𝝁subscript𝜎𝑖𝝁subscript𝜎𝑖subscriptΣ𝑑({\boldsymbol{x}}_{i},{\boldsymbol{y}}_{i})\sim\mathcal{N}\left(\left[\begin{%
matrix}\boldsymbol{\mu}\sigma_{i}&\boldsymbol{\mu}\sigma_{i}\end{matrix}\right%
],\Sigma_{d}\right),( bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ∼ caligraphic_N ( [ start_ARG start_ROW start_CELL bold_italic_μ italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_CELL start_CELL bold_italic_μ italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_CELL end_ROW end_ARG ] , roman_Σ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) ,. , 2 = . , 3 = (3)
where
, 1 = Σd:=[𝑰ddiag(ρ)diag(ρ)𝑰d].assignsubscriptΣ𝑑delimited-[]matrixsubscript𝑰𝑑diag𝜌diag𝜌subscript𝑰𝑑\Sigma_{d}:=\left[\begin{matrix}{\boldsymbol{I}}_{d}&\text{diag}(\rho)\\
\text{diag}(\rho)&{\boldsymbol{I}}_{d}\\
\end{matrix}\right].roman_Σ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT := [ start_ARG start_ROW start_CELL bold_italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT end_CELL start_CELL diag ( italic_ρ ) end_CELL end_ROW start_ROW start_CELL diag ( italic_ρ ) end_CELL start_CELL bold_italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT end_CELL end_ROW end_ARG ] .. , 2 = . , 3 = (4)
| 1,732
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
I Introduction
I-A Models
I-A1 Correlated Gaussian Mixture Models
First, we assign d𝑑ditalic_d dimensional features (or attributes) to n𝑛nitalic_n nodes. Let V1:=[n]assignsubscript𝑉1delimited-[]𝑛V_{1}:=[n]italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT := [ italic_n ] denote the set of nodes in the first database, and for each node i∈V1𝑖subscript𝑉1i\in V_{1}italic_i ∈ italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, the node attribute is given by
, 1 = 𝒙i=𝝁σi+𝒛i,subscript𝒙𝑖𝝁subscript𝜎𝑖subscript𝒛𝑖{\boldsymbol{x}}_{i}=\boldsymbol{\mu}\sigma_{i}+{\boldsymbol{z}}_{i},bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = bold_italic_μ italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + bold_italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ,. , 2 = . , 3 = (1)
where 𝝁∈ℝd𝝁superscriptℝ𝑑\boldsymbol{\mu}\in\mathbb{R}^{d}bold_italic_μ ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT, σi∈{−1,+1}subscript𝜎𝑖11\sigma_{i}\in\{-1,+1\}italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ { - 1 , + 1 } and 𝒛i∼𝒩(0,𝑰d)similar-tosubscript𝒛𝑖𝒩0subscript𝑰𝑑{\boldsymbol{z}}_{i}\sim\mathcal{N}(0,{\boldsymbol{I}}_{d})bold_italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∼ caligraphic_N ( 0 , bold_italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) for all i∈[n]𝑖delimited-[]𝑛i\in[n]italic_i ∈ [ italic_n ].
Let 𝝈:={σi}i=1nassign𝝈superscriptsubscriptsubscript𝜎𝑖𝑖1𝑛{\boldsymbol{\sigma}}:=\{\sigma_{i}\}_{i=1}^{n}bold_italic_σ := { italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT be the vector of community labels associated with the nodes in V1subscript𝑉1V_{1}italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT.
Next, for the node set V2:=[n]assignsubscript𝑉2delimited-[]𝑛V_{2}:=[n]italic_V start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT := [ italic_n ], we assign the attribute
, 1 = 𝒚i=𝝁σi+ρ𝒛i+1−ρ2𝒘i,subscript𝒚𝑖𝝁subscript𝜎𝑖𝜌subscript𝒛𝑖1superscript𝜌2subscript𝒘𝑖{\boldsymbol{y}}_{i}=\boldsymbol{\mu}\sigma_{i}+\rho{\boldsymbol{z}}_{i}+\sqrt%
{1-\rho^{2}}{\boldsymbol{w}}_{i},bold_italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = bold_italic_μ italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + italic_ρ bold_italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + square-root start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG bold_italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ,. , 2 = . , 3 = (2)
where ρ∈[0,1]𝜌01\rho\in[0,1]italic_ρ ∈ [ 0 , 1 ] and 𝒘i∼𝒩(0,𝑰d)similar-tosubscript𝒘𝑖𝒩0subscript𝑰𝑑{\boldsymbol{w}}_{i}\sim\mathcal{N}(0,{\boldsymbol{I}}_{d})bold_italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∼ caligraphic_N ( 0 , bold_italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) for all i∈[n]𝑖delimited-[]𝑛i\in[n]italic_i ∈ [ italic_n ].
Equivalently, for each i∈[n]𝑖delimited-[]𝑛i\in[n]italic_i ∈ [ italic_n ], the pair (𝒙i,𝒚i)subscript𝒙𝑖subscript𝒚𝑖({\boldsymbol{x}}_{i},{\boldsymbol{y}}_{i})( bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) can be represented as
, 1 = (𝒙i,𝒚i)∼𝒩([𝝁σi𝝁σi],Σd),similar-tosubscript𝒙𝑖subscript𝒚𝑖𝒩delimited-[]matrix𝝁subscript𝜎𝑖𝝁subscript𝜎𝑖subscriptΣ𝑑({\boldsymbol{x}}_{i},{\boldsymbol{y}}_{i})\sim\mathcal{N}\left(\left[\begin{%
matrix}\boldsymbol{\mu}\sigma_{i}&\boldsymbol{\mu}\sigma_{i}\end{matrix}\right%
],\Sigma_{d}\right),( bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ∼ caligraphic_N ( [ start_ARG start_ROW start_CELL bold_italic_μ italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_CELL start_CELL bold_italic_μ italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_CELL end_ROW end_ARG ] , roman_Σ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) ,. , 2 = . , 3 = (3)
where
, 1 = Σd:=[𝑰ddiag(ρ)diag(ρ)𝑰d].assignsubscriptΣ𝑑delimited-[]matrixsubscript𝑰𝑑diag𝜌diag𝜌subscript𝑰𝑑\Sigma_{d}:=\left[\begin{matrix}{\boldsymbol{I}}_{d}&\text{diag}(\rho)\\
\text{diag}(\rho)&{\boldsymbol{I}}_{d}\\
\end{matrix}\right].roman_Σ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT := [ start_ARG start_ROW start_CELL bold_italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT end_CELL start_CELL diag ( italic_ρ ) end_CELL end_ROW start_ROW start_CELL diag ( italic_ρ ) end_CELL start_CELL bold_italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT end_CELL end_ROW end_ARG ] .. , 2 = . , 3 = (4)
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We view these assigned attributes as two “databases,” which can be represented by the matrices X:=[𝒙1,𝒙2,…,𝒙n]⊤∈ℝn×dassign𝑋superscriptsubscript𝒙1subscript𝒙2…subscript𝒙𝑛topsuperscriptℝ𝑛𝑑X:=[{\boldsymbol{x}}_{1},{\boldsymbol{x}}_{2},\ldots,{\boldsymbol{x}}_{n}]^{%
\top}\in\mathbb{R}^{n\times d}italic_X := [ bold_italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , bold_italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … , bold_italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ] start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_n × italic_d end_POSTSUPERSCRIPT and Y′:=[𝒚1,𝒚2,…,𝒚n]⊤∈ℝn×dassignsuperscript𝑌′superscriptsubscript𝒚1subscript𝒚2…subscript𝒚𝑛topsuperscriptℝ𝑛𝑑Y^{\prime}:=[{\boldsymbol{y}}_{1},{\boldsymbol{y}}_{2},\ldots,{\boldsymbol{y}}%
_{n}]^{\top}\in\mathbb{R}^{n\times d}italic_Y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT := [ bold_italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , bold_italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … , bold_italic_y start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ] start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_n × italic_d end_POSTSUPERSCRIPT. Finally, for a permutation π∗:[n]→[n]:subscript𝜋→delimited-[]𝑛delimited-[]𝑛\pi_{*}:[n]\to[n]italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT : [ italic_n ] → [ italic_n ], define Y=[𝒚π∗(1),𝒚π∗(2),…,𝒚π∗(n)]⊤∈ℝn×d𝑌superscriptsubscript𝒚subscript𝜋1subscript𝒚subscript𝜋2…subscript𝒚subscript𝜋𝑛topsuperscriptℝ𝑛𝑑Y=[{\boldsymbol{y}}_{\pi_{*}(1)},{\boldsymbol{y}}_{\pi_{*}(2)},\ldots,{%
\boldsymbol{y}}_{\pi_{*}(n)}]^{\top}\in\mathbb{R}^{n\times d}italic_Y = [ bold_italic_y start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT , bold_italic_y start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( 2 ) end_POSTSUBSCRIPT , … , bold_italic_y start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_n ) end_POSTSUBSCRIPT ] start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_n × italic_d end_POSTSUPERSCRIPT. We assume π∗subscript𝜋\pi_{*}italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT is uniformly distributed over Snsubscript𝑆𝑛S_{n}italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, the set of all permutations on n𝑛nitalic_n elements. The community label vectors for X𝑋Xitalic_X and Y𝑌Yitalic_Y are given by 𝝈1:=𝝈assignsuperscript𝝈1𝝈{\boldsymbol{\sigma}}^{1}:={\boldsymbol{\sigma}}bold_italic_σ start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT := bold_italic_σ and 𝝈2:=𝝈∘π∗−1assignsuperscript𝝈2𝝈superscriptsubscript𝜋1{\boldsymbol{\sigma}}^{2}:={\boldsymbol{\sigma}}\circ\pi_{*}^{-1}bold_italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT := bold_italic_σ ∘ italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT, respectively. We write the resulting pair of databases as
, 1 = (X,Y)∼CGMMs(n,𝝁,d,ρ).similar-to𝑋𝑌CGMMs𝑛𝝁𝑑𝜌(X,\,Y)\;\sim\;\text{CGMMs}\bigl{(}n,\boldsymbol{\mu},d,\rho\bigr{)}.( italic_X , italic_Y ) ∼ CGMMs ( italic_n , bold_italic_μ , italic_d , italic_ρ ) .. , 2 =
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
I Introduction
I-A Models
I-A1 Correlated Gaussian Mixture Models
We view these assigned attributes as two “databases,” which can be represented by the matrices X:=[𝒙1,𝒙2,…,𝒙n]⊤∈ℝn×dassign𝑋superscriptsubscript𝒙1subscript𝒙2…subscript𝒙𝑛topsuperscriptℝ𝑛𝑑X:=[{\boldsymbol{x}}_{1},{\boldsymbol{x}}_{2},\ldots,{\boldsymbol{x}}_{n}]^{%
\top}\in\mathbb{R}^{n\times d}italic_X := [ bold_italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , bold_italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … , bold_italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ] start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_n × italic_d end_POSTSUPERSCRIPT and Y′:=[𝒚1,𝒚2,…,𝒚n]⊤∈ℝn×dassignsuperscript𝑌′superscriptsubscript𝒚1subscript𝒚2…subscript𝒚𝑛topsuperscriptℝ𝑛𝑑Y^{\prime}:=[{\boldsymbol{y}}_{1},{\boldsymbol{y}}_{2},\ldots,{\boldsymbol{y}}%
_{n}]^{\top}\in\mathbb{R}^{n\times d}italic_Y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT := [ bold_italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , bold_italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … , bold_italic_y start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ] start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_n × italic_d end_POSTSUPERSCRIPT. Finally, for a permutation π∗:[n]→[n]:subscript𝜋→delimited-[]𝑛delimited-[]𝑛\pi_{*}:[n]\to[n]italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT : [ italic_n ] → [ italic_n ], define Y=[𝒚π∗(1),𝒚π∗(2),…,𝒚π∗(n)]⊤∈ℝn×d𝑌superscriptsubscript𝒚subscript𝜋1subscript𝒚subscript𝜋2…subscript𝒚subscript𝜋𝑛topsuperscriptℝ𝑛𝑑Y=[{\boldsymbol{y}}_{\pi_{*}(1)},{\boldsymbol{y}}_{\pi_{*}(2)},\ldots,{%
\boldsymbol{y}}_{\pi_{*}(n)}]^{\top}\in\mathbb{R}^{n\times d}italic_Y = [ bold_italic_y start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT , bold_italic_y start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( 2 ) end_POSTSUBSCRIPT , … , bold_italic_y start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_n ) end_POSTSUBSCRIPT ] start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_n × italic_d end_POSTSUPERSCRIPT. We assume π∗subscript𝜋\pi_{*}italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT is uniformly distributed over Snsubscript𝑆𝑛S_{n}italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, the set of all permutations on n𝑛nitalic_n elements. The community label vectors for X𝑋Xitalic_X and Y𝑌Yitalic_Y are given by 𝝈1:=𝝈assignsuperscript𝝈1𝝈{\boldsymbol{\sigma}}^{1}:={\boldsymbol{\sigma}}bold_italic_σ start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT := bold_italic_σ and 𝝈2:=𝝈∘π∗−1assignsuperscript𝝈2𝝈superscriptsubscript𝜋1{\boldsymbol{\sigma}}^{2}:={\boldsymbol{\sigma}}\circ\pi_{*}^{-1}bold_italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT := bold_italic_σ ∘ italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT, respectively. We write the resulting pair of databases as
, 1 = (X,Y)∼CGMMs(n,𝝁,d,ρ).similar-to𝑋𝑌CGMMs𝑛𝝁𝑑𝜌(X,\,Y)\;\sim\;\text{CGMMs}\bigl{(}n,\boldsymbol{\mu},d,\rho\bigr{)}.( italic_X , italic_Y ) ∼ CGMMs ( italic_n , bold_italic_μ , italic_d , italic_ρ ) .. , 2 =
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Let V=[n]𝑉delimited-[]𝑛V=[n]italic_V = [ italic_n ] be the vertex set, and let 𝝈:={σi}i=1nassign𝝈superscriptsubscriptsubscript𝜎𝑖𝑖1𝑛{\boldsymbol{\sigma}}:=\{\sigma_{i}\}_{i=1}^{n}bold_italic_σ := { italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT be the community labels, where each σi∈{−1,+1}subscript𝜎𝑖11\sigma_{i}\in\{-1,+1\}italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ { - 1 , + 1 } is drawn independently and uniformly at random. We generate a “parent” graph G∼SBM(n,p,q)similar-to𝐺SBM𝑛𝑝𝑞G\sim\text{SBM}(n,p,q)italic_G ∼ SBM ( italic_n , italic_p , italic_q ) with p,q∈[0,1]𝑝𝑞01p,q\in[0,1]italic_p , italic_q ∈ [ 0 , 1 ], p>q𝑝𝑞p>qitalic_p > italic_q, and q=Θ(p)𝑞Θ𝑝q=\Theta(p)italic_q = roman_Θ ( italic_p ) in the following manner. Partition V𝑉Vitalic_V into V+:={i∈[n]:σi=+1}assignsuperscript𝑉conditional-set𝑖delimited-[]𝑛subscript𝜎𝑖1V^{+}:=\{i\in[n]:\sigma_{i}=+1\}italic_V start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT := { italic_i ∈ [ italic_n ] : italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = + 1 } and V−:={i∈[n]:σi=−1}assignsuperscript𝑉conditional-set𝑖delimited-[]𝑛subscript𝜎𝑖1V^{-}:=\{i\in[n]:\sigma_{i}=-1\}italic_V start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT := { italic_i ∈ [ italic_n ] : italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = - 1 }. If σuσv=+1subscript𝜎𝑢subscript𝜎𝑣1\sigma_{u}\sigma_{v}=+1italic_σ start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT = + 1, an edge (u,v)𝑢𝑣(u,v)( italic_u , italic_v ) is placed with probability p𝑝pitalic_p; if σuσv=−1subscript𝜎𝑢subscript𝜎𝑣1\sigma_{u}\sigma_{v}=-1italic_σ start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT = - 1, an edge is placed with probability q𝑞qitalic_q. The graph G1subscript𝐺1G_{1}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is then obtained by sampling every edge of G𝐺Gitalic_G independently with probability s𝑠sitalic_s. Similarly, G2′subscriptsuperscript𝐺′2G^{\prime}_{2}italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT is generated by the same sampling procedure, ensuring that G1subscript𝐺1G_{1}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and G2′subscriptsuperscript𝐺′2G^{\prime}_{2}italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT are subgraphs of G𝐺Gitalic_G with
ℙ{(u,v)∈ℰ(G2′)|(u,v)∈ℰ(G1)}=sℙconditional-set𝑢𝑣ℰsubscriptsuperscript𝐺′2𝑢𝑣ℰsubscript𝐺1𝑠\mathbb{P}\left\{(u,v)\in\mathcal{E}(G^{\prime}_{2})|(u,v)\in\mathcal{E}(G_{1}%
)\right\}=sblackboard_P { ( italic_u , italic_v ) ∈ caligraphic_E ( italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) | ( italic_u , italic_v ) ∈ caligraphic_E ( italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) } = italic_s
for all distinct u,v∈[n]𝑢𝑣delimited-[]𝑛u,v\in[n]italic_u , italic_v ∈ [ italic_n ], where ℰ(G)ℰ𝐺\mathcal{E}(G)caligraphic_E ( italic_G ) denotes the edge set of graph G𝐺Gitalic_G.
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
I Introduction
I-A Models
I-A2 Correlated Contextual Stochastic Block Models
Let V=[n]𝑉delimited-[]𝑛V=[n]italic_V = [ italic_n ] be the vertex set, and let 𝝈:={σi}i=1nassign𝝈superscriptsubscriptsubscript𝜎𝑖𝑖1𝑛{\boldsymbol{\sigma}}:=\{\sigma_{i}\}_{i=1}^{n}bold_italic_σ := { italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT be the community labels, where each σi∈{−1,+1}subscript𝜎𝑖11\sigma_{i}\in\{-1,+1\}italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ { - 1 , + 1 } is drawn independently and uniformly at random. We generate a “parent” graph G∼SBM(n,p,q)similar-to𝐺SBM𝑛𝑝𝑞G\sim\text{SBM}(n,p,q)italic_G ∼ SBM ( italic_n , italic_p , italic_q ) with p,q∈[0,1]𝑝𝑞01p,q\in[0,1]italic_p , italic_q ∈ [ 0 , 1 ], p>q𝑝𝑞p>qitalic_p > italic_q, and q=Θ(p)𝑞Θ𝑝q=\Theta(p)italic_q = roman_Θ ( italic_p ) in the following manner. Partition V𝑉Vitalic_V into V+:={i∈[n]:σi=+1}assignsuperscript𝑉conditional-set𝑖delimited-[]𝑛subscript𝜎𝑖1V^{+}:=\{i\in[n]:\sigma_{i}=+1\}italic_V start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT := { italic_i ∈ [ italic_n ] : italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = + 1 } and V−:={i∈[n]:σi=−1}assignsuperscript𝑉conditional-set𝑖delimited-[]𝑛subscript𝜎𝑖1V^{-}:=\{i\in[n]:\sigma_{i}=-1\}italic_V start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT := { italic_i ∈ [ italic_n ] : italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = - 1 }. If σuσv=+1subscript𝜎𝑢subscript𝜎𝑣1\sigma_{u}\sigma_{v}=+1italic_σ start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT = + 1, an edge (u,v)𝑢𝑣(u,v)( italic_u , italic_v ) is placed with probability p𝑝pitalic_p; if σuσv=−1subscript𝜎𝑢subscript𝜎𝑣1\sigma_{u}\sigma_{v}=-1italic_σ start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT = - 1, an edge is placed with probability q𝑞qitalic_q. The graph G1subscript𝐺1G_{1}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is then obtained by sampling every edge of G𝐺Gitalic_G independently with probability s𝑠sitalic_s. Similarly, G2′subscriptsuperscript𝐺′2G^{\prime}_{2}italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT is generated by the same sampling procedure, ensuring that G1subscript𝐺1G_{1}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and G2′subscriptsuperscript𝐺′2G^{\prime}_{2}italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT are subgraphs of G𝐺Gitalic_G with
ℙ{(u,v)∈ℰ(G2′)|(u,v)∈ℰ(G1)}=sℙconditional-set𝑢𝑣ℰsubscriptsuperscript𝐺′2𝑢𝑣ℰsubscript𝐺1𝑠\mathbb{P}\left\{(u,v)\in\mathcal{E}(G^{\prime}_{2})|(u,v)\in\mathcal{E}(G_{1}%
)\right\}=sblackboard_P { ( italic_u , italic_v ) ∈ caligraphic_E ( italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) | ( italic_u , italic_v ) ∈ caligraphic_E ( italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) } = italic_s
for all distinct u,v∈[n]𝑢𝑣delimited-[]𝑛u,v\in[n]italic_u , italic_v ∈ [ italic_n ], where ℰ(G)ℰ𝐺\mathcal{E}(G)caligraphic_E ( italic_G ) denotes the edge set of graph G𝐺Gitalic_G.
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Each node in G1subscript𝐺1G_{1}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and G2′subscriptsuperscript𝐺′2G^{\prime}_{2}italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT is assigned correlated Gaussian attributes {𝒙i}subscript𝒙𝑖\{{\boldsymbol{x}}_{i}\}{ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT } and {𝒚i}subscript𝒚𝑖\{{\boldsymbol{y}}_{i}\}{ bold_italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT } as defined in (3), where 𝝁𝝁\boldsymbol{\mu}bold_italic_μ is uniformly distributed over the set
{𝝁∈ℝd:‖𝝁‖2=R}conditional-set𝝁superscriptℝ𝑑superscriptnorm𝝁2𝑅\bigl{\{}\boldsymbol{\mu}\in\mathbb{R}^{d}:\|\boldsymbol{\mu}\|^{2}=R\bigr{\}}{ bold_italic_μ ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT : ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = italic_R }
for some R>0𝑅0R>0italic_R > 0. Finally, G2subscript𝐺2G_{2}italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT is obtained by permuting the nodes of G2′subscriptsuperscript𝐺′2G^{\prime}_{2}italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT using π∗:[n]→[n]:subscript𝜋→delimited-[]𝑛delimited-[]𝑛\pi_{*}:[n]\to[n]italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT : [ italic_n ] → [ italic_n ]. The community labels for G1subscript𝐺1G_{1}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and G2subscript𝐺2G_{2}italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT are given by 𝝈1:=𝝈assignsuperscript𝝈1𝝈{\boldsymbol{\sigma}}^{1}:={\boldsymbol{\sigma}}bold_italic_σ start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT := bold_italic_σ and 𝝈2:=𝝈∘π∗−1assignsuperscript𝝈2𝝈superscriptsubscript𝜋1{\boldsymbol{\sigma}}^{2}:={\boldsymbol{\sigma}}\circ\pi_{*}^{-1}bold_italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT := bold_italic_σ ∘ italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT, respectively. Denoting the database (attribute) matrices by X,Y′𝑋superscript𝑌′X,Y^{\prime}italic_X , italic_Y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT, and Y𝑌Yitalic_Y, and the adjacency matrices by A,B′𝐴superscript𝐵′A,B^{\prime}italic_A , italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT, and B𝐵Bitalic_B, we denote the resulting graphs by
, 1 = (G1,G2)∼CCSBMs(n,p,q,s;R,d,ρ).similar-tosubscript𝐺1subscript𝐺2CCSBMs𝑛𝑝𝑞𝑠𝑅𝑑𝜌(G_{1},G_{2})\;\sim\;\text{CCSBMs}\bigl{(}n,p,q,s;\,R,d,\rho\bigr{)}.( italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ∼ CCSBMs ( italic_n , italic_p , italic_q , italic_s ; italic_R , italic_d , italic_ρ ) .. , 2 =
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
I Introduction
I-A Models
I-A2 Correlated Contextual Stochastic Block Models
Each node in G1subscript𝐺1G_{1}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and G2′subscriptsuperscript𝐺′2G^{\prime}_{2}italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT is assigned correlated Gaussian attributes {𝒙i}subscript𝒙𝑖\{{\boldsymbol{x}}_{i}\}{ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT } and {𝒚i}subscript𝒚𝑖\{{\boldsymbol{y}}_{i}\}{ bold_italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT } as defined in (3), where 𝝁𝝁\boldsymbol{\mu}bold_italic_μ is uniformly distributed over the set
{𝝁∈ℝd:‖𝝁‖2=R}conditional-set𝝁superscriptℝ𝑑superscriptnorm𝝁2𝑅\bigl{\{}\boldsymbol{\mu}\in\mathbb{R}^{d}:\|\boldsymbol{\mu}\|^{2}=R\bigr{\}}{ bold_italic_μ ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT : ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = italic_R }
for some R>0𝑅0R>0italic_R > 0. Finally, G2subscript𝐺2G_{2}italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT is obtained by permuting the nodes of G2′subscriptsuperscript𝐺′2G^{\prime}_{2}italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT using π∗:[n]→[n]:subscript𝜋→delimited-[]𝑛delimited-[]𝑛\pi_{*}:[n]\to[n]italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT : [ italic_n ] → [ italic_n ]. The community labels for G1subscript𝐺1G_{1}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and G2subscript𝐺2G_{2}italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT are given by 𝝈1:=𝝈assignsuperscript𝝈1𝝈{\boldsymbol{\sigma}}^{1}:={\boldsymbol{\sigma}}bold_italic_σ start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT := bold_italic_σ and 𝝈2:=𝝈∘π∗−1assignsuperscript𝝈2𝝈superscriptsubscript𝜋1{\boldsymbol{\sigma}}^{2}:={\boldsymbol{\sigma}}\circ\pi_{*}^{-1}bold_italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT := bold_italic_σ ∘ italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT, respectively. Denoting the database (attribute) matrices by X,Y′𝑋superscript𝑌′X,Y^{\prime}italic_X , italic_Y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT, and Y𝑌Yitalic_Y, and the adjacency matrices by A,B′𝐴superscript𝐵′A,B^{\prime}italic_A , italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT, and B𝐵Bitalic_B, we denote the resulting graphs by
, 1 = (G1,G2)∼CCSBMs(n,p,q,s;R,d,ρ).similar-tosubscript𝐺1subscript𝐺2CCSBMs𝑛𝑝𝑞𝑠𝑅𝑑𝜌(G_{1},G_{2})\;\sim\;\text{CCSBMs}\bigl{(}n,p,q,s;\,R,d,\rho\bigr{)}.( italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ∼ CCSBMs ( italic_n , italic_p , italic_q , italic_s ; italic_R , italic_d , italic_ρ ) .. , 2 =
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In Table I, we present various graph models–including our newly introduced ones–classified according to whether they incorporate community structure, edges, node attributes, or correlated graphs. Table II provides a summary of information-theoretic limits for graph matching in correlated graphs and community recovery in graphs with community structure, highlighting the performance gains achievable in our proposed models by incorporating correlated edges and/or node attributes.
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
I Introduction
I-B Prior Works
In Table I, we present various graph models–including our newly introduced ones–classified according to whether they incorporate community structure, edges, node attributes, or correlated graphs. Table II provides a summary of information-theoretic limits for graph matching in correlated graphs and community recovery in graphs with community structure, highlighting the performance gains achievable in our proposed models by incorporating correlated edges and/or node attributes.
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One of the most extensively studied settings for graph matching is the correlated Erdős–Rényi (ER) model, first proposed in [15]. In this model, the parent graph G𝐺Gitalic_G is drawn from 𝒢(n,p)𝒢𝑛𝑝\mathcal{G}(n,p)caligraphic_G ( italic_n , italic_p ) (an ER graph), and G1subscript𝐺1G_{1}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and G2′subscriptsuperscript𝐺′2G^{\prime}_{2}italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT are obtained by independently sampling every edge of G𝐺Gitalic_G with probability s𝑠sitalic_s twice. Cullina and Kiyavash [16, 17] provided the first information-theoretic limits for exact matching, showing that exact matching is possible if nps2≥logn+ω(1)𝑛𝑝superscript𝑠2𝑛𝜔1nps^{2}\geq\log n+\omega(1)italic_n italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ roman_log italic_n + italic_ω ( 1 ) under the condition p≤O(1/(logn)3)𝑝𝑂1superscript𝑛3p\leq O(1/(\log n)^{3})italic_p ≤ italic_O ( 1 / ( roman_log italic_n ) start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ). More recently, Wu et al. [12] showed that exact matching remains feasible whenever p=o(1)𝑝𝑜1p=o(1)italic_p = italic_o ( 1 ) and nps2≥(1+ϵ)logn𝑛𝑝superscript𝑠21italic-ϵ𝑛nps^{2}\geq(1+\epsilon)\log nitalic_n italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ ( 1 + italic_ϵ ) roman_log italic_n for any fixed ϵ>0italic-ϵ0\epsilon>0italic_ϵ > 0. However, these proofs rely on checking all permutations, yielding time complexity on the order of Θ(n!)Θ𝑛\Theta(n!)roman_Θ ( italic_n ! ). Consequently, a significant effort has focused on more efficient algorithms. Quasi-polynomial time (nO(logn)superscript𝑛𝑂𝑛n^{O(\log n)}italic_n start_POSTSUPERSCRIPT italic_O ( roman_log italic_n ) end_POSTSUPERSCRIPT) approaches were proposed in [18, 19], while polynomial-time algorithms in [20, 21, 22] achieve exact matching under s=1−o(1)𝑠1𝑜1s=1-o(1)italic_s = 1 - italic_o ( 1 ). Recently, the first polynomial-time algorithms for constant correlation s≥α𝑠𝛼s\geq\alphaitalic_s ≥ italic_α (for a suitable constant α𝛼\alphaitalic_α) appeared in [23, 24], using subgraph counting or large-neighborhood statistics.
Graph matching under correlated Stochastic Block Models (SBMs), where the parent graph is an SBM, has also been investigated [25, 26]. Assuming known community labels in each graph, Onaran et al. [25] showed that exact matching is possible when s(1−1−s2)p+q2≥3logn𝑠11superscript𝑠2𝑝𝑞23𝑛s(1-\sqrt{1-s^{2}})\frac{p+q}{2}\geq 3\log nitalic_s ( 1 - square-root start_ARG 1 - italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) divide start_ARG italic_p + italic_q end_ARG start_ARG 2 end_ARG ≥ 3 roman_log italic_n for two communities. Cullina et al. [26] extended this to r𝑟ritalic_r communities, where p=alognn𝑝𝑎𝑛𝑛p=\frac{a\log n}{n}italic_p = divide start_ARG italic_a roman_log italic_n end_ARG start_ARG italic_n end_ARG and q=blognn𝑞𝑏𝑛𝑛q=\frac{b\log n}{n}italic_q = divide start_ARG italic_b roman_log italic_n end_ARG start_ARG italic_n end_ARG, demonstrating that exact matching holds if s2a+(r−1)br>2superscript𝑠2𝑎𝑟1𝑏𝑟2s^{2}\frac{a+(r-1)b}{r}>2italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divide start_ARG italic_a + ( italic_r - 1 ) italic_b end_ARG start_ARG italic_r end_ARG > 2. Notably, in the special case p=q𝑝𝑞p=qitalic_p = italic_q, correlated SBMs reduce to correlated ER graphs; even under known labels, the bounds in [25, 26] differ from the information-theoretic limit in the correlated ER setting. Rácz and Sridhar [6] refined these results for r=2𝑟2r=2italic_r = 2, proving that exact matching is possible if s2a+b2>1superscript𝑠2𝑎𝑏21s^{2}\,\frac{a+b}{2}>1italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divide start_ARG italic_a + italic_b end_ARG start_ARG 2 end_ARG > 1. Yang and Chung [7] generalized these findings to SBMs with r𝑟ritalic_r communities, showing that exact matching holds if ns2p+(r−1)qr≥(1+ϵ)logn𝑛superscript𝑠2𝑝𝑟1𝑞𝑟1italic-ϵ𝑛ns^{2}\,\frac{p+(r-1)q}{r}\geq(1+\epsilon)\log nitalic_n italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divide start_ARG italic_p + ( italic_r - 1 ) italic_q end_ARG start_ARG italic_r end_ARG ≥ ( 1 + italic_ϵ ) roman_log italic_n, under mild assumptions. As before, achieving this bound requires a time complexity of Θ(n!)Θ𝑛\Theta(n!)roman_Θ ( italic_n ! ). Yang et al. [27] designed a polynomial-time algorithm under constant correlation when community labels are known, and Chai and Rácz [28] recently devised a polynomial-time method that obviates label information.
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
I Introduction
I-B Prior Works
I-B1 Graph Matching
Matching correlated random graphs
One of the most extensively studied settings for graph matching is the correlated Erdős–Rényi (ER) model, first proposed in [15]. In this model, the parent graph G𝐺Gitalic_G is drawn from 𝒢(n,p)𝒢𝑛𝑝\mathcal{G}(n,p)caligraphic_G ( italic_n , italic_p ) (an ER graph), and G1subscript𝐺1G_{1}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and G2′subscriptsuperscript𝐺′2G^{\prime}_{2}italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT are obtained by independently sampling every edge of G𝐺Gitalic_G with probability s𝑠sitalic_s twice. Cullina and Kiyavash [16, 17] provided the first information-theoretic limits for exact matching, showing that exact matching is possible if nps2≥logn+ω(1)𝑛𝑝superscript𝑠2𝑛𝜔1nps^{2}\geq\log n+\omega(1)italic_n italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ roman_log italic_n + italic_ω ( 1 ) under the condition p≤O(1/(logn)3)𝑝𝑂1superscript𝑛3p\leq O(1/(\log n)^{3})italic_p ≤ italic_O ( 1 / ( roman_log italic_n ) start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ). More recently, Wu et al. [12] showed that exact matching remains feasible whenever p=o(1)𝑝𝑜1p=o(1)italic_p = italic_o ( 1 ) and nps2≥(1+ϵ)logn𝑛𝑝superscript𝑠21italic-ϵ𝑛nps^{2}\geq(1+\epsilon)\log nitalic_n italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ ( 1 + italic_ϵ ) roman_log italic_n for any fixed ϵ>0italic-ϵ0\epsilon>0italic_ϵ > 0. However, these proofs rely on checking all permutations, yielding time complexity on the order of Θ(n!)Θ𝑛\Theta(n!)roman_Θ ( italic_n ! ). Consequently, a significant effort has focused on more efficient algorithms. Quasi-polynomial time (nO(logn)superscript𝑛𝑂𝑛n^{O(\log n)}italic_n start_POSTSUPERSCRIPT italic_O ( roman_log italic_n ) end_POSTSUPERSCRIPT) approaches were proposed in [18, 19], while polynomial-time algorithms in [20, 21, 22] achieve exact matching under s=1−o(1)𝑠1𝑜1s=1-o(1)italic_s = 1 - italic_o ( 1 ). Recently, the first polynomial-time algorithms for constant correlation s≥α𝑠𝛼s\geq\alphaitalic_s ≥ italic_α (for a suitable constant α𝛼\alphaitalic_α) appeared in [23, 24], using subgraph counting or large-neighborhood statistics.
Graph matching under correlated Stochastic Block Models (SBMs), where the parent graph is an SBM, has also been investigated [25, 26]. Assuming known community labels in each graph, Onaran et al. [25] showed that exact matching is possible when s(1−1−s2)p+q2≥3logn𝑠11superscript𝑠2𝑝𝑞23𝑛s(1-\sqrt{1-s^{2}})\frac{p+q}{2}\geq 3\log nitalic_s ( 1 - square-root start_ARG 1 - italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) divide start_ARG italic_p + italic_q end_ARG start_ARG 2 end_ARG ≥ 3 roman_log italic_n for two communities. Cullina et al. [26] extended this to r𝑟ritalic_r communities, where p=alognn𝑝𝑎𝑛𝑛p=\frac{a\log n}{n}italic_p = divide start_ARG italic_a roman_log italic_n end_ARG start_ARG italic_n end_ARG and q=blognn𝑞𝑏𝑛𝑛q=\frac{b\log n}{n}italic_q = divide start_ARG italic_b roman_log italic_n end_ARG start_ARG italic_n end_ARG, demonstrating that exact matching holds if s2a+(r−1)br>2superscript𝑠2𝑎𝑟1𝑏𝑟2s^{2}\frac{a+(r-1)b}{r}>2italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divide start_ARG italic_a + ( italic_r - 1 ) italic_b end_ARG start_ARG italic_r end_ARG > 2. Notably, in the special case p=q𝑝𝑞p=qitalic_p = italic_q, correlated SBMs reduce to correlated ER graphs; even under known labels, the bounds in [25, 26] differ from the information-theoretic limit in the correlated ER setting. Rácz and Sridhar [6] refined these results for r=2𝑟2r=2italic_r = 2, proving that exact matching is possible if s2a+b2>1superscript𝑠2𝑎𝑏21s^{2}\,\frac{a+b}{2}>1italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divide start_ARG italic_a + italic_b end_ARG start_ARG 2 end_ARG > 1. Yang and Chung [7] generalized these findings to SBMs with r𝑟ritalic_r communities, showing that exact matching holds if ns2p+(r−1)qr≥(1+ϵ)logn𝑛superscript𝑠2𝑝𝑟1𝑞𝑟1italic-ϵ𝑛ns^{2}\,\frac{p+(r-1)q}{r}\geq(1+\epsilon)\log nitalic_n italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divide start_ARG italic_p + ( italic_r - 1 ) italic_q end_ARG start_ARG italic_r end_ARG ≥ ( 1 + italic_ϵ ) roman_log italic_n, under mild assumptions. As before, achieving this bound requires a time complexity of Θ(n!)Θ𝑛\Theta(n!)roman_Θ ( italic_n ! ). Yang et al. [27] designed a polynomial-time algorithm under constant correlation when community labels are known, and Chai and Rácz [28] recently devised a polynomial-time method that obviates label information.
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Database alignment [9, 29, 30, 31] addresses the problem of finding a one-to-one correspondence between nodes in two “databases,” where each node is associated with correlated attributes. Similar to graph matching, various models have been proposed, among which the correlated Gaussian database model is popular. In this model, each pair of corresponding nodes (𝒙i,𝒚i)subscript𝒙𝑖subscript𝒚𝑖({\boldsymbol{x}}_{i},{\boldsymbol{y}}_{i})( bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) is drawn i.i.d. from 𝒩(𝝁,Σd)𝒩𝝁subscriptΣ𝑑\mathcal{N}(\boldsymbol{\mu},\Sigma_{d})caligraphic_N ( bold_italic_μ , roman_Σ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ), where 𝝁∈ℝ2d𝝁superscriptℝ2𝑑\boldsymbol{\mu}\in\mathbb{R}^{2d}bold_italic_μ ∈ blackboard_R start_POSTSUPERSCRIPT 2 italic_d end_POSTSUPERSCRIPT and
Σd=[𝑰ddiag(ρ)diag(ρ)𝑰d].subscriptΣ𝑑matrixsubscript𝑰𝑑diag𝜌diag𝜌subscript𝑰𝑑\Sigma_{d}=\begin{bmatrix}{\boldsymbol{I}}_{d}&\text{diag}(\rho)\\
\text{diag}(\rho)&{\boldsymbol{I}}_{d}\end{bmatrix}.roman_Σ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT = [ start_ARG start_ROW start_CELL bold_italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT end_CELL start_CELL diag ( italic_ρ ) end_CELL end_ROW start_ROW start_CELL diag ( italic_ρ ) end_CELL start_CELL bold_italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT end_CELL end_ROW end_ARG ] .
Dai et al. [9] showed that exact alignment is possible if d4log11−ρ2≥logn+ω(1)𝑑411superscript𝜌2𝑛𝜔1\tfrac{d}{4}\log\tfrac{1}{1-\rho^{2}}\geq\log n+\omega(1)divide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≥ roman_log italic_n + italic_ω ( 1 ). Their method uses the maximum a posteriori (MAP) estimator, with a time complexity of O(n2d+n3)𝑂superscript𝑛2𝑑superscript𝑛3O(n^{2}d+n^{3})italic_O ( italic_n start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_d + italic_n start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ).
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
I Introduction
I-B Prior Works
I-B1 Graph Matching
Database alignment
Database alignment [9, 29, 30, 31] addresses the problem of finding a one-to-one correspondence between nodes in two “databases,” where each node is associated with correlated attributes. Similar to graph matching, various models have been proposed, among which the correlated Gaussian database model is popular. In this model, each pair of corresponding nodes (𝒙i,𝒚i)subscript𝒙𝑖subscript𝒚𝑖({\boldsymbol{x}}_{i},{\boldsymbol{y}}_{i})( bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) is drawn i.i.d. from 𝒩(𝝁,Σd)𝒩𝝁subscriptΣ𝑑\mathcal{N}(\boldsymbol{\mu},\Sigma_{d})caligraphic_N ( bold_italic_μ , roman_Σ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ), where 𝝁∈ℝ2d𝝁superscriptℝ2𝑑\boldsymbol{\mu}\in\mathbb{R}^{2d}bold_italic_μ ∈ blackboard_R start_POSTSUPERSCRIPT 2 italic_d end_POSTSUPERSCRIPT and
Σd=[𝑰ddiag(ρ)diag(ρ)𝑰d].subscriptΣ𝑑matrixsubscript𝑰𝑑diag𝜌diag𝜌subscript𝑰𝑑\Sigma_{d}=\begin{bmatrix}{\boldsymbol{I}}_{d}&\text{diag}(\rho)\\
\text{diag}(\rho)&{\boldsymbol{I}}_{d}\end{bmatrix}.roman_Σ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT = [ start_ARG start_ROW start_CELL bold_italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT end_CELL start_CELL diag ( italic_ρ ) end_CELL end_ROW start_ROW start_CELL diag ( italic_ρ ) end_CELL start_CELL bold_italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT end_CELL end_ROW end_ARG ] .
Dai et al. [9] showed that exact alignment is possible if d4log11−ρ2≥logn+ω(1)𝑑411superscript𝜌2𝑛𝜔1\tfrac{d}{4}\log\tfrac{1}{1-\rho^{2}}\geq\log n+\omega(1)divide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≥ roman_log italic_n + italic_ω ( 1 ). Their method uses the maximum a posteriori (MAP) estimator, with a time complexity of O(n2d+n3)𝑂superscript𝑛2𝑑superscript𝑛3O(n^{2}d+n^{3})italic_O ( italic_n start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_d + italic_n start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ).
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In many social networks, users (nodes) have both connections (edges) and personal attributes. The attributed graph alignment problem aims to match nodes across two correlated graphs while exploiting both edge structure and node features. In the correlated Gaussian-attributed ER model [13], the edges come from correlated ER graphs, and node attributes come from correlated Gaussian databases. It was shown that exact matching is possible if
, 1 = nps2+d4log11−ρ2≥(1+ϵ)logn,𝑛𝑝superscript𝑠2𝑑411superscript𝜌21italic-ϵ𝑛nps^{2}+\frac{d}{4}\log\frac{1}{1-\rho^{2}}\geq(1+\epsilon)\log n,italic_n italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + divide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≥ ( 1 + italic_ϵ ) roman_log italic_n ,. , 2 = . , 3 = (5)
indicating that the effective SNR is an additive combination of edge- and attribute-based signals.
Zhang et al. [32] introduced an alternative attributed ER pair model with n𝑛nitalic_n user nodes and m𝑚mitalic_m attribute nodes, assuming that the m𝑚mitalic_m attribute nodes are pre-aligned. Edges between user nodes appear with probability p𝑝pitalic_p, while edges between users and attribute nodes appear with probability q𝑞qitalic_q. Similar to the correlated ER framework, edges are independently subsampled with probabilities spsubscript𝑠𝑝s_{p}italic_s start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT and sqsubscript𝑠𝑞s_{q}italic_s start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT for user-user and user-attribute edges, respectively, and a permutation π∗subscript𝜋\pi_{*}italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT is applied to yield G1subscript𝐺1G_{1}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and G2subscript𝐺2G_{2}italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. Exact matching is possible if npsp2+mqsq2≥logn+ω(1)𝑛𝑝superscriptsubscript𝑠𝑝2𝑚𝑞superscriptsubscript𝑠𝑞2𝑛𝜔1nps_{p}^{2}+mqs_{q}^{2}\geq\log n+\omega(1)italic_n italic_p italic_s start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_m italic_q italic_s start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ roman_log italic_n + italic_ω ( 1 ). Polynomial-time algorithms for recovering π∗subscript𝜋\pi_{*}italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT in this setting have been explored in [33].
| 746
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
I Introduction
I-B Prior Works
I-B1 Graph Matching
Attributed graph matching
In many social networks, users (nodes) have both connections (edges) and personal attributes. The attributed graph alignment problem aims to match nodes across two correlated graphs while exploiting both edge structure and node features. In the correlated Gaussian-attributed ER model [13], the edges come from correlated ER graphs, and node attributes come from correlated Gaussian databases. It was shown that exact matching is possible if
, 1 = nps2+d4log11−ρ2≥(1+ϵ)logn,𝑛𝑝superscript𝑠2𝑑411superscript𝜌21italic-ϵ𝑛nps^{2}+\frac{d}{4}\log\frac{1}{1-\rho^{2}}\geq(1+\epsilon)\log n,italic_n italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + divide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≥ ( 1 + italic_ϵ ) roman_log italic_n ,. , 2 = . , 3 = (5)
indicating that the effective SNR is an additive combination of edge- and attribute-based signals.
Zhang et al. [32] introduced an alternative attributed ER pair model with n𝑛nitalic_n user nodes and m𝑚mitalic_m attribute nodes, assuming that the m𝑚mitalic_m attribute nodes are pre-aligned. Edges between user nodes appear with probability p𝑝pitalic_p, while edges between users and attribute nodes appear with probability q𝑞qitalic_q. Similar to the correlated ER framework, edges are independently subsampled with probabilities spsubscript𝑠𝑝s_{p}italic_s start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT and sqsubscript𝑠𝑞s_{q}italic_s start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT for user-user and user-attribute edges, respectively, and a permutation π∗subscript𝜋\pi_{*}italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT is applied to yield G1subscript𝐺1G_{1}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and G2subscript𝐺2G_{2}italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. Exact matching is possible if npsp2+mqsq2≥logn+ω(1)𝑛𝑝superscriptsubscript𝑠𝑝2𝑚𝑞superscriptsubscript𝑠𝑞2𝑛𝜔1nps_{p}^{2}+mqs_{q}^{2}\geq\log n+\omega(1)italic_n italic_p italic_s start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_m italic_q italic_s start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ roman_log italic_n + italic_ω ( 1 ). Polynomial-time algorithms for recovering π∗subscript𝜋\pi_{*}italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT in this setting have been explored in [33].
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13
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Rácz and Sridhar [6] first investigated exact community recovery in the presence of two or more correlated networks. Focusing on correlated SBMs with p=alognn𝑝𝑎𝑛𝑛p=\frac{a\log n}{n}italic_p = divide start_ARG italic_a roman_log italic_n end_ARG start_ARG italic_n end_ARG and q=blognn𝑞𝑏𝑛𝑛q=\frac{b\log n}{n}italic_q = divide start_ARG italic_b roman_log italic_n end_ARG start_ARG italic_n end_ARG (for a,b>0𝑎𝑏0a,b>0italic_a , italic_b > 0) and two communities, they established conditions under which exact matching is possible. Once the exact matching is achieved, they construct a union graph G1∨π∗G2∼SBM(n,p(1−(1−s)2),q(1−(1−s)2))similar-tosubscriptsubscript𝜋subscript𝐺1subscript𝐺2SBM𝑛𝑝1superscript1𝑠2𝑞1superscript1𝑠2G_{1}\vee_{\pi_{*}}G_{2}\sim\text{SBM}\bigl{(}n,p(1-(1-s)^{2}),q(1-(1-s)^{2})%
\bigr{)}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∨ start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∼ SBM ( italic_n , italic_p ( 1 - ( 1 - italic_s ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) , italic_q ( 1 - ( 1 - italic_s ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ), which is denser than the individual graphs G1,G2∼SBM(n,ps,qs)similar-tosubscript𝐺1subscript𝐺2SBM𝑛𝑝𝑠𝑞𝑠G_{1},G_{2}\sim\text{SBM}\bigl{(}n,ps,qs\bigr{)}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∼ SBM ( italic_n , italic_p italic_s , italic_q italic_s ). By doing so, the threshold for exact community recovery becomes less stringent: while a single graph G1subscript𝐺1G_{1}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT or G2subscript𝐺2G_{2}italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT requires a−b>2s𝑎𝑏2𝑠\sqrt{a}-\sqrt{b}>\tfrac{2}{s}square-root start_ARG italic_a end_ARG - square-root start_ARG italic_b end_ARG > divide start_ARG 2 end_ARG start_ARG italic_s end_ARG, the union graph only needs a−b>21−(1−s)2𝑎𝑏21superscript1𝑠2\sqrt{a}-\sqrt{b}>\tfrac{2}{\sqrt{1-(1-s)^{2}}}square-root start_ARG italic_a end_ARG - square-root start_ARG italic_b end_ARG > divide start_ARG 2 end_ARG start_ARG square-root start_ARG 1 - ( 1 - italic_s ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG end_ARG, illustrating a regime in which exact recovery is infeasible with a single graph but feasible with two correlated ones.
Although [6] narrowed the gap between achievability and impossibility for two-community correlated SBMs, Gaudio et al. [8] completely characterized the information-theoretic limit for exact recovery by leveraging partial matching, even in cases where perfect matching is not possible. Subsequent work has extended these ideas to correlated SBMs with more communities or more than two correlated graphs. Yang and Chung [7] generalized the results of [6] to SBMs with r𝑟ritalic_r communities that may scale with n𝑛nitalic_n, while Rácz and Zhang [34] built on [8] to determine the exact information-theoretic threshold for community recovery in scenarios involving more than two correlated SBM graphs.
| 995
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
I Introduction
I-B Prior Works
I-B2 Community Recovery in Correlated Random Graphs
Rácz and Sridhar [6] first investigated exact community recovery in the presence of two or more correlated networks. Focusing on correlated SBMs with p=alognn𝑝𝑎𝑛𝑛p=\frac{a\log n}{n}italic_p = divide start_ARG italic_a roman_log italic_n end_ARG start_ARG italic_n end_ARG and q=blognn𝑞𝑏𝑛𝑛q=\frac{b\log n}{n}italic_q = divide start_ARG italic_b roman_log italic_n end_ARG start_ARG italic_n end_ARG (for a,b>0𝑎𝑏0a,b>0italic_a , italic_b > 0) and two communities, they established conditions under which exact matching is possible. Once the exact matching is achieved, they construct a union graph G1∨π∗G2∼SBM(n,p(1−(1−s)2),q(1−(1−s)2))similar-tosubscriptsubscript𝜋subscript𝐺1subscript𝐺2SBM𝑛𝑝1superscript1𝑠2𝑞1superscript1𝑠2G_{1}\vee_{\pi_{*}}G_{2}\sim\text{SBM}\bigl{(}n,p(1-(1-s)^{2}),q(1-(1-s)^{2})%
\bigr{)}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∨ start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∼ SBM ( italic_n , italic_p ( 1 - ( 1 - italic_s ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) , italic_q ( 1 - ( 1 - italic_s ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ), which is denser than the individual graphs G1,G2∼SBM(n,ps,qs)similar-tosubscript𝐺1subscript𝐺2SBM𝑛𝑝𝑠𝑞𝑠G_{1},G_{2}\sim\text{SBM}\bigl{(}n,ps,qs\bigr{)}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∼ SBM ( italic_n , italic_p italic_s , italic_q italic_s ). By doing so, the threshold for exact community recovery becomes less stringent: while a single graph G1subscript𝐺1G_{1}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT or G2subscript𝐺2G_{2}italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT requires a−b>2s𝑎𝑏2𝑠\sqrt{a}-\sqrt{b}>\tfrac{2}{s}square-root start_ARG italic_a end_ARG - square-root start_ARG italic_b end_ARG > divide start_ARG 2 end_ARG start_ARG italic_s end_ARG, the union graph only needs a−b>21−(1−s)2𝑎𝑏21superscript1𝑠2\sqrt{a}-\sqrt{b}>\tfrac{2}{\sqrt{1-(1-s)^{2}}}square-root start_ARG italic_a end_ARG - square-root start_ARG italic_b end_ARG > divide start_ARG 2 end_ARG start_ARG square-root start_ARG 1 - ( 1 - italic_s ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG end_ARG, illustrating a regime in which exact recovery is infeasible with a single graph but feasible with two correlated ones.
Although [6] narrowed the gap between achievability and impossibility for two-community correlated SBMs, Gaudio et al. [8] completely characterized the information-theoretic limit for exact recovery by leveraging partial matching, even in cases where perfect matching is not possible. Subsequent work has extended these ideas to correlated SBMs with more communities or more than two correlated graphs. Yang and Chung [7] generalized the results of [6] to SBMs with r𝑟ritalic_r communities that may scale with n𝑛nitalic_n, while Rácz and Zhang [34] built on [8] to determine the exact information-theoretic threshold for community recovery in scenarios involving more than two correlated SBM graphs.
| 1,029
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14
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This paper introduces and analyzes two new models that jointly consider correlated graphs and correlated node attributes to better reflect hidden community structures. Specifically, we focus on correlated Gaussian Mixture Models (GMMs) and correlated Contextual Stochastic Block Models (CSBMs), as defined in Section I-A, with the ultimate goal of determining the conditions for exact community recovery. A key preliminary step is to establish exact matching–that is, to recover the one-to-one correspondence between the nodes of two correlated graphs–so that the second graph (or database) can serve as side information for community detection. We characterize the regimes under which exact matching is achievable or impossible in each proposed model.
In the correlated GMM setting, we adopt an estimator that minimizes the sum of squared distances between node attributes,
, 1 = π^:=argminπ∈Sn∑i=1n∥𝒙i−𝒚π(i)∥2,assign^𝜋subscriptargmin𝜋subscript𝑆𝑛superscriptsubscript𝑖1𝑛superscriptdelimited-∥∥subscript𝒙𝑖subscript𝒚𝜋𝑖2\hat{\pi}:=\operatorname*{arg\,min}_{\pi\in S_{n}}\sum_{i=1}^{n}\bigl{\lVert}{%
\boldsymbol{x}}_{i}-{\boldsymbol{y}}_{\pi(i)}\bigr{\rVert}^{2},over^ start_ARG italic_π end_ARG := start_OPERATOR roman_arg roman_min end_OPERATOR start_POSTSUBSCRIPT italic_π ∈ italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ∥ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_italic_y start_POSTSUBSCRIPT italic_π ( italic_i ) end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ,. , 2 = . , 3 = (6)
and establish threshold conditions for exact alignment. For the correlated CSBMs, we develop a two-step algorithm that first performs k𝑘kitalic_k-core matching using only edge information to match the majority of nodes, and then applies the distance-based attribute estimator (6) to align the remaining unmatched nodes. This two-step strategy, previously employed for correlated Gaussian-attributed Erdős–Rényi graphs [13], is shown here to be effective even when community labels are unknown. In particular, the k𝑘kitalic_k-core matching [11, 8, 35, 13], which selects the largest matching (the number of matched nodes) with a minimum degree of at least k𝑘kitalic_k in the intersection graph under a particular permutation, turns out to be successfully recovery n−o(n)𝑛𝑜𝑛n-o(n)italic_n - italic_o ( italic_n ) nodes with a proper choice of k𝑘kitalic_k. We also analyze the MAP estimator to characterize the regimes in which exact matching becomes information-theoretically impossible.
Having established the conditions for exact matching, we then investigate exact community recovery in these correlated models. When matching is successful, one can merge the two correlated graphs by taking their union, thereby creating a denser graph, and average their correlated Gaussian attributes to reduce variance. Consequently, the achievable range for exact community detection expands relative to the scenario of having only a single graph or database, as illustrated in Figures 1 and 3. In particular, for correlated GMMs (X,Y)∼CGMMs(n,𝝁,d,ρ)similar-to𝑋𝑌CGMMs𝑛𝝁𝑑𝜌(X,Y)\sim\text{CGMMs}\bigl{(}n,\boldsymbol{\mu},d,\rho\bigr{)}( italic_X , italic_Y ) ∼ CGMMs ( italic_n , bold_italic_μ , italic_d , italic_ρ ) with ‖𝝁‖2=Rsuperscriptnorm𝝁2𝑅\|\boldsymbol{\mu}\|^{2}=R∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = italic_R, the effective signal-to-noise ratio for exact recovery increases from
, 1 = R2R+dnto(21+ρR)221+ρR+dn,superscript𝑅2𝑅𝑑𝑛tosuperscript21𝜌𝑅221𝜌𝑅𝑑𝑛\frac{R^{2}}{\,R+\tfrac{d}{n}\,}\quad\text{to}\quad\frac{\bigl{(}\tfrac{2}{1+%
\rho}\,R\bigr{)}^{2}}{\,\tfrac{2}{1+\rho}\,R+\tfrac{d}{n}\,},divide start_ARG italic_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_R + divide start_ARG italic_d end_ARG start_ARG italic_n end_ARG end_ARG to divide start_ARG ( divide start_ARG 2 end_ARG start_ARG 1 + italic_ρ end_ARG italic_R ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG divide start_ARG 2 end_ARG start_ARG 1 + italic_ρ end_ARG italic_R + divide start_ARG italic_d end_ARG start_ARG italic_n end_ARG end_ARG ,. , 2 =
while for the contextual SBMs (G1,G2)∼CCSBMs(n,p,q,s;R,d,ρ)similar-tosubscript𝐺1subscript𝐺2CCSBMs𝑛𝑝𝑞𝑠𝑅𝑑𝜌(G_{1},G_{2})\sim\text{CCSBMs}\bigl{(}n,p,q,s;R,d,\rho\bigr{)}( italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ∼ CCSBMs ( italic_n , italic_p , italic_q , italic_s ; italic_R , italic_d , italic_ρ ), the corresponding SNR improves from
, 1 = s(a−b)2+c 2to(1−(1−s)2)(a−b)2+c′ 2,𝑠superscript𝑎𝑏2𝑐2to1superscript1𝑠2superscript𝑎𝑏2superscript𝑐′2\frac{s\bigl{(}\sqrt{a}-\sqrt{b}\bigr{)}^{2}+c}{\,2\,}\quad\text{to}\quad\frac%
{\bigl{(}1-(1-s)^{2}\bigr{)}\bigl{(}\sqrt{a}-\sqrt{b}\bigr{)}^{2}+c^{\prime}}{%
\,2\,},divide start_ARG italic_s ( square-root start_ARG italic_a end_ARG - square-root start_ARG italic_b end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_c end_ARG start_ARG 2 end_ARG to divide start_ARG ( 1 - ( 1 - italic_s ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ( square-root start_ARG italic_a end_ARG - square-root start_ARG italic_b end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG start_ARG 2 end_ARG ,. , 2 =
| 1,685
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
I Introduction
I-C Our Contributions
This paper introduces and analyzes two new models that jointly consider correlated graphs and correlated node attributes to better reflect hidden community structures. Specifically, we focus on correlated Gaussian Mixture Models (GMMs) and correlated Contextual Stochastic Block Models (CSBMs), as defined in Section I-A, with the ultimate goal of determining the conditions for exact community recovery. A key preliminary step is to establish exact matching–that is, to recover the one-to-one correspondence between the nodes of two correlated graphs–so that the second graph (or database) can serve as side information for community detection. We characterize the regimes under which exact matching is achievable or impossible in each proposed model.
In the correlated GMM setting, we adopt an estimator that minimizes the sum of squared distances between node attributes,
, 1 = π^:=argminπ∈Sn∑i=1n∥𝒙i−𝒚π(i)∥2,assign^𝜋subscriptargmin𝜋subscript𝑆𝑛superscriptsubscript𝑖1𝑛superscriptdelimited-∥∥subscript𝒙𝑖subscript𝒚𝜋𝑖2\hat{\pi}:=\operatorname*{arg\,min}_{\pi\in S_{n}}\sum_{i=1}^{n}\bigl{\lVert}{%
\boldsymbol{x}}_{i}-{\boldsymbol{y}}_{\pi(i)}\bigr{\rVert}^{2},over^ start_ARG italic_π end_ARG := start_OPERATOR roman_arg roman_min end_OPERATOR start_POSTSUBSCRIPT italic_π ∈ italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ∥ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_italic_y start_POSTSUBSCRIPT italic_π ( italic_i ) end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ,. , 2 = . , 3 = (6)
and establish threshold conditions for exact alignment. For the correlated CSBMs, we develop a two-step algorithm that first performs k𝑘kitalic_k-core matching using only edge information to match the majority of nodes, and then applies the distance-based attribute estimator (6) to align the remaining unmatched nodes. This two-step strategy, previously employed for correlated Gaussian-attributed Erdős–Rényi graphs [13], is shown here to be effective even when community labels are unknown. In particular, the k𝑘kitalic_k-core matching [11, 8, 35, 13], which selects the largest matching (the number of matched nodes) with a minimum degree of at least k𝑘kitalic_k in the intersection graph under a particular permutation, turns out to be successfully recovery n−o(n)𝑛𝑜𝑛n-o(n)italic_n - italic_o ( italic_n ) nodes with a proper choice of k𝑘kitalic_k. We also analyze the MAP estimator to characterize the regimes in which exact matching becomes information-theoretically impossible.
Having established the conditions for exact matching, we then investigate exact community recovery in these correlated models. When matching is successful, one can merge the two correlated graphs by taking their union, thereby creating a denser graph, and average their correlated Gaussian attributes to reduce variance. Consequently, the achievable range for exact community detection expands relative to the scenario of having only a single graph or database, as illustrated in Figures 1 and 3. In particular, for correlated GMMs (X,Y)∼CGMMs(n,𝝁,d,ρ)similar-to𝑋𝑌CGMMs𝑛𝝁𝑑𝜌(X,Y)\sim\text{CGMMs}\bigl{(}n,\boldsymbol{\mu},d,\rho\bigr{)}( italic_X , italic_Y ) ∼ CGMMs ( italic_n , bold_italic_μ , italic_d , italic_ρ ) with ‖𝝁‖2=Rsuperscriptnorm𝝁2𝑅\|\boldsymbol{\mu}\|^{2}=R∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = italic_R, the effective signal-to-noise ratio for exact recovery increases from
, 1 = R2R+dnto(21+ρR)221+ρR+dn,superscript𝑅2𝑅𝑑𝑛tosuperscript21𝜌𝑅221𝜌𝑅𝑑𝑛\frac{R^{2}}{\,R+\tfrac{d}{n}\,}\quad\text{to}\quad\frac{\bigl{(}\tfrac{2}{1+%
\rho}\,R\bigr{)}^{2}}{\,\tfrac{2}{1+\rho}\,R+\tfrac{d}{n}\,},divide start_ARG italic_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_R + divide start_ARG italic_d end_ARG start_ARG italic_n end_ARG end_ARG to divide start_ARG ( divide start_ARG 2 end_ARG start_ARG 1 + italic_ρ end_ARG italic_R ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG divide start_ARG 2 end_ARG start_ARG 1 + italic_ρ end_ARG italic_R + divide start_ARG italic_d end_ARG start_ARG italic_n end_ARG end_ARG ,. , 2 =
while for the contextual SBMs (G1,G2)∼CCSBMs(n,p,q,s;R,d,ρ)similar-tosubscript𝐺1subscript𝐺2CCSBMs𝑛𝑝𝑞𝑠𝑅𝑑𝜌(G_{1},G_{2})\sim\text{CCSBMs}\bigl{(}n,p,q,s;R,d,\rho\bigr{)}( italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ∼ CCSBMs ( italic_n , italic_p , italic_q , italic_s ; italic_R , italic_d , italic_ρ ), the corresponding SNR improves from
, 1 = s(a−b)2+c 2to(1−(1−s)2)(a−b)2+c′ 2,𝑠superscript𝑎𝑏2𝑐2to1superscript1𝑠2superscript𝑎𝑏2superscript𝑐′2\frac{s\bigl{(}\sqrt{a}-\sqrt{b}\bigr{)}^{2}+c}{\,2\,}\quad\text{to}\quad\frac%
{\bigl{(}1-(1-s)^{2}\bigr{)}\bigl{(}\sqrt{a}-\sqrt{b}\bigr{)}^{2}+c^{\prime}}{%
\,2\,},divide start_ARG italic_s ( square-root start_ARG italic_a end_ARG - square-root start_ARG italic_b end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_c end_ARG start_ARG 2 end_ARG to divide start_ARG ( 1 - ( 1 - italic_s ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ( square-root start_ARG italic_a end_ARG - square-root start_ARG italic_b end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG start_ARG 2 end_ARG ,. , 2 =
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15
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where clogn=R2R+d/n𝑐𝑛superscript𝑅2𝑅𝑑𝑛c\log n=\frac{R^{2}}{\,R+d/n\,}italic_c roman_log italic_n = divide start_ARG italic_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_R + italic_d / italic_n end_ARG and c′logn=(21+ρR)221+ρR+d/n,superscript𝑐′𝑛superscript21𝜌𝑅221𝜌𝑅𝑑𝑛c^{\prime}\log n=\frac{\bigl{(}\tfrac{2}{1+\rho}\,R\bigr{)}^{2}}{\,\tfrac{2}{1%
+\rho}\,R+d/n\,},italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT roman_log italic_n = divide start_ARG ( divide start_ARG 2 end_ARG start_ARG 1 + italic_ρ end_ARG italic_R ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG divide start_ARG 2 end_ARG start_ARG 1 + italic_ρ end_ARG italic_R + italic_d / italic_n end_ARG , compared to having only one contextual SBM (G1subscript𝐺1G_{1}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT or G2subscript𝐺2G_{2}italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT).
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
I Introduction
I-C Our Contributions
where clogn=R2R+d/n𝑐𝑛superscript𝑅2𝑅𝑑𝑛c\log n=\frac{R^{2}}{\,R+d/n\,}italic_c roman_log italic_n = divide start_ARG italic_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_R + italic_d / italic_n end_ARG and c′logn=(21+ρR)221+ρR+d/n,superscript𝑐′𝑛superscript21𝜌𝑅221𝜌𝑅𝑑𝑛c^{\prime}\log n=\frac{\bigl{(}\tfrac{2}{1+\rho}\,R\bigr{)}^{2}}{\,\tfrac{2}{1%
+\rho}\,R+d/n\,},italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT roman_log italic_n = divide start_ARG ( divide start_ARG 2 end_ARG start_ARG 1 + italic_ρ end_ARG italic_R ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG divide start_ARG 2 end_ARG start_ARG 1 + italic_ρ end_ARG italic_R + italic_d / italic_n end_ARG , compared to having only one contextual SBM (G1subscript𝐺1G_{1}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT or G2subscript𝐺2G_{2}italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT).
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For a positive integer n𝑛nitalic_n, write [n]:={1,2,…,n}assigndelimited-[]𝑛12…𝑛[n]:=\{1,2,\ldots,n\}[ italic_n ] := { 1 , 2 , … , italic_n }. For a graph G𝐺Gitalic_G on vertex set [n]delimited-[]𝑛[n][ italic_n ], let degG(i)subscriptdegree𝐺𝑖\deg_{G}(i)roman_deg start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_i ) be the degree of node i𝑖iitalic_i, and let G{M}𝐺𝑀G\{M\}italic_G { italic_M } be the subgraph induced by M⊆[n]𝑀delimited-[]𝑛M\subseteq[n]italic_M ⊆ [ italic_n ]. Define dmin(G)subscript𝑑𝐺d_{\min}(G)italic_d start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( italic_G ) as the minimum degree of G𝐺Gitalic_G. Let ℰ:={{i,j}:i,j∈[n],i≠j}assignℰconditional-set𝑖𝑗formulae-sequence𝑖𝑗delimited-[]𝑛𝑖𝑗\mathcal{E}:=\{\{i,j\}:i,j\in[n],i\neq j\}caligraphic_E := { { italic_i , italic_j } : italic_i , italic_j ∈ [ italic_n ] , italic_i ≠ italic_j } be the set of all unordered vertex pairs. For a community label vector 𝝈𝝈{\boldsymbol{\sigma}}bold_italic_σ, define
, 1 = ℰ+(𝝈):={{i,j}∈ℰ:σiσj=+1}andℰ−(𝝈):={{i,j}∈ℰ:σiσj=−1}.formulae-sequenceassignsuperscriptℰ𝝈conditional-set𝑖𝑗ℰsubscript𝜎𝑖subscript𝜎𝑗1andassignsuperscriptℰ𝝈conditional-set𝑖𝑗ℰsubscript𝜎𝑖subscript𝜎𝑗1\mathcal{E}^{+}({\boldsymbol{\sigma}}):=\{\{i,j\}\in\mathcal{E}:\sigma_{i}%
\sigma_{j}=+1\}\quad\text{and}\quad\mathcal{E}^{-}({\boldsymbol{\sigma}}):=\{%
\{i,j\}\in\mathcal{E}:\sigma_{i}\sigma_{j}=-1\}.caligraphic_E start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ( bold_italic_σ ) := { { italic_i , italic_j } ∈ caligraphic_E : italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = + 1 } and caligraphic_E start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ( bold_italic_σ ) := { { italic_i , italic_j } ∈ caligraphic_E : italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = - 1 } .. , 2 =
Then ℰ+(𝝈)superscriptℰ𝝈\mathcal{E}^{+}({\boldsymbol{\sigma}})caligraphic_E start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ( bold_italic_σ ) and ℰ−(𝝈)superscriptℰ𝝈\mathcal{E}^{-}({\boldsymbol{\sigma}})caligraphic_E start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ( bold_italic_σ ) partition ℰℰ\mathcal{E}caligraphic_E into intra- and inter-community node pairs. Let A,B′,B𝐴superscript𝐵′𝐵A,B^{\prime},Bitalic_A , italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_B be the adjacency matrices of G1,G2′,G2subscript𝐺1subscriptsuperscript𝐺′2subscript𝐺2G_{1},G^{\prime}_{2},G_{2}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, respectively, and let X,Y′,Y𝑋superscript𝑌′𝑌X,Y^{\prime},Yitalic_X , italic_Y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_Y be the corresponding databases of node attributes. Denote by ∨\vee∨ and ∧\wedge∧ the entrywise max and min, respectively. For a permutation π𝜋\piitalic_π, define
, 1 = (A∨πB)i,j=max{Ai,j,Bπ(i),π(j)}and(A∧πB)i,j=min{Ai,j,Bπ(i),π(j)}.formulae-sequencesubscriptsubscript𝜋𝐴𝐵𝑖𝑗subscript𝐴𝑖𝑗subscript𝐵𝜋𝑖𝜋𝑗andsubscriptsubscript𝜋𝐴𝐵𝑖𝑗subscript𝐴𝑖𝑗subscript𝐵𝜋𝑖𝜋𝑗(A\vee_{\pi}B)_{i,j}=\max\{A_{i,j},\,B_{\pi(i),\pi(j)}\}\quad\text{and}\quad(A%
\wedge_{\pi}B)_{i,j}=\min\{A_{i,j},\,B_{\pi(i),\pi(j)}\}.( italic_A ∨ start_POSTSUBSCRIPT italic_π end_POSTSUBSCRIPT italic_B ) start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT = roman_max { italic_A start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT , italic_B start_POSTSUBSCRIPT italic_π ( italic_i ) , italic_π ( italic_j ) end_POSTSUBSCRIPT } and ( italic_A ∧ start_POSTSUBSCRIPT italic_π end_POSTSUBSCRIPT italic_B ) start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT = roman_min { italic_A start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT , italic_B start_POSTSUBSCRIPT italic_π ( italic_i ) , italic_π ( italic_j ) end_POSTSUBSCRIPT } .. , 2 =
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
I Introduction
I-D Notation
For a positive integer n𝑛nitalic_n, write [n]:={1,2,…,n}assigndelimited-[]𝑛12…𝑛[n]:=\{1,2,\ldots,n\}[ italic_n ] := { 1 , 2 , … , italic_n }. For a graph G𝐺Gitalic_G on vertex set [n]delimited-[]𝑛[n][ italic_n ], let degG(i)subscriptdegree𝐺𝑖\deg_{G}(i)roman_deg start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_i ) be the degree of node i𝑖iitalic_i, and let G{M}𝐺𝑀G\{M\}italic_G { italic_M } be the subgraph induced by M⊆[n]𝑀delimited-[]𝑛M\subseteq[n]italic_M ⊆ [ italic_n ]. Define dmin(G)subscript𝑑𝐺d_{\min}(G)italic_d start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( italic_G ) as the minimum degree of G𝐺Gitalic_G. Let ℰ:={{i,j}:i,j∈[n],i≠j}assignℰconditional-set𝑖𝑗formulae-sequence𝑖𝑗delimited-[]𝑛𝑖𝑗\mathcal{E}:=\{\{i,j\}:i,j\in[n],i\neq j\}caligraphic_E := { { italic_i , italic_j } : italic_i , italic_j ∈ [ italic_n ] , italic_i ≠ italic_j } be the set of all unordered vertex pairs. For a community label vector 𝝈𝝈{\boldsymbol{\sigma}}bold_italic_σ, define
, 1 = ℰ+(𝝈):={{i,j}∈ℰ:σiσj=+1}andℰ−(𝝈):={{i,j}∈ℰ:σiσj=−1}.formulae-sequenceassignsuperscriptℰ𝝈conditional-set𝑖𝑗ℰsubscript𝜎𝑖subscript𝜎𝑗1andassignsuperscriptℰ𝝈conditional-set𝑖𝑗ℰsubscript𝜎𝑖subscript𝜎𝑗1\mathcal{E}^{+}({\boldsymbol{\sigma}}):=\{\{i,j\}\in\mathcal{E}:\sigma_{i}%
\sigma_{j}=+1\}\quad\text{and}\quad\mathcal{E}^{-}({\boldsymbol{\sigma}}):=\{%
\{i,j\}\in\mathcal{E}:\sigma_{i}\sigma_{j}=-1\}.caligraphic_E start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ( bold_italic_σ ) := { { italic_i , italic_j } ∈ caligraphic_E : italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = + 1 } and caligraphic_E start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ( bold_italic_σ ) := { { italic_i , italic_j } ∈ caligraphic_E : italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = - 1 } .. , 2 =
Then ℰ+(𝝈)superscriptℰ𝝈\mathcal{E}^{+}({\boldsymbol{\sigma}})caligraphic_E start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ( bold_italic_σ ) and ℰ−(𝝈)superscriptℰ𝝈\mathcal{E}^{-}({\boldsymbol{\sigma}})caligraphic_E start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ( bold_italic_σ ) partition ℰℰ\mathcal{E}caligraphic_E into intra- and inter-community node pairs. Let A,B′,B𝐴superscript𝐵′𝐵A,B^{\prime},Bitalic_A , italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_B be the adjacency matrices of G1,G2′,G2subscript𝐺1subscriptsuperscript𝐺′2subscript𝐺2G_{1},G^{\prime}_{2},G_{2}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, respectively, and let X,Y′,Y𝑋superscript𝑌′𝑌X,Y^{\prime},Yitalic_X , italic_Y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_Y be the corresponding databases of node attributes. Denote by ∨\vee∨ and ∧\wedge∧ the entrywise max and min, respectively. For a permutation π𝜋\piitalic_π, define
, 1 = (A∨πB)i,j=max{Ai,j,Bπ(i),π(j)}and(A∧πB)i,j=min{Ai,j,Bπ(i),π(j)}.formulae-sequencesubscriptsubscript𝜋𝐴𝐵𝑖𝑗subscript𝐴𝑖𝑗subscript𝐵𝜋𝑖𝜋𝑗andsubscriptsubscript𝜋𝐴𝐵𝑖𝑗subscript𝐴𝑖𝑗subscript𝐵𝜋𝑖𝜋𝑗(A\vee_{\pi}B)_{i,j}=\max\{A_{i,j},\,B_{\pi(i),\pi(j)}\}\quad\text{and}\quad(A%
\wedge_{\pi}B)_{i,j}=\min\{A_{i,j},\,B_{\pi(i),\pi(j)}\}.( italic_A ∨ start_POSTSUBSCRIPT italic_π end_POSTSUBSCRIPT italic_B ) start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT = roman_max { italic_A start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT , italic_B start_POSTSUBSCRIPT italic_π ( italic_i ) , italic_π ( italic_j ) end_POSTSUBSCRIPT } and ( italic_A ∧ start_POSTSUBSCRIPT italic_π end_POSTSUBSCRIPT italic_B ) start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT = roman_min { italic_A start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT , italic_B start_POSTSUBSCRIPT italic_π ( italic_i ) , italic_π ( italic_j ) end_POSTSUBSCRIPT } .. , 2 =
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For v=[v1,…,vk]⊤∈ℝk𝑣superscriptsubscript𝑣1…subscript𝑣𝑘topsuperscriptℝ𝑘v=[v_{1},\ldots,v_{k}]^{\top}\in\mathbb{R}^{k}italic_v = [ italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_v start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ] start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT and i∈[k]𝑖delimited-[]𝑘i\in[k]italic_i ∈ [ italic_k ], let v−isubscript𝑣𝑖v_{-i}italic_v start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT be the vector obtained by removing visubscript𝑣𝑖v_{i}italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT. For an event E𝐸Eitalic_E, let 𝟙(E)1𝐸\mathds{1}(E)blackboard_1 ( italic_E ) be its indicator. Write Φ(⋅)Φ⋅\Phi(\cdot)roman_Φ ( ⋅ ) for the tail distribution of a standard Gaussian. For two functions f,g:[n]→[m]:𝑓𝑔→delimited-[]𝑛delimited-[]𝑚f,g:[n]\to[m]italic_f , italic_g : [ italic_n ] → [ italic_m ], define their overlap as 𝐨𝐯(f,g):=1n∑i=1n𝟙(f(i)=g(i))assign𝐨𝐯𝑓𝑔1𝑛superscriptsubscript𝑖1𝑛1𝑓𝑖𝑔𝑖\mathbf{ov}(f,g):=\tfrac{1}{n}\sum_{i=1}^{n}\mathds{1}\bigl{(}f(i)=g(i)\bigr{)}bold_ov ( italic_f , italic_g ) := divide start_ARG 1 end_ARG start_ARG italic_n end_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT blackboard_1 ( italic_f ( italic_i ) = italic_g ( italic_i ) ). Lastly, asymptotic notation O(⋅),o(⋅),Ω(⋅),ω(⋅),Θ(⋅)𝑂⋅𝑜⋅Ω⋅𝜔⋅Θ⋅O(\cdot),o(\cdot),\Omega(\cdot),\omega(\cdot),\Theta(\cdot)italic_O ( ⋅ ) , italic_o ( ⋅ ) , roman_Ω ( ⋅ ) , italic_ω ( ⋅ ) , roman_Θ ( ⋅ ) is used with n→∞→𝑛n\to\inftyitalic_n → ∞.
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
I Introduction
I-D Notation
For v=[v1,…,vk]⊤∈ℝk𝑣superscriptsubscript𝑣1…subscript𝑣𝑘topsuperscriptℝ𝑘v=[v_{1},\ldots,v_{k}]^{\top}\in\mathbb{R}^{k}italic_v = [ italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_v start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ] start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT and i∈[k]𝑖delimited-[]𝑘i\in[k]italic_i ∈ [ italic_k ], let v−isubscript𝑣𝑖v_{-i}italic_v start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT be the vector obtained by removing visubscript𝑣𝑖v_{i}italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT. For an event E𝐸Eitalic_E, let 𝟙(E)1𝐸\mathds{1}(E)blackboard_1 ( italic_E ) be its indicator. Write Φ(⋅)Φ⋅\Phi(\cdot)roman_Φ ( ⋅ ) for the tail distribution of a standard Gaussian. For two functions f,g:[n]→[m]:𝑓𝑔→delimited-[]𝑛delimited-[]𝑚f,g:[n]\to[m]italic_f , italic_g : [ italic_n ] → [ italic_m ], define their overlap as 𝐨𝐯(f,g):=1n∑i=1n𝟙(f(i)=g(i))assign𝐨𝐯𝑓𝑔1𝑛superscriptsubscript𝑖1𝑛1𝑓𝑖𝑔𝑖\mathbf{ov}(f,g):=\tfrac{1}{n}\sum_{i=1}^{n}\mathds{1}\bigl{(}f(i)=g(i)\bigr{)}bold_ov ( italic_f , italic_g ) := divide start_ARG 1 end_ARG start_ARG italic_n end_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT blackboard_1 ( italic_f ( italic_i ) = italic_g ( italic_i ) ). Lastly, asymptotic notation O(⋅),o(⋅),Ω(⋅),ω(⋅),Θ(⋅)𝑂⋅𝑜⋅Ω⋅𝜔⋅Θ⋅O(\cdot),o(\cdot),\Omega(\cdot),\omega(\cdot),\Theta(\cdot)italic_O ( ⋅ ) , italic_o ( ⋅ ) , roman_Ω ( ⋅ ) , italic_ω ( ⋅ ) , roman_Θ ( ⋅ ) is used with n→∞→𝑛n\to\inftyitalic_n → ∞.
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In this section, we investigate the correlated Gaussian Mixture Models (GMMs) introduced in Section I-A1, with a primary goal of determining conditions for exact community recovery when two correlated databases are provided. Our approach consists of two steps: (i) establishing exact matching between the two databases, and (ii) merging the matched databases to identify regimes in which exact community recovery is significantly more tractable compared to using only a single database.
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
II Correlated Gaussian Mixture Models
In this section, we investigate the correlated Gaussian Mixture Models (GMMs) introduced in Section I-A1, with a primary goal of determining conditions for exact community recovery when two correlated databases are provided. Our approach consists of two steps: (i) establishing exact matching between the two databases, and (ii) merging the matched databases to identify regimes in which exact community recovery is significantly more tractable compared to using only a single database.
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We begin by examining the requirements for exact matching in correlated GMMs. Theorem 1 below provides sufficient conditions under which an estimator (6) achieves perfect alignment with high probability.
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
II Correlated Gaussian Mixture Models
II-A Exact Matching on Correlated Gaussian Mixture Models
We begin by examining the requirements for exact matching in correlated GMMs. Theorem 1 below provides sufficient conditions under which an estimator (6) achieves perfect alignment with high probability.
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Let (X,Y)∼CGMMs(n,𝛍,d,ρ)similar-to𝑋𝑌CGMMs𝑛𝛍𝑑𝜌(X,Y)\sim\textnormal{CGMMs}(n,\boldsymbol{\mu},d,\rho)( italic_X , italic_Y ) ∼ CGMMs ( italic_n , bold_italic_μ , italic_d , italic_ρ ) as defined in Section I-A1. Suppose that either
, 1 = d4log11−ρ2≥logn+ω(1)and‖𝝁‖2≥2logn+ω(1),formulae-sequence𝑑411superscript𝜌2𝑛𝜔1andsuperscriptnorm𝝁22𝑛𝜔1\frac{d}{4}\log\frac{1}{1-\rho^{2}}\geq\log n+\omega(1)\quad\text{and}\quad\|%
\boldsymbol{\mu}\|^{2}\geq 2\log n+\omega(1),divide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≥ roman_log italic_n + italic_ω ( 1 ) and ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ 2 roman_log italic_n + italic_ω ( 1 ) ,. , 2 = . , 3 = (7)
or
, 1 = d4log11−ρ2≥(1+ϵ)lognandd=ω(logn)formulae-sequence𝑑411superscript𝜌21italic-ϵ𝑛and𝑑𝜔𝑛\frac{d}{4}\log\frac{1}{1-\rho^{2}}\geq(1+\epsilon)\log n\quad\text{and}\quad d%
=\omega(\log n)divide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≥ ( 1 + italic_ϵ ) roman_log italic_n and italic_d = italic_ω ( roman_log italic_n ). , 2 = . , 3 = (8)
holds for an arbitrarily small constant ϵ>0italic-ϵ0\epsilon>0italic_ϵ > 0. Then there exists an estimator π^(X,Y)^𝜋𝑋𝑌\hat{\pi}(X,Y)over^ start_ARG italic_π end_ARG ( italic_X , italic_Y ) such that π^(X,Y)=π∗^𝜋𝑋𝑌subscript𝜋\hat{\pi}(X,Y)=\pi_{*}over^ start_ARG italic_π end_ARG ( italic_X , italic_Y ) = italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT with high probability.
In correlated GMMs, the MAP estimator can be written as
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
II Correlated Gaussian Mixture Models
II-A Exact Matching on Correlated Gaussian Mixture Models
Theorem 1 (Achievability for Exact Matching).
Let (X,Y)∼CGMMs(n,𝛍,d,ρ)similar-to𝑋𝑌CGMMs𝑛𝛍𝑑𝜌(X,Y)\sim\textnormal{CGMMs}(n,\boldsymbol{\mu},d,\rho)( italic_X , italic_Y ) ∼ CGMMs ( italic_n , bold_italic_μ , italic_d , italic_ρ ) as defined in Section I-A1. Suppose that either
, 1 = d4log11−ρ2≥logn+ω(1)and‖𝝁‖2≥2logn+ω(1),formulae-sequence𝑑411superscript𝜌2𝑛𝜔1andsuperscriptnorm𝝁22𝑛𝜔1\frac{d}{4}\log\frac{1}{1-\rho^{2}}\geq\log n+\omega(1)\quad\text{and}\quad\|%
\boldsymbol{\mu}\|^{2}\geq 2\log n+\omega(1),divide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≥ roman_log italic_n + italic_ω ( 1 ) and ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ 2 roman_log italic_n + italic_ω ( 1 ) ,. , 2 = . , 3 = (7)
or
, 1 = d4log11−ρ2≥(1+ϵ)lognandd=ω(logn)formulae-sequence𝑑411superscript𝜌21italic-ϵ𝑛and𝑑𝜔𝑛\frac{d}{4}\log\frac{1}{1-\rho^{2}}\geq(1+\epsilon)\log n\quad\text{and}\quad d%
=\omega(\log n)divide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≥ ( 1 + italic_ϵ ) roman_log italic_n and italic_d = italic_ω ( roman_log italic_n ). , 2 = . , 3 = (8)
holds for an arbitrarily small constant ϵ>0italic-ϵ0\epsilon>0italic_ϵ > 0. Then there exists an estimator π^(X,Y)^𝜋𝑋𝑌\hat{\pi}(X,Y)over^ start_ARG italic_π end_ARG ( italic_X , italic_Y ) such that π^(X,Y)=π∗^𝜋𝑋𝑌subscript𝜋\hat{\pi}(X,Y)=\pi_{*}over^ start_ARG italic_π end_ARG ( italic_X , italic_Y ) = italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT with high probability.
In correlated GMMs, the MAP estimator can be written as
| 770
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21
|
, 1 = π^^𝜋\displaystyle\hat{\pi}over^ start_ARG italic_π end_ARG. , 2 = =argmaxπ∈Snℙ(π∗=π|X,Y)absentsubscriptargmax𝜋subscript𝑆𝑛ℙsubscript𝜋conditional𝜋𝑋𝑌\displaystyle=\operatorname*{arg\,max}_{\pi\in S_{n}}\mathbb{P}(\pi_{*}=\pi|X,Y)= start_OPERATOR roman_arg roman_max end_OPERATOR start_POSTSUBSCRIPT italic_π ∈ italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT blackboard_P ( italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT = italic_π | italic_X , italic_Y ). , 3 = . , 4 = (9). , 1 = . , 2 = =(a)argmaxπ∈Snℙ(X,Y|π∗=π)superscript𝑎absentsubscriptargmax𝜋subscript𝑆𝑛ℙ𝑋conditional𝑌subscript𝜋𝜋\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\operatorname*{arg\,max}_{\pi\in
S%
_{n}}\mathbb{P}(X,Y|\pi_{*}=\pi)start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG ( italic_a ) end_ARG end_RELOP start_OPERATOR roman_arg roman_max end_OPERATOR start_POSTSUBSCRIPT italic_π ∈ italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT blackboard_P ( italic_X , italic_Y | italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT = italic_π ). , 3 = . , 4 = (9). , 1 = . , 2 = =argmaxπ∈Sn∑i=1n(−‖𝒙i−𝝁σi‖2+2ρ⟨𝒙i−𝝁σi,𝒚π(i)−𝝁σπ(i)⟩−‖𝒚π(i)−𝝁σπ(i)‖2)absentsubscriptargmax𝜋subscript𝑆𝑛subscriptsuperscript𝑛𝑖1superscriptnormsubscript𝒙𝑖𝝁subscript𝜎𝑖22𝜌subscript𝒙𝑖𝝁subscript𝜎𝑖subscript𝒚𝜋𝑖𝝁subscript𝜎𝜋𝑖superscriptnormsubscript𝒚𝜋𝑖𝝁subscript𝜎𝜋𝑖2\displaystyle=\operatorname*{arg\,max}_{\pi\in S_{n}}\sum^{n}_{i=1}\left(-\|{%
\boldsymbol{x}}_{i}-\boldsymbol{\mu}\sigma_{i}\|^{2}+2\rho\langle{\boldsymbol{%
x}}_{i}-\boldsymbol{\mu}\sigma_{i},{\boldsymbol{y}}_{\pi(i)}-\boldsymbol{\mu}%
\sigma_{\pi(i)}\rangle-\|{\boldsymbol{y}}_{\pi(i)}-\boldsymbol{\mu}\sigma_{\pi%
(i)}\|^{2}\right)= start_OPERATOR roman_arg roman_max end_OPERATOR start_POSTSUBSCRIPT italic_π ∈ italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∑ start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT ( - ∥ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_italic_μ italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 2 italic_ρ ⟨ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_italic_μ italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_italic_y start_POSTSUBSCRIPT italic_π ( italic_i ) end_POSTSUBSCRIPT - bold_italic_μ italic_σ start_POSTSUBSCRIPT italic_π ( italic_i ) end_POSTSUBSCRIPT ⟩ - ∥ bold_italic_y start_POSTSUBSCRIPT italic_π ( italic_i ) end_POSTSUBSCRIPT - bold_italic_μ italic_σ start_POSTSUBSCRIPT italic_π ( italic_i ) end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ). , 3 = . , 4 = (9). , 1 = . , 2 = =argmaxπ∈Sn∑i=1n(−ρ‖𝒙i−𝒚π(i)‖2−(1−ρ)(‖𝒙i‖2+‖𝒚π(i)‖2))+f(𝝈,𝝈π)absentsubscriptargmax𝜋subscript𝑆𝑛subscriptsuperscript𝑛𝑖1𝜌superscriptnormsubscript𝒙𝑖subscript𝒚𝜋𝑖21𝜌superscriptnormsubscript𝒙𝑖2superscriptnormsubscript𝒚𝜋𝑖2𝑓𝝈subscript𝝈𝜋\displaystyle=\operatorname*{arg\,max}_{\pi\in S_{n}}\sum^{n}_{i=1}\left(-\rho%
\|{\boldsymbol{x}}_{i}-{\boldsymbol{y}}_{\pi(i)}\|^{2}-(1-\rho)(\|{\boldsymbol%
{x}}_{i}\|^{2}+\|{\boldsymbol{y}}_{\pi(i)}\|^{2})\right)+f({\boldsymbol{\sigma%
}},{\boldsymbol{\sigma}}_{\pi})= start_OPERATOR roman_arg roman_max end_OPERATOR start_POSTSUBSCRIPT italic_π ∈ italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∑ start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT ( - italic_ρ ∥ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_italic_y start_POSTSUBSCRIPT italic_π ( italic_i ) end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - ( 1 - italic_ρ ) ( ∥ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ∥ bold_italic_y start_POSTSUBSCRIPT italic_π ( italic_i ) end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ) + italic_f ( bold_italic_σ , bold_italic_σ start_POSTSUBSCRIPT italic_π end_POSTSUBSCRIPT ). , 3 = . , 4 = (9). , 1 = . , 2 = =argminπ∈Sn∑i=1n‖𝒙i−𝒚π(i)‖2−1ρf(𝝈,𝝈π),absentsubscriptargmin𝜋subscript𝑆𝑛subscriptsuperscript𝑛𝑖1superscriptnormsubscript𝒙𝑖subscript𝒚𝜋𝑖21𝜌𝑓𝝈subscript𝝈𝜋\displaystyle=\operatorname*{arg\,min}_{\pi\in S_{n}}\sum^{n}_{i=1}\|{%
\boldsymbol{x}}_{i}-{\boldsymbol{y}}_{\pi(i)}\|^{2}-\frac{1}{\rho}f({%
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
II Correlated Gaussian Mixture Models
II-A Exact Matching on Correlated Gaussian Mixture Models
Theorem 1 (Achievability for Exact Matching).
, 1 = π^^𝜋\displaystyle\hat{\pi}over^ start_ARG italic_π end_ARG. , 2 = =argmaxπ∈Snℙ(π∗=π|X,Y)absentsubscriptargmax𝜋subscript𝑆𝑛ℙsubscript𝜋conditional𝜋𝑋𝑌\displaystyle=\operatorname*{arg\,max}_{\pi\in S_{n}}\mathbb{P}(\pi_{*}=\pi|X,Y)= start_OPERATOR roman_arg roman_max end_OPERATOR start_POSTSUBSCRIPT italic_π ∈ italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT blackboard_P ( italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT = italic_π | italic_X , italic_Y ). , 3 = . , 4 = (9). , 1 = . , 2 = =(a)argmaxπ∈Snℙ(X,Y|π∗=π)superscript𝑎absentsubscriptargmax𝜋subscript𝑆𝑛ℙ𝑋conditional𝑌subscript𝜋𝜋\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\operatorname*{arg\,max}_{\pi\in
S%
_{n}}\mathbb{P}(X,Y|\pi_{*}=\pi)start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG ( italic_a ) end_ARG end_RELOP start_OPERATOR roman_arg roman_max end_OPERATOR start_POSTSUBSCRIPT italic_π ∈ italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT blackboard_P ( italic_X , italic_Y | italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT = italic_π ). , 3 = . , 4 = (9). , 1 = . , 2 = =argmaxπ∈Sn∑i=1n(−‖𝒙i−𝝁σi‖2+2ρ⟨𝒙i−𝝁σi,𝒚π(i)−𝝁σπ(i)⟩−‖𝒚π(i)−𝝁σπ(i)‖2)absentsubscriptargmax𝜋subscript𝑆𝑛subscriptsuperscript𝑛𝑖1superscriptnormsubscript𝒙𝑖𝝁subscript𝜎𝑖22𝜌subscript𝒙𝑖𝝁subscript𝜎𝑖subscript𝒚𝜋𝑖𝝁subscript𝜎𝜋𝑖superscriptnormsubscript𝒚𝜋𝑖𝝁subscript𝜎𝜋𝑖2\displaystyle=\operatorname*{arg\,max}_{\pi\in S_{n}}\sum^{n}_{i=1}\left(-\|{%
\boldsymbol{x}}_{i}-\boldsymbol{\mu}\sigma_{i}\|^{2}+2\rho\langle{\boldsymbol{%
x}}_{i}-\boldsymbol{\mu}\sigma_{i},{\boldsymbol{y}}_{\pi(i)}-\boldsymbol{\mu}%
\sigma_{\pi(i)}\rangle-\|{\boldsymbol{y}}_{\pi(i)}-\boldsymbol{\mu}\sigma_{\pi%
(i)}\|^{2}\right)= start_OPERATOR roman_arg roman_max end_OPERATOR start_POSTSUBSCRIPT italic_π ∈ italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∑ start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT ( - ∥ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_italic_μ italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 2 italic_ρ ⟨ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_italic_μ italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_italic_y start_POSTSUBSCRIPT italic_π ( italic_i ) end_POSTSUBSCRIPT - bold_italic_μ italic_σ start_POSTSUBSCRIPT italic_π ( italic_i ) end_POSTSUBSCRIPT ⟩ - ∥ bold_italic_y start_POSTSUBSCRIPT italic_π ( italic_i ) end_POSTSUBSCRIPT - bold_italic_μ italic_σ start_POSTSUBSCRIPT italic_π ( italic_i ) end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ). , 3 = . , 4 = (9). , 1 = . , 2 = =argmaxπ∈Sn∑i=1n(−ρ‖𝒙i−𝒚π(i)‖2−(1−ρ)(‖𝒙i‖2+‖𝒚π(i)‖2))+f(𝝈,𝝈π)absentsubscriptargmax𝜋subscript𝑆𝑛subscriptsuperscript𝑛𝑖1𝜌superscriptnormsubscript𝒙𝑖subscript𝒚𝜋𝑖21𝜌superscriptnormsubscript𝒙𝑖2superscriptnormsubscript𝒚𝜋𝑖2𝑓𝝈subscript𝝈𝜋\displaystyle=\operatorname*{arg\,max}_{\pi\in S_{n}}\sum^{n}_{i=1}\left(-\rho%
\|{\boldsymbol{x}}_{i}-{\boldsymbol{y}}_{\pi(i)}\|^{2}-(1-\rho)(\|{\boldsymbol%
{x}}_{i}\|^{2}+\|{\boldsymbol{y}}_{\pi(i)}\|^{2})\right)+f({\boldsymbol{\sigma%
}},{\boldsymbol{\sigma}}_{\pi})= start_OPERATOR roman_arg roman_max end_OPERATOR start_POSTSUBSCRIPT italic_π ∈ italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∑ start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT ( - italic_ρ ∥ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_italic_y start_POSTSUBSCRIPT italic_π ( italic_i ) end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - ( 1 - italic_ρ ) ( ∥ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ∥ bold_italic_y start_POSTSUBSCRIPT italic_π ( italic_i ) end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ) + italic_f ( bold_italic_σ , bold_italic_σ start_POSTSUBSCRIPT italic_π end_POSTSUBSCRIPT ). , 3 = . , 4 = (9). , 1 = . , 2 = =argminπ∈Sn∑i=1n‖𝒙i−𝒚π(i)‖2−1ρf(𝝈,𝝈π),absentsubscriptargmin𝜋subscript𝑆𝑛subscriptsuperscript𝑛𝑖1superscriptnormsubscript𝒙𝑖subscript𝒚𝜋𝑖21𝜌𝑓𝝈subscript𝝈𝜋\displaystyle=\operatorname*{arg\,min}_{\pi\in S_{n}}\sum^{n}_{i=1}\|{%
\boldsymbol{x}}_{i}-{\boldsymbol{y}}_{\pi(i)}\|^{2}-\frac{1}{\rho}f({%
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22
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\boldsymbol{\sigma}},{\boldsymbol{\sigma}}_{\pi}),= start_OPERATOR roman_arg roman_min end_OPERATOR start_POSTSUBSCRIPT italic_π ∈ italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∑ start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT ∥ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_italic_y start_POSTSUBSCRIPT italic_π ( italic_i ) end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - divide start_ARG 1 end_ARG start_ARG italic_ρ end_ARG italic_f ( bold_italic_σ , bold_italic_σ start_POSTSUBSCRIPT italic_π end_POSTSUBSCRIPT ) ,. , 3 = . , 4 = (9)
where
, 1 = f(𝝈,𝝈π)=∑i=1n(2⟨𝒙i,𝝁σi⟩+2⟨𝒚π(i),𝝁σπ(i)⟩−2ρ⟨𝒙i,𝝁σπ(i)⟩−2ρ⟨𝒚π(i),𝝁σi⟩+2ρ‖𝝁‖2σiσπ(i)).𝑓𝝈subscript𝝈𝜋superscriptsubscript𝑖1𝑛2subscript𝒙𝑖𝝁subscript𝜎𝑖2subscript𝒚𝜋𝑖𝝁subscript𝜎𝜋𝑖2𝜌subscript𝒙𝑖𝝁subscript𝜎𝜋𝑖2𝜌subscript𝒚𝜋𝑖𝝁subscript𝜎𝑖2𝜌superscriptnorm𝝁2subscript𝜎𝑖subscript𝜎𝜋𝑖f\bigl{(}{\boldsymbol{\sigma}},{\boldsymbol{\sigma}}_{\pi}\bigr{)}=\sum_{i=1}^%
{n}\Bigl{(}2\langle{\boldsymbol{x}}_{i},\boldsymbol{\mu}\sigma_{i}\rangle+2%
\langle{\boldsymbol{y}}_{\pi(i)},\boldsymbol{\mu}\sigma_{\pi(i)}\rangle-2\rho%
\langle{\boldsymbol{x}}_{i},\boldsymbol{\mu}\sigma_{\pi(i)}\rangle-2\rho%
\langle{\boldsymbol{y}}_{\pi(i)},\boldsymbol{\mu}\sigma_{i}\rangle+2\rho\|%
\boldsymbol{\mu}\|^{2}\sigma_{i}\sigma_{\pi(i)}\Bigr{)}.italic_f ( bold_italic_σ , bold_italic_σ start_POSTSUBSCRIPT italic_π end_POSTSUBSCRIPT ) = ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ( 2 ⟨ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_italic_μ italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⟩ + 2 ⟨ bold_italic_y start_POSTSUBSCRIPT italic_π ( italic_i ) end_POSTSUBSCRIPT , bold_italic_μ italic_σ start_POSTSUBSCRIPT italic_π ( italic_i ) end_POSTSUBSCRIPT ⟩ - 2 italic_ρ ⟨ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_italic_μ italic_σ start_POSTSUBSCRIPT italic_π ( italic_i ) end_POSTSUBSCRIPT ⟩ - 2 italic_ρ ⟨ bold_italic_y start_POSTSUBSCRIPT italic_π ( italic_i ) end_POSTSUBSCRIPT , bold_italic_μ italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⟩ + 2 italic_ρ ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_π ( italic_i ) end_POSTSUBSCRIPT ) .. , 2 =
Step (a)𝑎(a)( italic_a ) uses the fact that π∗subscript𝜋\pi_{*}italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT is uniformly distributed over Snsubscript𝑆𝑛S_{n}italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT. Note also that
∑i=1n‖𝒙i‖2+‖𝒚π(i)‖2superscriptsubscript𝑖1𝑛superscriptnormsubscript𝒙𝑖2superscriptnormsubscript𝒚𝜋𝑖2\sum_{i=1}^{n}\|{\boldsymbol{x}}_{i}\|^{2}+\|{\boldsymbol{y}}_{\pi(i)}\|^{2}∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ∥ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ∥ bold_italic_y start_POSTSUBSCRIPT italic_π ( italic_i ) end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT
is invariant under permutation π𝜋\piitalic_π, ensuring this part does not affect the alignment decision. If the community labels 𝝈𝝈{\boldsymbol{\sigma}}bold_italic_σ and 𝝈πsubscript𝝈𝜋{\boldsymbol{\sigma}}_{\pi}bold_italic_σ start_POSTSUBSCRIPT italic_π end_POSTSUBSCRIPT are known, then f(𝝈,𝝈π)𝑓𝝈subscript𝝈𝜋f\bigl{(}{\boldsymbol{\sigma}},{\boldsymbol{\sigma}}_{\pi}\bigr{)}italic_f ( bold_italic_σ , bold_italic_σ start_POSTSUBSCRIPT italic_π end_POSTSUBSCRIPT ) is also fixed, so the MAP estimator reduces exactly to the simpler distance-based estimator in (6). Even without known labels, the proof of Theorem 1 shows that (6) attains tight bounds under the conditions ‖𝝁‖2≥2logn+ω(1)superscriptnorm𝝁22𝑛𝜔1\|\boldsymbol{\mu}\|^{2}\geq 2\log n+\omega(1)∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ 2 roman_log italic_n + italic_ω ( 1 ) or d=ω(logn)𝑑𝜔𝑛d=\omega(\log n)italic_d = italic_ω ( roman_log italic_n ); see Section VI for details.
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
II Correlated Gaussian Mixture Models
II-A Exact Matching on Correlated Gaussian Mixture Models
Theorem 1 (Achievability for Exact Matching).
\boldsymbol{\sigma}},{\boldsymbol{\sigma}}_{\pi}),= start_OPERATOR roman_arg roman_min end_OPERATOR start_POSTSUBSCRIPT italic_π ∈ italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∑ start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT ∥ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_italic_y start_POSTSUBSCRIPT italic_π ( italic_i ) end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - divide start_ARG 1 end_ARG start_ARG italic_ρ end_ARG italic_f ( bold_italic_σ , bold_italic_σ start_POSTSUBSCRIPT italic_π end_POSTSUBSCRIPT ) ,. , 3 = . , 4 = (9)
where
, 1 = f(𝝈,𝝈π)=∑i=1n(2⟨𝒙i,𝝁σi⟩+2⟨𝒚π(i),𝝁σπ(i)⟩−2ρ⟨𝒙i,𝝁σπ(i)⟩−2ρ⟨𝒚π(i),𝝁σi⟩+2ρ‖𝝁‖2σiσπ(i)).𝑓𝝈subscript𝝈𝜋superscriptsubscript𝑖1𝑛2subscript𝒙𝑖𝝁subscript𝜎𝑖2subscript𝒚𝜋𝑖𝝁subscript𝜎𝜋𝑖2𝜌subscript𝒙𝑖𝝁subscript𝜎𝜋𝑖2𝜌subscript𝒚𝜋𝑖𝝁subscript𝜎𝑖2𝜌superscriptnorm𝝁2subscript𝜎𝑖subscript𝜎𝜋𝑖f\bigl{(}{\boldsymbol{\sigma}},{\boldsymbol{\sigma}}_{\pi}\bigr{)}=\sum_{i=1}^%
{n}\Bigl{(}2\langle{\boldsymbol{x}}_{i},\boldsymbol{\mu}\sigma_{i}\rangle+2%
\langle{\boldsymbol{y}}_{\pi(i)},\boldsymbol{\mu}\sigma_{\pi(i)}\rangle-2\rho%
\langle{\boldsymbol{x}}_{i},\boldsymbol{\mu}\sigma_{\pi(i)}\rangle-2\rho%
\langle{\boldsymbol{y}}_{\pi(i)},\boldsymbol{\mu}\sigma_{i}\rangle+2\rho\|%
\boldsymbol{\mu}\|^{2}\sigma_{i}\sigma_{\pi(i)}\Bigr{)}.italic_f ( bold_italic_σ , bold_italic_σ start_POSTSUBSCRIPT italic_π end_POSTSUBSCRIPT ) = ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ( 2 ⟨ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_italic_μ italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⟩ + 2 ⟨ bold_italic_y start_POSTSUBSCRIPT italic_π ( italic_i ) end_POSTSUBSCRIPT , bold_italic_μ italic_σ start_POSTSUBSCRIPT italic_π ( italic_i ) end_POSTSUBSCRIPT ⟩ - 2 italic_ρ ⟨ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_italic_μ italic_σ start_POSTSUBSCRIPT italic_π ( italic_i ) end_POSTSUBSCRIPT ⟩ - 2 italic_ρ ⟨ bold_italic_y start_POSTSUBSCRIPT italic_π ( italic_i ) end_POSTSUBSCRIPT , bold_italic_μ italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⟩ + 2 italic_ρ ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_π ( italic_i ) end_POSTSUBSCRIPT ) .. , 2 =
Step (a)𝑎(a)( italic_a ) uses the fact that π∗subscript𝜋\pi_{*}italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT is uniformly distributed over Snsubscript𝑆𝑛S_{n}italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT. Note also that
∑i=1n‖𝒙i‖2+‖𝒚π(i)‖2superscriptsubscript𝑖1𝑛superscriptnormsubscript𝒙𝑖2superscriptnormsubscript𝒚𝜋𝑖2\sum_{i=1}^{n}\|{\boldsymbol{x}}_{i}\|^{2}+\|{\boldsymbol{y}}_{\pi(i)}\|^{2}∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ∥ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ∥ bold_italic_y start_POSTSUBSCRIPT italic_π ( italic_i ) end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT
is invariant under permutation π𝜋\piitalic_π, ensuring this part does not affect the alignment decision. If the community labels 𝝈𝝈{\boldsymbol{\sigma}}bold_italic_σ and 𝝈πsubscript𝝈𝜋{\boldsymbol{\sigma}}_{\pi}bold_italic_σ start_POSTSUBSCRIPT italic_π end_POSTSUBSCRIPT are known, then f(𝝈,𝝈π)𝑓𝝈subscript𝝈𝜋f\bigl{(}{\boldsymbol{\sigma}},{\boldsymbol{\sigma}}_{\pi}\bigr{)}italic_f ( bold_italic_σ , bold_italic_σ start_POSTSUBSCRIPT italic_π end_POSTSUBSCRIPT ) is also fixed, so the MAP estimator reduces exactly to the simpler distance-based estimator in (6). Even without known labels, the proof of Theorem 1 shows that (6) attains tight bounds under the conditions ‖𝝁‖2≥2logn+ω(1)superscriptnorm𝝁22𝑛𝜔1\|\boldsymbol{\mu}\|^{2}\geq 2\log n+\omega(1)∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ 2 roman_log italic_n + italic_ω ( 1 ) or d=ω(logn)𝑑𝜔𝑛d=\omega(\log n)italic_d = italic_ω ( roman_log italic_n ); see Section VI for details.
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Assume d=o(nlogn)𝑑𝑜𝑛𝑛d=o(n\log n)italic_d = italic_o ( italic_n roman_log italic_n ). If ‖𝛍‖2≥(2+ϵ)lognsuperscriptnorm𝛍22italic-ϵ𝑛\|\boldsymbol{\mu}\|^{2}\geq(2+\epsilon)\log n∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ ( 2 + italic_ϵ ) roman_log italic_n for some ϵ>0italic-ϵ0\epsilon>0italic_ϵ > 0, then by [14], exact community recovery is already possible in each individual database X𝑋Xitalic_X or Y𝑌Yitalic_Y. In this situation, we only need to align nodes within the same community. Consequently, the distance-based estimator (6), which coincides with the MAP rule under known labels, matches nodes optimally. This explains the sufficiency of condition (7) for exact alignment when communities can first be recovered.
On the other hand, if ‖𝛍‖2=O(logn)superscriptnorm𝛍2𝑂𝑛\|\boldsymbol{\mu}\|^{2}=O(\log n)∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = italic_O ( roman_log italic_n ) is not large enough to recover community labels, exact matching can still be achieved by taking the condition (8). If d=ω(logn)𝑑𝜔𝑛d=\omega(\log n)italic_d = italic_ω ( roman_log italic_n ), ‖𝐱i−𝐲i‖2superscriptnormsubscript𝐱𝑖subscript𝐲𝑖2\|{\boldsymbol{x}}_{i}-{\boldsymbol{y}}_{i}\|^{2}∥ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, which originally follows the scaled chi-squared distribution, can be approximated as a normal distribution as follows:
, 1 = ‖𝒙i−𝒚i‖2≈(d)2𝒩((1−ρ)d,2(1−ρ)2d).superscript𝑑superscriptnormsubscript𝒙𝑖subscript𝒚𝑖22𝒩1𝜌𝑑2superscript1𝜌2𝑑\|{\boldsymbol{x}}_{i}-{\boldsymbol{y}}_{i}\|^{2}\stackrel{{\scriptstyle(d)}}{%
{\approx}}2\mathcal{N}((1-\rho)d,2(1-\rho)^{2}d).∥ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_RELOP SUPERSCRIPTOP start_ARG ≈ end_ARG start_ARG ( italic_d ) end_ARG end_RELOP 2 caligraphic_N ( ( 1 - italic_ρ ) italic_d , 2 ( 1 - italic_ρ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_d ) .. , 2 = . , 3 = (10)
Similarly, for different nodes within the same community, we obtain
, 1 = ‖𝒙i−𝒚j‖2≈(d)2𝒩(d,2d) for i≠j and σi=σj,superscript𝑑superscriptnormsubscript𝒙𝑖subscript𝒚𝑗22𝒩𝑑2𝑑 for 𝑖𝑗 and subscript𝜎𝑖subscript𝜎𝑗\|{\boldsymbol{x}}_{i}-{\boldsymbol{y}}_{j}\|^{2}\stackrel{{\scriptstyle(d)}}{%
{\approx}}2\mathcal{N}(d,2d)\;\text{ for }i\neq j\text{ and }\sigma_{i}=\sigma%
_{j},∥ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_italic_y start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_RELOP SUPERSCRIPTOP start_ARG ≈ end_ARG start_ARG ( italic_d ) end_ARG end_RELOP 2 caligraphic_N ( italic_d , 2 italic_d ) for italic_i ≠ italic_j and italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ,. , 2 = . , 3 = (11)
and for nodes in different communities,
, 1 = ‖𝒙i−𝒚k‖2≈(d)2𝒩(2‖𝝁‖2+d,8‖𝝁‖2+2d) for i≠k and σi≠σk.superscript𝑑superscriptnormsubscript𝒙𝑖subscript𝒚𝑘22𝒩2superscriptnorm𝝁2𝑑8superscriptnorm𝝁22𝑑 for 𝑖𝑘 and subscript𝜎𝑖subscript𝜎𝑘\|{\boldsymbol{x}}_{i}-{\boldsymbol{y}}_{k}\|^{2}\stackrel{{\scriptstyle(d)}}{%
{\approx}}2\mathcal{N}(2\|\boldsymbol{\mu}\|^{2}+d,8\|\boldsymbol{\mu}\|^{2}+2%
d)\;\text{ for }i\neq k\text{ and }\sigma_{i}\neq\sigma_{k}.∥ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_italic_y start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_RELOP SUPERSCRIPTOP start_ARG ≈ end_ARG start_ARG ( italic_d ) end_ARG end_RELOP 2 caligraphic_N ( 2 ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_d , 8 ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 2 italic_d ) for italic_i ≠ italic_k and italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≠ italic_σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT .. , 2 = . , 3 = (12)
If d≫‖𝛍‖2much-greater-than𝑑superscriptnorm𝛍2d\gg\|\boldsymbol{\mu}\|^{2}italic_d ≫ ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, then we can consider ‖𝐱i−𝐲k‖2≈(d)2𝒩(d,2d)superscript𝑑superscriptnormsubscript𝐱𝑖subscript𝐲𝑘22𝒩𝑑2𝑑\|{\boldsymbol{x}}_{i}-{\boldsymbol{y}}_{k}\|^{2}\stackrel{{\scriptstyle(d)}}{%
{\approx}}2\mathcal{N}(d,2d)∥ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_italic_y start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_RELOP SUPERSCRIPTOP start_ARG ≈ end_ARG start_ARG ( italic_d ) end_ARG end_RELOP 2 caligraphic_N ( italic_d , 2 italic_d ). Thus, through the approximations, for a fixed i𝑖iitalic_i, we can obtain that
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
II Correlated Gaussian Mixture Models
II-A Exact Matching on Correlated Gaussian Mixture Models
Remark 1 (Interpretation of Conditions for Exact Matching).
Assume d=o(nlogn)𝑑𝑜𝑛𝑛d=o(n\log n)italic_d = italic_o ( italic_n roman_log italic_n ). If ‖𝛍‖2≥(2+ϵ)lognsuperscriptnorm𝛍22italic-ϵ𝑛\|\boldsymbol{\mu}\|^{2}\geq(2+\epsilon)\log n∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ ( 2 + italic_ϵ ) roman_log italic_n for some ϵ>0italic-ϵ0\epsilon>0italic_ϵ > 0, then by [14], exact community recovery is already possible in each individual database X𝑋Xitalic_X or Y𝑌Yitalic_Y. In this situation, we only need to align nodes within the same community. Consequently, the distance-based estimator (6), which coincides with the MAP rule under known labels, matches nodes optimally. This explains the sufficiency of condition (7) for exact alignment when communities can first be recovered.
On the other hand, if ‖𝛍‖2=O(logn)superscriptnorm𝛍2𝑂𝑛\|\boldsymbol{\mu}\|^{2}=O(\log n)∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = italic_O ( roman_log italic_n ) is not large enough to recover community labels, exact matching can still be achieved by taking the condition (8). If d=ω(logn)𝑑𝜔𝑛d=\omega(\log n)italic_d = italic_ω ( roman_log italic_n ), ‖𝐱i−𝐲i‖2superscriptnormsubscript𝐱𝑖subscript𝐲𝑖2\|{\boldsymbol{x}}_{i}-{\boldsymbol{y}}_{i}\|^{2}∥ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, which originally follows the scaled chi-squared distribution, can be approximated as a normal distribution as follows:
, 1 = ‖𝒙i−𝒚i‖2≈(d)2𝒩((1−ρ)d,2(1−ρ)2d).superscript𝑑superscriptnormsubscript𝒙𝑖subscript𝒚𝑖22𝒩1𝜌𝑑2superscript1𝜌2𝑑\|{\boldsymbol{x}}_{i}-{\boldsymbol{y}}_{i}\|^{2}\stackrel{{\scriptstyle(d)}}{%
{\approx}}2\mathcal{N}((1-\rho)d,2(1-\rho)^{2}d).∥ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_RELOP SUPERSCRIPTOP start_ARG ≈ end_ARG start_ARG ( italic_d ) end_ARG end_RELOP 2 caligraphic_N ( ( 1 - italic_ρ ) italic_d , 2 ( 1 - italic_ρ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_d ) .. , 2 = . , 3 = (10)
Similarly, for different nodes within the same community, we obtain
, 1 = ‖𝒙i−𝒚j‖2≈(d)2𝒩(d,2d) for i≠j and σi=σj,superscript𝑑superscriptnormsubscript𝒙𝑖subscript𝒚𝑗22𝒩𝑑2𝑑 for 𝑖𝑗 and subscript𝜎𝑖subscript𝜎𝑗\|{\boldsymbol{x}}_{i}-{\boldsymbol{y}}_{j}\|^{2}\stackrel{{\scriptstyle(d)}}{%
{\approx}}2\mathcal{N}(d,2d)\;\text{ for }i\neq j\text{ and }\sigma_{i}=\sigma%
_{j},∥ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_italic_y start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_RELOP SUPERSCRIPTOP start_ARG ≈ end_ARG start_ARG ( italic_d ) end_ARG end_RELOP 2 caligraphic_N ( italic_d , 2 italic_d ) for italic_i ≠ italic_j and italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ,. , 2 = . , 3 = (11)
and for nodes in different communities,
, 1 = ‖𝒙i−𝒚k‖2≈(d)2𝒩(2‖𝝁‖2+d,8‖𝝁‖2+2d) for i≠k and σi≠σk.superscript𝑑superscriptnormsubscript𝒙𝑖subscript𝒚𝑘22𝒩2superscriptnorm𝝁2𝑑8superscriptnorm𝝁22𝑑 for 𝑖𝑘 and subscript𝜎𝑖subscript𝜎𝑘\|{\boldsymbol{x}}_{i}-{\boldsymbol{y}}_{k}\|^{2}\stackrel{{\scriptstyle(d)}}{%
{\approx}}2\mathcal{N}(2\|\boldsymbol{\mu}\|^{2}+d,8\|\boldsymbol{\mu}\|^{2}+2%
d)\;\text{ for }i\neq k\text{ and }\sigma_{i}\neq\sigma_{k}.∥ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_italic_y start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_RELOP SUPERSCRIPTOP start_ARG ≈ end_ARG start_ARG ( italic_d ) end_ARG end_RELOP 2 caligraphic_N ( 2 ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_d , 8 ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 2 italic_d ) for italic_i ≠ italic_k and italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≠ italic_σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT .. , 2 = . , 3 = (12)
If d≫‖𝛍‖2much-greater-than𝑑superscriptnorm𝛍2d\gg\|\boldsymbol{\mu}\|^{2}italic_d ≫ ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, then we can consider ‖𝐱i−𝐲k‖2≈(d)2𝒩(d,2d)superscript𝑑superscriptnormsubscript𝐱𝑖subscript𝐲𝑘22𝒩𝑑2𝑑\|{\boldsymbol{x}}_{i}-{\boldsymbol{y}}_{k}\|^{2}\stackrel{{\scriptstyle(d)}}{%
{\approx}}2\mathcal{N}(d,2d)∥ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_italic_y start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_RELOP SUPERSCRIPTOP start_ARG ≈ end_ARG start_ARG ( italic_d ) end_ARG end_RELOP 2 caligraphic_N ( italic_d , 2 italic_d ). Thus, through the approximations, for a fixed i𝑖iitalic_i, we can obtain that
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, 1 = . , 2 = ℙ(‖𝒙i−𝒚i‖2≥‖𝒙i−𝒚j‖2,∀j≠i)ℙformulae-sequencesuperscriptnormsubscript𝒙𝑖subscript𝒚𝑖2superscriptnormsubscript𝒙𝑖subscript𝒚𝑗2for-all𝑗𝑖\displaystyle\mathbb{P}\left(\|{\boldsymbol{x}}_{i}-{\boldsymbol{y}}_{i}\|^{2}%
\geq\|{\boldsymbol{x}}_{i}-{\boldsymbol{y}}_{j}\|^{2},\;\forall j\neq i\right)blackboard_P ( ∥ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ ∥ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_italic_y start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , ∀ italic_j ≠ italic_i ). , 3 = . , 4 = (13). , 1 = . , 2 = ≤(a)nℙ(𝒩((1−ρ)d,2(1−ρ)2d)≥𝒩(d,2d))superscript𝑎absent𝑛ℙ𝒩1𝜌𝑑2superscript1𝜌2𝑑𝒩𝑑2𝑑\displaystyle\stackrel{{\scriptstyle(a)}}{{\leq}}n\mathbb{P}\left(\mathcal{N}(%
(1-\rho)d,2(1-\rho)^{2}d)\geq\mathcal{N}(d,2d)\right)start_RELOP SUPERSCRIPTOP start_ARG ≤ end_ARG start_ARG ( italic_a ) end_ARG end_RELOP italic_n blackboard_P ( caligraphic_N ( ( 1 - italic_ρ ) italic_d , 2 ( 1 - italic_ρ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_d ) ≥ caligraphic_N ( italic_d , 2 italic_d ) ). , 3 = . , 4 = (13). , 1 = . , 2 = ≤nℙ(𝒩(0,2d(1+(1−ρ)2))≥ρd)absent𝑛ℙ𝒩02𝑑1superscript1𝜌2𝜌𝑑\displaystyle\leq n\mathbb{P}\left(\mathcal{N}(0,2d(1+(1-\rho)^{2}))\geq\rho d\right)≤ italic_n blackboard_P ( caligraphic_N ( 0 , 2 italic_d ( 1 + ( 1 - italic_ρ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ) ≥ italic_ρ italic_d ). , 3 = . , 4 = (13). , 1 = . , 2 = ≤(b)exp(logn−14dρ2⋅11+(1−ρ)2).superscript𝑏absent𝑛⋅14𝑑superscript𝜌211superscript1𝜌2\displaystyle\stackrel{{\scriptstyle(b)}}{{\leq}}\exp\left(\log n-\frac{1}{4}d%
\rho^{2}\cdot\frac{1}{1+(1-\rho)^{2}}\right).start_RELOP SUPERSCRIPTOP start_ARG ≤ end_ARG start_ARG ( italic_b ) end_ARG end_RELOP roman_exp ( roman_log italic_n - divide start_ARG 1 end_ARG start_ARG 4 end_ARG italic_d italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ⋅ divide start_ARG 1 end_ARG start_ARG 1 + ( 1 - italic_ρ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) .. , 3 = . , 4 = (13)
The inequality (a)𝑎(a)( italic_a ) holds by approximating (12) as (11) and taking a union bound over j≠i𝑗𝑖j\neq iitalic_j ≠ italic_i, and the inequality (b)𝑏(b)( italic_b ) holds by the tail bound of normal distribution (Lemma 19).
Therefore, if 14dρ2≥(1+(1−ρ)2)logn+ω(1)14𝑑superscript𝜌21superscript1𝜌2𝑛𝜔1\frac{1}{4}d\rho^{2}\geq(1+(1-\rho)^{2})\log n+\omega(1)divide start_ARG 1 end_ARG start_ARG 4 end_ARG italic_d italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ ( 1 + ( 1 - italic_ρ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) roman_log italic_n + italic_ω ( 1 ), exact matching becomes achievable even with a greedy algorithm, which attempts to match each point to its nearest neighbor.
If we consider the case where ρ=o(1)𝜌𝑜1\rho=o(1)italic_ρ = italic_o ( 1 ), the above condition becomes 14dρ2≥2logn+ω(1)14𝑑superscript𝜌22𝑛𝜔1\frac{1}{4}d\rho^{2}\geq 2\log n+\omega(1)divide start_ARG 1 end_ARG start_ARG 4 end_ARG italic_d italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ 2 roman_log italic_n + italic_ω ( 1 ), while the bound in (8) is approximately d4log11−ρ2≈14dρ2≥(1+ϵ)logn𝑑411superscript𝜌214𝑑superscript𝜌21italic-ϵ𝑛\frac{d}{4}\log\frac{1}{1-\rho^{2}}\approx\frac{1}{4}d\rho^{2}\geq(1+\epsilon)\log
ndivide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≈ divide start_ARG 1 end_ARG start_ARG 4 end_ARG italic_d italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ ( 1 + italic_ϵ ) roman_log italic_n. The tighter bound in (8) is obtained by carefully analyzing the distance-based estimator (6).
We now state the matching impossibility result for correlated GMMs.
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
II Correlated Gaussian Mixture Models
II-A Exact Matching on Correlated Gaussian Mixture Models
Remark 1 (Interpretation of Conditions for Exact Matching).
, 1 = . , 2 = ℙ(‖𝒙i−𝒚i‖2≥‖𝒙i−𝒚j‖2,∀j≠i)ℙformulae-sequencesuperscriptnormsubscript𝒙𝑖subscript𝒚𝑖2superscriptnormsubscript𝒙𝑖subscript𝒚𝑗2for-all𝑗𝑖\displaystyle\mathbb{P}\left(\|{\boldsymbol{x}}_{i}-{\boldsymbol{y}}_{i}\|^{2}%
\geq\|{\boldsymbol{x}}_{i}-{\boldsymbol{y}}_{j}\|^{2},\;\forall j\neq i\right)blackboard_P ( ∥ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ ∥ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_italic_y start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , ∀ italic_j ≠ italic_i ). , 3 = . , 4 = (13). , 1 = . , 2 = ≤(a)nℙ(𝒩((1−ρ)d,2(1−ρ)2d)≥𝒩(d,2d))superscript𝑎absent𝑛ℙ𝒩1𝜌𝑑2superscript1𝜌2𝑑𝒩𝑑2𝑑\displaystyle\stackrel{{\scriptstyle(a)}}{{\leq}}n\mathbb{P}\left(\mathcal{N}(%
(1-\rho)d,2(1-\rho)^{2}d)\geq\mathcal{N}(d,2d)\right)start_RELOP SUPERSCRIPTOP start_ARG ≤ end_ARG start_ARG ( italic_a ) end_ARG end_RELOP italic_n blackboard_P ( caligraphic_N ( ( 1 - italic_ρ ) italic_d , 2 ( 1 - italic_ρ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_d ) ≥ caligraphic_N ( italic_d , 2 italic_d ) ). , 3 = . , 4 = (13). , 1 = . , 2 = ≤nℙ(𝒩(0,2d(1+(1−ρ)2))≥ρd)absent𝑛ℙ𝒩02𝑑1superscript1𝜌2𝜌𝑑\displaystyle\leq n\mathbb{P}\left(\mathcal{N}(0,2d(1+(1-\rho)^{2}))\geq\rho d\right)≤ italic_n blackboard_P ( caligraphic_N ( 0 , 2 italic_d ( 1 + ( 1 - italic_ρ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ) ≥ italic_ρ italic_d ). , 3 = . , 4 = (13). , 1 = . , 2 = ≤(b)exp(logn−14dρ2⋅11+(1−ρ)2).superscript𝑏absent𝑛⋅14𝑑superscript𝜌211superscript1𝜌2\displaystyle\stackrel{{\scriptstyle(b)}}{{\leq}}\exp\left(\log n-\frac{1}{4}d%
\rho^{2}\cdot\frac{1}{1+(1-\rho)^{2}}\right).start_RELOP SUPERSCRIPTOP start_ARG ≤ end_ARG start_ARG ( italic_b ) end_ARG end_RELOP roman_exp ( roman_log italic_n - divide start_ARG 1 end_ARG start_ARG 4 end_ARG italic_d italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ⋅ divide start_ARG 1 end_ARG start_ARG 1 + ( 1 - italic_ρ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) .. , 3 = . , 4 = (13)
The inequality (a)𝑎(a)( italic_a ) holds by approximating (12) as (11) and taking a union bound over j≠i𝑗𝑖j\neq iitalic_j ≠ italic_i, and the inequality (b)𝑏(b)( italic_b ) holds by the tail bound of normal distribution (Lemma 19).
Therefore, if 14dρ2≥(1+(1−ρ)2)logn+ω(1)14𝑑superscript𝜌21superscript1𝜌2𝑛𝜔1\frac{1}{4}d\rho^{2}\geq(1+(1-\rho)^{2})\log n+\omega(1)divide start_ARG 1 end_ARG start_ARG 4 end_ARG italic_d italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ ( 1 + ( 1 - italic_ρ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) roman_log italic_n + italic_ω ( 1 ), exact matching becomes achievable even with a greedy algorithm, which attempts to match each point to its nearest neighbor.
If we consider the case where ρ=o(1)𝜌𝑜1\rho=o(1)italic_ρ = italic_o ( 1 ), the above condition becomes 14dρ2≥2logn+ω(1)14𝑑superscript𝜌22𝑛𝜔1\frac{1}{4}d\rho^{2}\geq 2\log n+\omega(1)divide start_ARG 1 end_ARG start_ARG 4 end_ARG italic_d italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ 2 roman_log italic_n + italic_ω ( 1 ), while the bound in (8) is approximately d4log11−ρ2≈14dρ2≥(1+ϵ)logn𝑑411superscript𝜌214𝑑superscript𝜌21italic-ϵ𝑛\frac{d}{4}\log\frac{1}{1-\rho^{2}}\approx\frac{1}{4}d\rho^{2}\geq(1+\epsilon)\log
ndivide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≈ divide start_ARG 1 end_ARG start_ARG 4 end_ARG italic_d italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ ( 1 + italic_ϵ ) roman_log italic_n. The tighter bound in (8) is obtained by carefully analyzing the distance-based estimator (6).
We now state the matching impossibility result for correlated GMMs.
| 1,662
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25
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Let (X,Y)∼CGMMs(n,𝛍,d,ρ)similar-to𝑋𝑌CGMMs𝑛𝛍𝑑𝜌(X,Y)\sim\textnormal{CGMMs}(n,\boldsymbol{\mu},d,\rho)( italic_X , italic_Y ) ∼ CGMMs ( italic_n , bold_italic_μ , italic_d , italic_ρ ) be as in Section I-A1. Suppose either
, 1 = d4log11−ρ2≤(1−ϵ)lognand1≪d=O(logn),formulae-sequence𝑑411superscript𝜌21italic-ϵ𝑛andmuch-less-than1𝑑𝑂𝑛\frac{d}{4}\log\frac{1}{1-\rho^{2}}\leq(1-\epsilon)\log n\quad\text{and}\quad 1%
\ll d=O(\log n),divide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≤ ( 1 - italic_ϵ ) roman_log italic_n and 1 ≪ italic_d = italic_O ( roman_log italic_n ) ,. , 2 = . , 3 = (14)
for any small ϵ>0italic-ϵ0\epsilon>0italic_ϵ > 0, or
, 1 = d4log11−ρ2≤logn−logd+Cand1ρ2−1≤d40formulae-sequence𝑑411superscript𝜌2𝑛𝑑𝐶and1superscript𝜌21𝑑40\frac{d}{4}\log\frac{1}{1-\rho^{2}}\leq\log n-\log d+C\quad\text{and}\quad%
\frac{1}{\rho^{2}}-1\leq\frac{d}{40}divide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≤ roman_log italic_n - roman_log italic_d + italic_C and divide start_ARG 1 end_ARG start_ARG italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG - 1 ≤ divide start_ARG italic_d end_ARG start_ARG 40 end_ARG. , 2 = . , 3 = (15)
for some positive constant C>0𝐶0C>0italic_C > 0. Under either set of conditions, there is no estimator that can achieve exact matching with high probability.
| 628
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
II Correlated Gaussian Mixture Models
II-A Exact Matching on Correlated Gaussian Mixture Models
Theorem 2 (Impossibility for Exact Matching).
Let (X,Y)∼CGMMs(n,𝛍,d,ρ)similar-to𝑋𝑌CGMMs𝑛𝛍𝑑𝜌(X,Y)\sim\textnormal{CGMMs}(n,\boldsymbol{\mu},d,\rho)( italic_X , italic_Y ) ∼ CGMMs ( italic_n , bold_italic_μ , italic_d , italic_ρ ) be as in Section I-A1. Suppose either
, 1 = d4log11−ρ2≤(1−ϵ)lognand1≪d=O(logn),formulae-sequence𝑑411superscript𝜌21italic-ϵ𝑛andmuch-less-than1𝑑𝑂𝑛\frac{d}{4}\log\frac{1}{1-\rho^{2}}\leq(1-\epsilon)\log n\quad\text{and}\quad 1%
\ll d=O(\log n),divide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≤ ( 1 - italic_ϵ ) roman_log italic_n and 1 ≪ italic_d = italic_O ( roman_log italic_n ) ,. , 2 = . , 3 = (14)
for any small ϵ>0italic-ϵ0\epsilon>0italic_ϵ > 0, or
, 1 = d4log11−ρ2≤logn−logd+Cand1ρ2−1≤d40formulae-sequence𝑑411superscript𝜌2𝑛𝑑𝐶and1superscript𝜌21𝑑40\frac{d}{4}\log\frac{1}{1-\rho^{2}}\leq\log n-\log d+C\quad\text{and}\quad%
\frac{1}{\rho^{2}}-1\leq\frac{d}{40}divide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≤ roman_log italic_n - roman_log italic_d + italic_C and divide start_ARG 1 end_ARG start_ARG italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG - 1 ≤ divide start_ARG italic_d end_ARG start_ARG 40 end_ARG. , 2 = . , 3 = (15)
for some positive constant C>0𝐶0C>0italic_C > 0. Under either set of conditions, there is no estimator that can achieve exact matching with high probability.
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Comparing Theorems 1 and 2 shows that the limiting condition for exact matching is roughly
, 1 = d4log11−ρ2≥(1+ϵ)logn.𝑑411superscript𝜌21italic-ϵ𝑛\frac{d}{4}\log\frac{1}{1-\rho^{2}}\geq(1+\epsilon)\log n.divide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≥ ( 1 + italic_ϵ ) roman_log italic_n .. , 2 = . , 3 = (16)
However, to attain achievability, we also require ‖𝛍‖2≥2logn+ω(1)superscriptnorm𝛍22𝑛𝜔1\|\boldsymbol{\mu}\|^{2}\geq 2\log n+\omega(1)∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ 2 roman_log italic_n + italic_ω ( 1 ) or d=ω(logn)𝑑𝜔𝑛d=\omega(\log n)italic_d = italic_ω ( roman_log italic_n ). This gap may arise because the distance-based estimator (6) ignores the term f(𝛔,𝛔π)𝑓𝛔subscript𝛔𝜋f({\boldsymbol{\sigma}},{\boldsymbol{\sigma}}_{\pi})italic_f ( bold_italic_σ , bold_italic_σ start_POSTSUBSCRIPT italic_π end_POSTSUBSCRIPT ) in (9). This simplification enables us to recover the matching without relying on the knowledge of the community labels 𝛔𝛔{\boldsymbol{\sigma}}bold_italic_σ and 𝛔πsubscript𝛔𝜋{\boldsymbol{\sigma}}_{\pi}bold_italic_σ start_POSTSUBSCRIPT italic_π end_POSTSUBSCRIPT, but possibly at the cost of stricter requirements than the full MAP approach.
To prove the converse in Theorem 2, we show that the stated conditions imply failure of exact matching even if the community labels 𝝈1superscript𝝈1{\boldsymbol{\sigma}}^{1}bold_italic_σ start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT and 𝝈2superscript𝝈2{\boldsymbol{\sigma}}^{2}bold_italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT are known, and a partial matching for all but |V′|superscript𝑉′|V^{\prime}|| italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT | nodes is revealed (where V′:={i∈[n]:σi=+1}assignsuperscript𝑉′conditional-set𝑖delimited-[]𝑛subscript𝜎𝑖1V^{\prime}:=\{\,i\in[n]:\sigma_{i}=+1\}italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT := { italic_i ∈ [ italic_n ] : italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = + 1 }). This reduces the task to a database alignment problem with |V′|superscript𝑉′|V^{\prime}|| italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT | nodes in correlated Gaussian databases, for which previous impossibility arguments in [9, 36] directly apply. A complete proof is given in Section VII.
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
II Correlated Gaussian Mixture Models
II-A Exact Matching on Correlated Gaussian Mixture Models
Remark 2 (Gaps in Achievability and Converse Results).
Comparing Theorems 1 and 2 shows that the limiting condition for exact matching is roughly
, 1 = d4log11−ρ2≥(1+ϵ)logn.𝑑411superscript𝜌21italic-ϵ𝑛\frac{d}{4}\log\frac{1}{1-\rho^{2}}\geq(1+\epsilon)\log n.divide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≥ ( 1 + italic_ϵ ) roman_log italic_n .. , 2 = . , 3 = (16)
However, to attain achievability, we also require ‖𝛍‖2≥2logn+ω(1)superscriptnorm𝛍22𝑛𝜔1\|\boldsymbol{\mu}\|^{2}\geq 2\log n+\omega(1)∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ 2 roman_log italic_n + italic_ω ( 1 ) or d=ω(logn)𝑑𝜔𝑛d=\omega(\log n)italic_d = italic_ω ( roman_log italic_n ). This gap may arise because the distance-based estimator (6) ignores the term f(𝛔,𝛔π)𝑓𝛔subscript𝛔𝜋f({\boldsymbol{\sigma}},{\boldsymbol{\sigma}}_{\pi})italic_f ( bold_italic_σ , bold_italic_σ start_POSTSUBSCRIPT italic_π end_POSTSUBSCRIPT ) in (9). This simplification enables us to recover the matching without relying on the knowledge of the community labels 𝛔𝛔{\boldsymbol{\sigma}}bold_italic_σ and 𝛔πsubscript𝛔𝜋{\boldsymbol{\sigma}}_{\pi}bold_italic_σ start_POSTSUBSCRIPT italic_π end_POSTSUBSCRIPT, but possibly at the cost of stricter requirements than the full MAP approach.
To prove the converse in Theorem 2, we show that the stated conditions imply failure of exact matching even if the community labels 𝝈1superscript𝝈1{\boldsymbol{\sigma}}^{1}bold_italic_σ start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT and 𝝈2superscript𝝈2{\boldsymbol{\sigma}}^{2}bold_italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT are known, and a partial matching for all but |V′|superscript𝑉′|V^{\prime}|| italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT | nodes is revealed (where V′:={i∈[n]:σi=+1}assignsuperscript𝑉′conditional-set𝑖delimited-[]𝑛subscript𝜎𝑖1V^{\prime}:=\{\,i\in[n]:\sigma_{i}=+1\}italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT := { italic_i ∈ [ italic_n ] : italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = + 1 }). This reduces the task to a database alignment problem with |V′|superscript𝑉′|V^{\prime}|| italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT | nodes in correlated Gaussian databases, for which previous impossibility arguments in [9, 36] directly apply. A complete proof is given in Section VII.
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In [6], it was demonstrated that for correlated SBMs, once exact matching is achieved, combining correlated edges to form a denser union graph can extend the regime where exact community recovery is possible. Similarly, we now identify the conditions under which exact community recovery becomes feasible in correlated Gaussian Mixture Models (GMMs) when two correlated node-attribute databases are available.
| 75
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
II Correlated Gaussian Mixture Models
II-B Exact Community Recovery in Correlated Gaussian Mixture Models
In [6], it was demonstrated that for correlated SBMs, once exact matching is achieved, combining correlated edges to form a denser union graph can extend the regime where exact community recovery is possible. Similarly, we now identify the conditions under which exact community recovery becomes feasible in correlated Gaussian Mixture Models (GMMs) when two correlated node-attribute databases are available.
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Let (X,Y)∼CGMMs(n,𝛍,d,ρ)similar-to𝑋𝑌CGMMs𝑛𝛍𝑑𝜌(X,Y)\sim\textnormal{CGMMs}(n,\boldsymbol{\mu},d,\rho)( italic_X , italic_Y ) ∼ CGMMs ( italic_n , bold_italic_μ , italic_d , italic_ρ ) be as defined in Section I-A1. Suppose either (7) or (8) holds. If
, 1 = ‖𝝁‖2≥(1+ϵ)1+ρ2(1+1+2dnlogn)lognsuperscriptnorm𝝁21italic-ϵ1𝜌2112𝑑𝑛𝑛𝑛\|\boldsymbol{\mu}\|^{2}\geq(1+\epsilon)\frac{1+\rho}{2}\biggl{(}1+\sqrt{1+%
\frac{2d}{n\log n}}\biggr{)}\log n∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ ( 1 + italic_ϵ ) divide start_ARG 1 + italic_ρ end_ARG start_ARG 2 end_ARG ( 1 + square-root start_ARG 1 + divide start_ARG 2 italic_d end_ARG start_ARG italic_n roman_log italic_n end_ARG end_ARG ) roman_log italic_n. , 2 = . , 3 = (17)
for an arbitrarily small constant ϵ>0italic-ϵ0\epsilon>0italic_ϵ > 0, then there exists an estimator 𝛔^(X,Y)^𝛔𝑋𝑌\hat{{\boldsymbol{\sigma}}}(X,Y)over^ start_ARG bold_italic_σ end_ARG ( italic_X , italic_Y ) such that 𝛔^(X,Y)=𝛔^𝛔𝑋𝑌𝛔\hat{{\boldsymbol{\sigma}}}(X,Y)={\boldsymbol{\sigma}}over^ start_ARG bold_italic_σ end_ARG ( italic_X , italic_Y ) = bold_italic_σ with high probability.
If exact matching is possible–or if π∗subscript𝜋\pi_{*}italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT is given–then the two correlated node attributes can be merged by taking their average. Specifically, if node i∈[n]𝑖delimited-[]𝑛i\in[n]italic_i ∈ [ italic_n ] has attributes 𝒙isubscript𝒙𝑖{\boldsymbol{x}}_{i}bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT and 𝒚π∗(i)subscript𝒚subscript𝜋𝑖{\boldsymbol{y}}_{\pi_{*}(i)}bold_italic_y start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_i ) end_POSTSUBSCRIPT, we form
, 1 = 𝒙i+𝒚π∗(i)2=𝝁σi+(1+ρ)𝒛i+1−ρ2𝒘i2,subscript𝒙𝑖subscript𝒚subscript𝜋𝑖2𝝁subscript𝜎𝑖1𝜌subscript𝒛𝑖1superscript𝜌2subscript𝒘𝑖2\frac{{\boldsymbol{x}}_{i}+{\boldsymbol{y}}_{\pi_{*}(i)}}{2}=\boldsymbol{\mu}%
\sigma_{i}+\frac{(1+\rho){\boldsymbol{z}}_{i}+\sqrt{1-\rho^{2}}{\boldsymbol{w}%
}_{i}}{2},divide start_ARG bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + bold_italic_y start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_i ) end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG = bold_italic_μ italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + divide start_ARG ( 1 + italic_ρ ) bold_italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + square-root start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG bold_italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG ,. , 2 = . , 3 = (18)
where 𝒙isubscript𝒙𝑖{\boldsymbol{x}}_{i}bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT and 𝒚isubscript𝒚𝑖{\boldsymbol{y}}_{i}bold_italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT are defined in (1) and (2), respectively. Since 𝒛isubscript𝒛𝑖{\boldsymbol{z}}_{i}bold_italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT and 𝒘isubscript𝒘𝑖{\boldsymbol{w}}_{i}bold_italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT are independent d𝑑ditalic_d-dimensional Gaussian vectors,
(1+ρ)𝒛i+1−ρ2𝒘i21𝜌subscript𝒛𝑖1superscript𝜌2subscript𝒘𝑖2\frac{(1+\rho){\boldsymbol{z}}_{i}+\sqrt{1-\rho^{2}}{\boldsymbol{w}}_{i}}{2}divide start_ARG ( 1 + italic_ρ ) bold_italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + square-root start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG bold_italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG
is a d𝑑ditalic_d-dimensional Gaussian with mean 00 and covariance 1+ρ2𝑰d1𝜌2subscript𝑰𝑑\frac{1+\rho}{2}\,{\boldsymbol{I}}_{d}divide start_ARG 1 + italic_ρ end_ARG start_ARG 2 end_ARG bold_italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT. Therefore, the averaged attribute has the same mean as the individual node attributes but a smaller variance. By applying known results for Gaussian Mixture Models [14, 5] to this averaged feature, we establish Theorem 3. A detailed proof is provided in Section VIII.
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
II Correlated Gaussian Mixture Models
II-B Exact Community Recovery in Correlated Gaussian Mixture Models
Theorem 3 (Achievability for Exact Community Recovery).
Let (X,Y)∼CGMMs(n,𝛍,d,ρ)similar-to𝑋𝑌CGMMs𝑛𝛍𝑑𝜌(X,Y)\sim\textnormal{CGMMs}(n,\boldsymbol{\mu},d,\rho)( italic_X , italic_Y ) ∼ CGMMs ( italic_n , bold_italic_μ , italic_d , italic_ρ ) be as defined in Section I-A1. Suppose either (7) or (8) holds. If
, 1 = ‖𝝁‖2≥(1+ϵ)1+ρ2(1+1+2dnlogn)lognsuperscriptnorm𝝁21italic-ϵ1𝜌2112𝑑𝑛𝑛𝑛\|\boldsymbol{\mu}\|^{2}\geq(1+\epsilon)\frac{1+\rho}{2}\biggl{(}1+\sqrt{1+%
\frac{2d}{n\log n}}\biggr{)}\log n∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ ( 1 + italic_ϵ ) divide start_ARG 1 + italic_ρ end_ARG start_ARG 2 end_ARG ( 1 + square-root start_ARG 1 + divide start_ARG 2 italic_d end_ARG start_ARG italic_n roman_log italic_n end_ARG end_ARG ) roman_log italic_n. , 2 = . , 3 = (17)
for an arbitrarily small constant ϵ>0italic-ϵ0\epsilon>0italic_ϵ > 0, then there exists an estimator 𝛔^(X,Y)^𝛔𝑋𝑌\hat{{\boldsymbol{\sigma}}}(X,Y)over^ start_ARG bold_italic_σ end_ARG ( italic_X , italic_Y ) such that 𝛔^(X,Y)=𝛔^𝛔𝑋𝑌𝛔\hat{{\boldsymbol{\sigma}}}(X,Y)={\boldsymbol{\sigma}}over^ start_ARG bold_italic_σ end_ARG ( italic_X , italic_Y ) = bold_italic_σ with high probability.
If exact matching is possible–or if π∗subscript𝜋\pi_{*}italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT is given–then the two correlated node attributes can be merged by taking their average. Specifically, if node i∈[n]𝑖delimited-[]𝑛i\in[n]italic_i ∈ [ italic_n ] has attributes 𝒙isubscript𝒙𝑖{\boldsymbol{x}}_{i}bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT and 𝒚π∗(i)subscript𝒚subscript𝜋𝑖{\boldsymbol{y}}_{\pi_{*}(i)}bold_italic_y start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_i ) end_POSTSUBSCRIPT, we form
, 1 = 𝒙i+𝒚π∗(i)2=𝝁σi+(1+ρ)𝒛i+1−ρ2𝒘i2,subscript𝒙𝑖subscript𝒚subscript𝜋𝑖2𝝁subscript𝜎𝑖1𝜌subscript𝒛𝑖1superscript𝜌2subscript𝒘𝑖2\frac{{\boldsymbol{x}}_{i}+{\boldsymbol{y}}_{\pi_{*}(i)}}{2}=\boldsymbol{\mu}%
\sigma_{i}+\frac{(1+\rho){\boldsymbol{z}}_{i}+\sqrt{1-\rho^{2}}{\boldsymbol{w}%
}_{i}}{2},divide start_ARG bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + bold_italic_y start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_i ) end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG = bold_italic_μ italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + divide start_ARG ( 1 + italic_ρ ) bold_italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + square-root start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG bold_italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG ,. , 2 = . , 3 = (18)
where 𝒙isubscript𝒙𝑖{\boldsymbol{x}}_{i}bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT and 𝒚isubscript𝒚𝑖{\boldsymbol{y}}_{i}bold_italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT are defined in (1) and (2), respectively. Since 𝒛isubscript𝒛𝑖{\boldsymbol{z}}_{i}bold_italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT and 𝒘isubscript𝒘𝑖{\boldsymbol{w}}_{i}bold_italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT are independent d𝑑ditalic_d-dimensional Gaussian vectors,
(1+ρ)𝒛i+1−ρ2𝒘i21𝜌subscript𝒛𝑖1superscript𝜌2subscript𝒘𝑖2\frac{(1+\rho){\boldsymbol{z}}_{i}+\sqrt{1-\rho^{2}}{\boldsymbol{w}}_{i}}{2}divide start_ARG ( 1 + italic_ρ ) bold_italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + square-root start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG bold_italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG
is a d𝑑ditalic_d-dimensional Gaussian with mean 00 and covariance 1+ρ2𝑰d1𝜌2subscript𝑰𝑑\frac{1+\rho}{2}\,{\boldsymbol{I}}_{d}divide start_ARG 1 + italic_ρ end_ARG start_ARG 2 end_ARG bold_italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT. Therefore, the averaged attribute has the same mean as the individual node attributes but a smaller variance. By applying known results for Gaussian Mixture Models [14, 5] to this averaged feature, we establish Theorem 3. A detailed proof is provided in Section VIII.
| 1,571
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When only X𝑋Xitalic_X is available, [14, 5] showed that exact community recovery in a Gaussian Mixture Model requires
, 1 = ∥𝝁∥2≥(1+ϵ)(1+1+2dnlogn)logn.superscriptdelimited-∥∥𝝁21italic-ϵ112𝑑𝑛𝑛𝑛\lVert\boldsymbol{\mu}\rVert^{2}\geq(1+\epsilon)\left(1+\sqrt{1+\frac{2d}{n%
\log n}}\right)\log n.∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ ( 1 + italic_ϵ ) ( 1 + square-root start_ARG 1 + divide start_ARG 2 italic_d end_ARG start_ARG italic_n roman_log italic_n end_ARG end_ARG ) roman_log italic_n .. , 2 = . , 3 = (19)
Comparing (17) with (19) reveals that ‖𝛍‖2superscriptnorm𝛍2\|\boldsymbol{\mu}\|^{2}∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT can be smaller by a factor of 1+ρ21𝜌2\tfrac{1+\rho}{2}divide start_ARG 1 + italic_ρ end_ARG start_ARG 2 end_ARG when the correlated database Y𝑌Yitalic_Y is available and we can form the combined feature (18) via exact matching. From (8), such a gain is particularly relevant in the regime d=ω(logn)𝑑𝜔𝑛d=\omega(\log n)italic_d = italic_ω ( roman_log italic_n ). If d=O(logn)𝑑𝑂𝑛d=O(\log n)italic_d = italic_O ( roman_log italic_n ), then to satisfy (7) we need ‖𝛍‖2≥2logn+ω(1)superscriptnorm𝛍22𝑛𝜔1\|\boldsymbol{\mu}\|^{2}\geq 2\log n+\omega(1)∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ 2 roman_log italic_n + italic_ω ( 1 ) for exact matching, which can exceed the threshold (17) for community recovery. Hence, in low-dimensional regimes, the advantage of incorporating the second database Y𝑌Yitalic_Y is constrained by the stricter requirement on ‖𝛍‖2superscriptnorm𝛍2\|\boldsymbol{\mu}\|^{2}∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT for alignment.
We now provide the converse result, stating when exact community recovery is not possible in correlated GMMs.
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II Correlated Gaussian Mixture Models
II-B Exact Community Recovery in Correlated Gaussian Mixture Models
Remark 3 (Comparison with Standard GMM Results).
When only X𝑋Xitalic_X is available, [14, 5] showed that exact community recovery in a Gaussian Mixture Model requires
, 1 = ∥𝝁∥2≥(1+ϵ)(1+1+2dnlogn)logn.superscriptdelimited-∥∥𝝁21italic-ϵ112𝑑𝑛𝑛𝑛\lVert\boldsymbol{\mu}\rVert^{2}\geq(1+\epsilon)\left(1+\sqrt{1+\frac{2d}{n%
\log n}}\right)\log n.∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ ( 1 + italic_ϵ ) ( 1 + square-root start_ARG 1 + divide start_ARG 2 italic_d end_ARG start_ARG italic_n roman_log italic_n end_ARG end_ARG ) roman_log italic_n .. , 2 = . , 3 = (19)
Comparing (17) with (19) reveals that ‖𝛍‖2superscriptnorm𝛍2\|\boldsymbol{\mu}\|^{2}∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT can be smaller by a factor of 1+ρ21𝜌2\tfrac{1+\rho}{2}divide start_ARG 1 + italic_ρ end_ARG start_ARG 2 end_ARG when the correlated database Y𝑌Yitalic_Y is available and we can form the combined feature (18) via exact matching. From (8), such a gain is particularly relevant in the regime d=ω(logn)𝑑𝜔𝑛d=\omega(\log n)italic_d = italic_ω ( roman_log italic_n ). If d=O(logn)𝑑𝑂𝑛d=O(\log n)italic_d = italic_O ( roman_log italic_n ), then to satisfy (7) we need ‖𝛍‖2≥2logn+ω(1)superscriptnorm𝛍22𝑛𝜔1\|\boldsymbol{\mu}\|^{2}\geq 2\log n+\omega(1)∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ 2 roman_log italic_n + italic_ω ( 1 ) for exact matching, which can exceed the threshold (17) for community recovery. Hence, in low-dimensional regimes, the advantage of incorporating the second database Y𝑌Yitalic_Y is constrained by the stricter requirement on ‖𝛍‖2superscriptnorm𝛍2\|\boldsymbol{\mu}\|^{2}∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT for alignment.
We now provide the converse result, stating when exact community recovery is not possible in correlated GMMs.
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Let (X,Y)∼CGMMs(n,𝛍,d,ρ)similar-to𝑋𝑌CGMMs𝑛𝛍𝑑𝜌(X,Y)\sim\textnormal{CGMMs}(n,\boldsymbol{\mu},d,\rho)( italic_X , italic_Y ) ∼ CGMMs ( italic_n , bold_italic_μ , italic_d , italic_ρ ) as in Section I-A1. Suppose
, 1 = ∥𝝁∥2≤(1−ϵ)1+ρ2(1+1+2dnlogn)lognsuperscriptdelimited-∥∥𝝁21italic-ϵ1𝜌2112𝑑𝑛𝑛𝑛\lVert\boldsymbol{\mu}\rVert^{2}\leq(1-\epsilon)\frac{1+\rho}{2}\left(1+\sqrt{%
1+\frac{2d}{n\log n}}\right)\log n∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≤ ( 1 - italic_ϵ ) divide start_ARG 1 + italic_ρ end_ARG start_ARG 2 end_ARG ( 1 + square-root start_ARG 1 + divide start_ARG 2 italic_d end_ARG start_ARG italic_n roman_log italic_n end_ARG end_ARG ) roman_log italic_n. , 2 = . , 3 = (20)
for an arbitrarily small constant ϵ>0italic-ϵ0\epsilon>0italic_ϵ > 0. Then, there is no estimator that achieves exact community recovery with high probability.
The proof of Theorem 4 follows the same reasoning as in Theorem 8 for the special case p,q=0𝑝𝑞0p,q=0italic_p , italic_q = 0, thus it can be viewed as a corollary of that result.
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II Correlated Gaussian Mixture Models
II-B Exact Community Recovery in Correlated Gaussian Mixture Models
Theorem 4 (Impossibility for Exact Community Recovery).
Let (X,Y)∼CGMMs(n,𝛍,d,ρ)similar-to𝑋𝑌CGMMs𝑛𝛍𝑑𝜌(X,Y)\sim\textnormal{CGMMs}(n,\boldsymbol{\mu},d,\rho)( italic_X , italic_Y ) ∼ CGMMs ( italic_n , bold_italic_μ , italic_d , italic_ρ ) as in Section I-A1. Suppose
, 1 = ∥𝝁∥2≤(1−ϵ)1+ρ2(1+1+2dnlogn)lognsuperscriptdelimited-∥∥𝝁21italic-ϵ1𝜌2112𝑑𝑛𝑛𝑛\lVert\boldsymbol{\mu}\rVert^{2}\leq(1-\epsilon)\frac{1+\rho}{2}\left(1+\sqrt{%
1+\frac{2d}{n\log n}}\right)\log n∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≤ ( 1 - italic_ϵ ) divide start_ARG 1 + italic_ρ end_ARG start_ARG 2 end_ARG ( 1 + square-root start_ARG 1 + divide start_ARG 2 italic_d end_ARG start_ARG italic_n roman_log italic_n end_ARG end_ARG ) roman_log italic_n. , 2 = . , 3 = (20)
for an arbitrarily small constant ϵ>0italic-ϵ0\epsilon>0italic_ϵ > 0. Then, there is no estimator that achieves exact community recovery with high probability.
The proof of Theorem 4 follows the same reasoning as in Theorem 8 for the special case p,q=0𝑝𝑞0p,q=0italic_p , italic_q = 0, thus it can be viewed as a corollary of that result.
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Theorem 3 assumes that either (7) or (8) is satisfied to ensure exact matching. This raises a natural question: if matching is not feasible (i.e., d4log11−ρ2<logn𝑑411superscript𝜌2𝑛\tfrac{d}{4}\log\tfrac{1}{1-\rho^{2}}<\log ndivide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG < roman_log italic_n), must exact community recovery also remain impossible, even if (17) holds? Addressing this question when exact matching is precluded is an interesting open problem, discussed further in Section V.
Figure 1 illustrates the threshold conditions for both exact matching and exact community recovery in correlated Gaussian Mixture Models. It visually highlights how the addition of correlated node attributes expands the parameter regime where community detection becomes feasible.
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II Correlated Gaussian Mixture Models
II-B Exact Community Recovery in Correlated Gaussian Mixture Models
Remark 4 (Information-Theoretic Gaps in Exact Community Recovery).
Theorem 3 assumes that either (7) or (8) is satisfied to ensure exact matching. This raises a natural question: if matching is not feasible (i.e., d4log11−ρ2<logn𝑑411superscript𝜌2𝑛\tfrac{d}{4}\log\tfrac{1}{1-\rho^{2}}<\log ndivide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG < roman_log italic_n), must exact community recovery also remain impossible, even if (17) holds? Addressing this question when exact matching is precluded is an interesting open problem, discussed further in Section V.
Figure 1 illustrates the threshold conditions for both exact matching and exact community recovery in correlated Gaussian Mixture Models. It visually highlights how the addition of correlated node attributes expands the parameter regime where community detection becomes feasible.
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In this section, we consider the correlated Contextual Stochastic Block Models (CSBMs) introduced in Section I-A2. Similar to Section II, we first establish the conditions for exact matching and then derive the conditions for exact community recovery, assuming we can perfectly match the nodes between the two correlated graphs.
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III Correlated Contextual Stochastic Block Models
In this section, we consider the correlated Contextual Stochastic Block Models (CSBMs) introduced in Section I-A2. Similar to Section II, we first establish the conditions for exact matching and then derive the conditions for exact community recovery, assuming we can perfectly match the nodes between the two correlated graphs.
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The following theorem provides sufficient conditions for exact matching in correlated CSBMs.
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
III Correlated Contextual Stochastic Block Models
III-A Exact Matching
The following theorem provides sufficient conditions for exact matching in correlated CSBMs.
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Let (G1,G2)∼CCSBMs(n,p,q,s;R,d,ρ)similar-tosubscript𝐺1subscript𝐺2CCSBMs𝑛𝑝𝑞𝑠𝑅𝑑𝜌(G_{1},G_{2})\sim\textnormal{CCSBMs}(n,p,q,s;R,d,\rho)( italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ∼ CCSBMs ( italic_n , italic_p , italic_q , italic_s ; italic_R , italic_d , italic_ρ ) be as defined in Section I-A2, and suppose
, 1 = p≤O(1e(loglogn)3),𝑝𝑂1superscript𝑒superscript𝑛3p\leq O\left(\frac{1}{e^{(\log\log n)^{3}}}\right),italic_p ≤ italic_O ( divide start_ARG 1 end_ARG start_ARG italic_e start_POSTSUPERSCRIPT ( roman_log roman_log italic_n ) start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT end_ARG ) ,. , 2 = . , 3 = (21)
as well as
, 1 = R≥2logn+ω(1) or d=ω(logn)𝑅2𝑛𝜔1 or 𝑑𝜔𝑛R\geq 2\log n+\omega(1)\text{ or }d=\omega(\log n)italic_R ≥ 2 roman_log italic_n + italic_ω ( 1 ) or italic_d = italic_ω ( roman_log italic_n ). , 2 = . , 3 = (22)
holds. If
, 1 = ns2p+q2+d4log11−ρ2≥(1+ϵ)logn𝑛superscript𝑠2𝑝𝑞2𝑑411superscript𝜌21italic-ϵ𝑛ns^{2}\frac{p+q}{2}+\frac{d}{4}\log\frac{1}{1-\rho^{2}}\geq(1+\epsilon)\log nitalic_n italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divide start_ARG italic_p + italic_q end_ARG start_ARG 2 end_ARG + divide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≥ ( 1 + italic_ϵ ) roman_log italic_n. , 2 = . , 3 = (23)
for an arbitrarily small constant ϵ>0italic-ϵ0\epsilon>0italic_ϵ > 0, then there exists an estimator π^(G1,G2)^𝜋subscript𝐺1subscript𝐺2\hat{\pi}(G_{1},G_{2})over^ start_ARG italic_π end_ARG ( italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) such that π^(G1,G2)=π∗^𝜋subscript𝐺1subscript𝐺2subscript𝜋\hat{\pi}(G_{1},G_{2})=\pi_{*}over^ start_ARG italic_π end_ARG ( italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT with high probability.
In [6, 35, 7], it was shown that for correlated SBMs, exact matching is possible if the average degree of the intersection graph ns2p+q2𝑛superscript𝑠2𝑝𝑞2ns^{2}\tfrac{p+q}{2}italic_n italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divide start_ARG italic_p + italic_q end_ARG start_ARG 2 end_ARG exceeds logn𝑛\log nroman_log italic_n. Hence, we focus on the regime ns2p+q2=O(logn)𝑛superscript𝑠2𝑝𝑞2𝑂𝑛ns^{2}\tfrac{p+q}{2}=O(\log n)italic_n italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divide start_ARG italic_p + italic_q end_ARG start_ARG 2 end_ARG = italic_O ( roman_log italic_n ), where we employ a two-step approach. First, we use k𝑘kitalic_k-core matching to align a large fraction of the nodes via edge information alone; the k𝑘kitalic_k-core approach has been extensively studied in [11, 8, 35] and is adapted here to handle more general regimes of p𝑝pitalic_p and q𝑞qitalic_q. By choosing an appropriate k𝑘kitalic_k, we obtain a partial matching that is both large (in terms of the number of matched nodes) and accurate (no mismatches). The remaining unmatched node pairs follow the correlated Gaussian Mixture Model framework, so we apply Theorem 1 to match those residual pairs via the node-attribute estimator.
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III Correlated Contextual Stochastic Block Models
III-A Exact Matching
Theorem 5 (Achievability for Exact Matching).
Let (G1,G2)∼CCSBMs(n,p,q,s;R,d,ρ)similar-tosubscript𝐺1subscript𝐺2CCSBMs𝑛𝑝𝑞𝑠𝑅𝑑𝜌(G_{1},G_{2})\sim\textnormal{CCSBMs}(n,p,q,s;R,d,\rho)( italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ∼ CCSBMs ( italic_n , italic_p , italic_q , italic_s ; italic_R , italic_d , italic_ρ ) be as defined in Section I-A2, and suppose
, 1 = p≤O(1e(loglogn)3),𝑝𝑂1superscript𝑒superscript𝑛3p\leq O\left(\frac{1}{e^{(\log\log n)^{3}}}\right),italic_p ≤ italic_O ( divide start_ARG 1 end_ARG start_ARG italic_e start_POSTSUPERSCRIPT ( roman_log roman_log italic_n ) start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT end_ARG ) ,. , 2 = . , 3 = (21)
as well as
, 1 = R≥2logn+ω(1) or d=ω(logn)𝑅2𝑛𝜔1 or 𝑑𝜔𝑛R\geq 2\log n+\omega(1)\text{ or }d=\omega(\log n)italic_R ≥ 2 roman_log italic_n + italic_ω ( 1 ) or italic_d = italic_ω ( roman_log italic_n ). , 2 = . , 3 = (22)
holds. If
, 1 = ns2p+q2+d4log11−ρ2≥(1+ϵ)logn𝑛superscript𝑠2𝑝𝑞2𝑑411superscript𝜌21italic-ϵ𝑛ns^{2}\frac{p+q}{2}+\frac{d}{4}\log\frac{1}{1-\rho^{2}}\geq(1+\epsilon)\log nitalic_n italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divide start_ARG italic_p + italic_q end_ARG start_ARG 2 end_ARG + divide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≥ ( 1 + italic_ϵ ) roman_log italic_n. , 2 = . , 3 = (23)
for an arbitrarily small constant ϵ>0italic-ϵ0\epsilon>0italic_ϵ > 0, then there exists an estimator π^(G1,G2)^𝜋subscript𝐺1subscript𝐺2\hat{\pi}(G_{1},G_{2})over^ start_ARG italic_π end_ARG ( italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) such that π^(G1,G2)=π∗^𝜋subscript𝐺1subscript𝐺2subscript𝜋\hat{\pi}(G_{1},G_{2})=\pi_{*}over^ start_ARG italic_π end_ARG ( italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT with high probability.
In [6, 35, 7], it was shown that for correlated SBMs, exact matching is possible if the average degree of the intersection graph ns2p+q2𝑛superscript𝑠2𝑝𝑞2ns^{2}\tfrac{p+q}{2}italic_n italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divide start_ARG italic_p + italic_q end_ARG start_ARG 2 end_ARG exceeds logn𝑛\log nroman_log italic_n. Hence, we focus on the regime ns2p+q2=O(logn)𝑛superscript𝑠2𝑝𝑞2𝑂𝑛ns^{2}\tfrac{p+q}{2}=O(\log n)italic_n italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divide start_ARG italic_p + italic_q end_ARG start_ARG 2 end_ARG = italic_O ( roman_log italic_n ), where we employ a two-step approach. First, we use k𝑘kitalic_k-core matching to align a large fraction of the nodes via edge information alone; the k𝑘kitalic_k-core approach has been extensively studied in [11, 8, 35] and is adapted here to handle more general regimes of p𝑝pitalic_p and q𝑞qitalic_q. By choosing an appropriate k𝑘kitalic_k, we obtain a partial matching that is both large (in terms of the number of matched nodes) and accurate (no mismatches). The remaining unmatched node pairs follow the correlated Gaussian Mixture Model framework, so we apply Theorem 1 to match those residual pairs via the node-attribute estimator.
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Condition (21) ensures that the k𝑘kitalic_k-core algorithm yields the correct matching for nodes within the k𝑘kitalic_k-core of G1∧π∗G2subscriptsubscript𝜋subscript𝐺1subscript𝐺2G_{1}\wedge_{\pi_{*}}G_{2}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∧ start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. Condition (22) is required for matching the remaining nodes using node attributes. The combined inequality (23) highlights the benefit of leveraging both edge and node-attribute correlations. As a baseline, if d=0𝑑0d=0italic_d = 0 (no node attributes), the model reduces to correlated SBMs, and (23) becomes ns2p+q2≥(1+ϵ)logn𝑛superscript𝑠2𝑝𝑞21italic-ϵ𝑛ns^{2}\tfrac{p+q}{2}\geq(1+\epsilon)\,\log nitalic_n italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divide start_ARG italic_p + italic_q end_ARG start_ARG 2 end_ARG ≥ ( 1 + italic_ϵ ) roman_log italic_n, matching the exact-matching threshold in [6, 35, 7]. Conversely, if p=q=0𝑝𝑞0p=q=0italic_p = italic_q = 0 (no edges), the model is just correlated GMMs, and (23) becomes d4log(11−ρ2)≥(1+ϵ)logn𝑑411superscript𝜌21italic-ϵ𝑛\frac{d}{4}\log\!\bigl{(}\tfrac{1}{1-\rho^{2}}\bigr{)}\geq(1+\epsilon)\log ndivide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log ( divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) ≥ ( 1 + italic_ϵ ) roman_log italic_n, consistent with Theorem 1.
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III Correlated Contextual Stochastic Block Models
III-A Exact Matching
Remark 5 (Interpretation of Conditions for Exact Matching).
Condition (21) ensures that the k𝑘kitalic_k-core algorithm yields the correct matching for nodes within the k𝑘kitalic_k-core of G1∧π∗G2subscriptsubscript𝜋subscript𝐺1subscript𝐺2G_{1}\wedge_{\pi_{*}}G_{2}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∧ start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. Condition (22) is required for matching the remaining nodes using node attributes. The combined inequality (23) highlights the benefit of leveraging both edge and node-attribute correlations. As a baseline, if d=0𝑑0d=0italic_d = 0 (no node attributes), the model reduces to correlated SBMs, and (23) becomes ns2p+q2≥(1+ϵ)logn𝑛superscript𝑠2𝑝𝑞21italic-ϵ𝑛ns^{2}\tfrac{p+q}{2}\geq(1+\epsilon)\,\log nitalic_n italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divide start_ARG italic_p + italic_q end_ARG start_ARG 2 end_ARG ≥ ( 1 + italic_ϵ ) roman_log italic_n, matching the exact-matching threshold in [6, 35, 7]. Conversely, if p=q=0𝑝𝑞0p=q=0italic_p = italic_q = 0 (no edges), the model is just correlated GMMs, and (23) becomes d4log(11−ρ2)≥(1+ϵ)logn𝑑411superscript𝜌21italic-ϵ𝑛\frac{d}{4}\log\!\bigl{(}\tfrac{1}{1-\rho^{2}}\bigr{)}\geq(1+\epsilon)\log ndivide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log ( divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) ≥ ( 1 + italic_ϵ ) roman_log italic_n, consistent with Theorem 1.
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Let (G1,G2)∼CCSBMs(n,p,q,s;R,d,ρ)similar-tosubscript𝐺1subscript𝐺2CCSBMs𝑛𝑝𝑞𝑠𝑅𝑑𝜌(G_{1},G_{2})\sim\textnormal{CCSBMs}(n,p,q,s;R,d,\rho)( italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ∼ CCSBMs ( italic_n , italic_p , italic_q , italic_s ; italic_R , italic_d , italic_ρ ) be as in Section I-A2, and assume ps2=o(1)𝑝superscript𝑠2𝑜1ps^{2}=o(1)italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = italic_o ( 1 ). Suppose either
, 1 = ns2p+q2+d4log11−ρ2≤(1−ϵ)logn and 1≪d=O(logn)𝑛superscript𝑠2𝑝𝑞2𝑑411superscript𝜌21italic-ϵ𝑛 and 1much-less-than𝑑𝑂𝑛ns^{2}\frac{p+q}{2}+\frac{d}{4}\log\frac{1}{1-\rho^{2}}\leq(1-\epsilon)\log n%
\text{ and }1\ll d=O(\log n)italic_n italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divide start_ARG italic_p + italic_q end_ARG start_ARG 2 end_ARG + divide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≤ ( 1 - italic_ϵ ) roman_log italic_n and 1 ≪ italic_d = italic_O ( roman_log italic_n ). , 2 = . , 3 = (24)
for arbitrarily small ϵ>0italic-ϵ0\epsilon>0italic_ϵ > 0, or
, 1 = ns2p+q2+d4log11−ρ2≤logn−logd−ω(1) and 1ρ2−1≤d40.𝑛superscript𝑠2𝑝𝑞2𝑑411superscript𝜌2𝑛𝑑𝜔1 and 1superscript𝜌21𝑑40ns^{2}\frac{p+q}{2}+\frac{d}{4}\log\frac{1}{1-\rho^{2}}\leq\log n-\log d-%
\omega(1)\text{ and }\frac{1}{\rho^{2}}-1\leq\frac{d}{40}.italic_n italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divide start_ARG italic_p + italic_q end_ARG start_ARG 2 end_ARG + divide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≤ roman_log italic_n - roman_log italic_d - italic_ω ( 1 ) and divide start_ARG 1 end_ARG start_ARG italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG - 1 ≤ divide start_ARG italic_d end_ARG start_ARG 40 end_ARG .. , 2 = . , 3 = (25)
Then no estimator can achieve exact matching with high probability.
To prove Theorem 6, we show that even a MAP estimator fails when we provide G1subscript𝐺1G_{1}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, G2subscript𝐺2G_{2}italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, the community labels 𝝈1superscript𝝈1{\boldsymbol{\sigma}}^{1}bold_italic_σ start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT and 𝝈2superscript𝝈2{\boldsymbol{\sigma}}^{2}bold_italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT for each graph, the mean vector 𝝁𝝁\boldsymbol{\mu}bold_italic_μ, and partial knowledge of π∗subscript𝜋\pi_{*}italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT on all but the set
, 1 = ℋ∗:={i∈[n]:∀j∈[n],AijBπ∗(i)π∗(j)=0}assignsubscriptℋconditional-set𝑖delimited-[]𝑛formulae-sequencefor-all𝑗delimited-[]𝑛subscript𝐴𝑖𝑗subscript𝐵subscript𝜋𝑖subscript𝜋𝑗0\mathcal{H}_{*}:=\left\{i\in[n]:\forall j\in[n],A_{ij}B_{\pi_{*}(i)\pi_{*}(j)}%
=0\right\}caligraphic_H start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT := { italic_i ∈ [ italic_n ] : ∀ italic_j ∈ [ italic_n ] , italic_A start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_i ) italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_j ) end_POSTSUBSCRIPT = 0 }. , 2 = . , 3 = (26)
which consists of isolated nodes in the intersection graph A∧π∗Bsubscriptsubscript𝜋𝐴𝐵A\wedge_{\pi_{*}}Bitalic_A ∧ start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_B. Section X contains the detailed argument.
The figure 2 illustrates the parameter regions where exact node matching is (blue area) and is not (gray area) achievable in correlated Contextual Stochastic Block Models. It also highlights how removing either edges (p,q=0𝑝𝑞0p,q=0italic_p , italic_q = 0) or node attributes (d=0𝑑0d=0italic_d = 0) reduces the region of feasible matching to the thresholds for correlated GMMs and correlated SBMs, respectively.
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
III Correlated Contextual Stochastic Block Models
III-A Exact Matching
Theorem 6 (Impossibility for Exact Matching).
Let (G1,G2)∼CCSBMs(n,p,q,s;R,d,ρ)similar-tosubscript𝐺1subscript𝐺2CCSBMs𝑛𝑝𝑞𝑠𝑅𝑑𝜌(G_{1},G_{2})\sim\textnormal{CCSBMs}(n,p,q,s;R,d,\rho)( italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ∼ CCSBMs ( italic_n , italic_p , italic_q , italic_s ; italic_R , italic_d , italic_ρ ) be as in Section I-A2, and assume ps2=o(1)𝑝superscript𝑠2𝑜1ps^{2}=o(1)italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = italic_o ( 1 ). Suppose either
, 1 = ns2p+q2+d4log11−ρ2≤(1−ϵ)logn and 1≪d=O(logn)𝑛superscript𝑠2𝑝𝑞2𝑑411superscript𝜌21italic-ϵ𝑛 and 1much-less-than𝑑𝑂𝑛ns^{2}\frac{p+q}{2}+\frac{d}{4}\log\frac{1}{1-\rho^{2}}\leq(1-\epsilon)\log n%
\text{ and }1\ll d=O(\log n)italic_n italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divide start_ARG italic_p + italic_q end_ARG start_ARG 2 end_ARG + divide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≤ ( 1 - italic_ϵ ) roman_log italic_n and 1 ≪ italic_d = italic_O ( roman_log italic_n ). , 2 = . , 3 = (24)
for arbitrarily small ϵ>0italic-ϵ0\epsilon>0italic_ϵ > 0, or
, 1 = ns2p+q2+d4log11−ρ2≤logn−logd−ω(1) and 1ρ2−1≤d40.𝑛superscript𝑠2𝑝𝑞2𝑑411superscript𝜌2𝑛𝑑𝜔1 and 1superscript𝜌21𝑑40ns^{2}\frac{p+q}{2}+\frac{d}{4}\log\frac{1}{1-\rho^{2}}\leq\log n-\log d-%
\omega(1)\text{ and }\frac{1}{\rho^{2}}-1\leq\frac{d}{40}.italic_n italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divide start_ARG italic_p + italic_q end_ARG start_ARG 2 end_ARG + divide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≤ roman_log italic_n - roman_log italic_d - italic_ω ( 1 ) and divide start_ARG 1 end_ARG start_ARG italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG - 1 ≤ divide start_ARG italic_d end_ARG start_ARG 40 end_ARG .. , 2 = . , 3 = (25)
Then no estimator can achieve exact matching with high probability.
To prove Theorem 6, we show that even a MAP estimator fails when we provide G1subscript𝐺1G_{1}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, G2subscript𝐺2G_{2}italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, the community labels 𝝈1superscript𝝈1{\boldsymbol{\sigma}}^{1}bold_italic_σ start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT and 𝝈2superscript𝝈2{\boldsymbol{\sigma}}^{2}bold_italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT for each graph, the mean vector 𝝁𝝁\boldsymbol{\mu}bold_italic_μ, and partial knowledge of π∗subscript𝜋\pi_{*}italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT on all but the set
, 1 = ℋ∗:={i∈[n]:∀j∈[n],AijBπ∗(i)π∗(j)=0}assignsubscriptℋconditional-set𝑖delimited-[]𝑛formulae-sequencefor-all𝑗delimited-[]𝑛subscript𝐴𝑖𝑗subscript𝐵subscript𝜋𝑖subscript𝜋𝑗0\mathcal{H}_{*}:=\left\{i\in[n]:\forall j\in[n],A_{ij}B_{\pi_{*}(i)\pi_{*}(j)}%
=0\right\}caligraphic_H start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT := { italic_i ∈ [ italic_n ] : ∀ italic_j ∈ [ italic_n ] , italic_A start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_i ) italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_j ) end_POSTSUBSCRIPT = 0 }. , 2 = . , 3 = (26)
which consists of isolated nodes in the intersection graph A∧π∗Bsubscriptsubscript𝜋𝐴𝐵A\wedge_{\pi_{*}}Bitalic_A ∧ start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_B. Section X contains the detailed argument.
The figure 2 illustrates the parameter regions where exact node matching is (blue area) and is not (gray area) achievable in correlated Contextual Stochastic Block Models. It also highlights how removing either edges (p,q=0𝑝𝑞0p,q=0italic_p , italic_q = 0) or node attributes (d=0𝑑0d=0italic_d = 0) reduces the region of feasible matching to the thresholds for correlated GMMs and correlated SBMs, respectively.
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In [4], it was shown that if p,q=ω(lognn)𝑝𝑞𝜔𝑛𝑛p,q=\omega\bigl{(}\tfrac{\log n}{n}\bigr{)}italic_p , italic_q = italic_ω ( divide start_ARG roman_log italic_n end_ARG start_ARG italic_n end_ARG ) (p>q𝑝𝑞p>qitalic_p > italic_q), then exact community recovery in a single CSBM graph G1subscript𝐺1G_{1}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is achievable using only edge information. Likewise, it was shown in [14, 5] that if the effective SNR
R2R+d/n=ω(logn)superscript𝑅2𝑅𝑑𝑛𝜔𝑛\frac{R^{2}}{R+d/n}=\omega(\log n)divide start_ARG italic_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_R + italic_d / italic_n end_ARG = italic_ω ( roman_log italic_n ) where
‖𝝁‖2=R,superscriptnorm𝝁2𝑅\|\boldsymbol{\mu}\|^{2}=R,∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = italic_R ,
then exact community recovery is possible using only node attributes in G1subscript𝐺1G_{1}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT. Accordingly, for correlated CSBMs, we focus on the setting
, 1 = p=alognn,q=blognn,R2R+d/n=clogn,(21+ρR)221+ρR+d/n=c′lognformulae-sequence𝑝𝑎𝑛𝑛formulae-sequence𝑞𝑏𝑛𝑛formulae-sequencesuperscript𝑅2𝑅𝑑𝑛𝑐𝑛superscript21𝜌𝑅221𝜌𝑅𝑑𝑛superscript𝑐′𝑛p=\frac{a\log n}{n},\quad q=\frac{b\log n}{n},\quad\frac{R^{2}}{R+d/n}=c\log n%
,\quad\frac{\left(\frac{2}{1+\rho}R\right)^{2}}{\frac{2}{1+\rho}R+d/n}=c^{%
\prime}\log nitalic_p = divide start_ARG italic_a roman_log italic_n end_ARG start_ARG italic_n end_ARG , italic_q = divide start_ARG italic_b roman_log italic_n end_ARG start_ARG italic_n end_ARG , divide start_ARG italic_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_R + italic_d / italic_n end_ARG = italic_c roman_log italic_n , divide start_ARG ( divide start_ARG 2 end_ARG start_ARG 1 + italic_ρ end_ARG italic_R ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG divide start_ARG 2 end_ARG start_ARG 1 + italic_ρ end_ARG italic_R + italic_d / italic_n end_ARG = italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT roman_log italic_n. , 2 = . , 3 = (27)
for positive constants a,b,c,c′𝑎𝑏𝑐superscript𝑐′a,b,c,c^{\prime}italic_a , italic_b , italic_c , italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT. The last term in (27) represents the SNR for the averaged correlated Gaussian attributes in (18). Under these assumptions, we have the following feasibility and infeasibility results for exact community recovery.
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
III Correlated Contextual Stochastic Block Models
III-B Exact Community Recovery
In [4], it was shown that if p,q=ω(lognn)𝑝𝑞𝜔𝑛𝑛p,q=\omega\bigl{(}\tfrac{\log n}{n}\bigr{)}italic_p , italic_q = italic_ω ( divide start_ARG roman_log italic_n end_ARG start_ARG italic_n end_ARG ) (p>q𝑝𝑞p>qitalic_p > italic_q), then exact community recovery in a single CSBM graph G1subscript𝐺1G_{1}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is achievable using only edge information. Likewise, it was shown in [14, 5] that if the effective SNR
R2R+d/n=ω(logn)superscript𝑅2𝑅𝑑𝑛𝜔𝑛\frac{R^{2}}{R+d/n}=\omega(\log n)divide start_ARG italic_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_R + italic_d / italic_n end_ARG = italic_ω ( roman_log italic_n ) where
‖𝝁‖2=R,superscriptnorm𝝁2𝑅\|\boldsymbol{\mu}\|^{2}=R,∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = italic_R ,
then exact community recovery is possible using only node attributes in G1subscript𝐺1G_{1}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT. Accordingly, for correlated CSBMs, we focus on the setting
, 1 = p=alognn,q=blognn,R2R+d/n=clogn,(21+ρR)221+ρR+d/n=c′lognformulae-sequence𝑝𝑎𝑛𝑛formulae-sequence𝑞𝑏𝑛𝑛formulae-sequencesuperscript𝑅2𝑅𝑑𝑛𝑐𝑛superscript21𝜌𝑅221𝜌𝑅𝑑𝑛superscript𝑐′𝑛p=\frac{a\log n}{n},\quad q=\frac{b\log n}{n},\quad\frac{R^{2}}{R+d/n}=c\log n%
,\quad\frac{\left(\frac{2}{1+\rho}R\right)^{2}}{\frac{2}{1+\rho}R+d/n}=c^{%
\prime}\log nitalic_p = divide start_ARG italic_a roman_log italic_n end_ARG start_ARG italic_n end_ARG , italic_q = divide start_ARG italic_b roman_log italic_n end_ARG start_ARG italic_n end_ARG , divide start_ARG italic_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_R + italic_d / italic_n end_ARG = italic_c roman_log italic_n , divide start_ARG ( divide start_ARG 2 end_ARG start_ARG 1 + italic_ρ end_ARG italic_R ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG divide start_ARG 2 end_ARG start_ARG 1 + italic_ρ end_ARG italic_R + italic_d / italic_n end_ARG = italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT roman_log italic_n. , 2 = . , 3 = (27)
for positive constants a,b,c,c′𝑎𝑏𝑐superscript𝑐′a,b,c,c^{\prime}italic_a , italic_b , italic_c , italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT. The last term in (27) represents the SNR for the averaged correlated Gaussian attributes in (18). Under these assumptions, we have the following feasibility and infeasibility results for exact community recovery.
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Let (G1,G2)∼CCSBMs(n,p,q,s;R,d,ρ)similar-tosubscript𝐺1subscript𝐺2CCSBMs𝑛𝑝𝑞𝑠𝑅𝑑𝜌(G_{1},G_{2})\sim\textnormal{CCSBMs}\bigl{(}n,p,q,s;R,d,\rho\bigr{)}( italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ∼ CCSBMs ( italic_n , italic_p , italic_q , italic_s ; italic_R , italic_d , italic_ρ ) be as in Section I-A2. Suppose conditions (21), (22), and (23) hold, and also assume (27). If
, 1 = (1−(1−s)2)(a−b)2+c′ 2> 1,1superscript1𝑠2superscript𝑎𝑏2superscript𝑐′21\frac{\bigl{(}1-(1-s)^{2}\bigr{)}\,\bigl{(}\sqrt{a}-\sqrt{b}\bigr{)}^{2}+c^{%
\prime}}{\,2\,}\;>\;1,divide start_ARG ( 1 - ( 1 - italic_s ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ( square-root start_ARG italic_a end_ARG - square-root start_ARG italic_b end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG start_ARG 2 end_ARG > 1 ,. , 2 = . , 3 = (28)
then there exists an estimator 𝛔^(G1,G2)^𝛔subscript𝐺1subscript𝐺2\hat{{\boldsymbol{\sigma}}}(G_{1},G_{2})over^ start_ARG bold_italic_σ end_ARG ( italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) such that 𝛔^(G1,G2)=𝛔^𝛔subscript𝐺1subscript𝐺2𝛔\hat{{\boldsymbol{\sigma}}}(G_{1},G_{2})={\boldsymbol{\sigma}}over^ start_ARG bold_italic_σ end_ARG ( italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = bold_italic_σ with high probability.
When exact matching is achievable, we can construct a new Contextual Stochastic Block Model by forming a denser union graph G1∨π∗G2subscriptsubscript𝜋subscript𝐺1subscript𝐺2G_{1}\vee_{\pi_{*}}G_{2}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∨ start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT whose edges follow
SBM(n,p(1−(1−s)2),q(1−(1−s)2))SBM𝑛𝑝1superscript1𝑠2𝑞1superscript1𝑠2\textnormal{SBM}\bigl{(}n,p(1-(1-s)^{2}),q(1-(1-s)^{2})\bigr{)}SBM ( italic_n , italic_p ( 1 - ( 1 - italic_s ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) , italic_q ( 1 - ( 1 - italic_s ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ), and by assigning to each node the averaged correlated attribute from (18). Applying the techniques of [5] to this merged structure establishes Theorem 7.
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
III Correlated Contextual Stochastic Block Models
III-B Exact Community Recovery
Theorem 7 (Achievability for Exact Community Recovery).
Let (G1,G2)∼CCSBMs(n,p,q,s;R,d,ρ)similar-tosubscript𝐺1subscript𝐺2CCSBMs𝑛𝑝𝑞𝑠𝑅𝑑𝜌(G_{1},G_{2})\sim\textnormal{CCSBMs}\bigl{(}n,p,q,s;R,d,\rho\bigr{)}( italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ∼ CCSBMs ( italic_n , italic_p , italic_q , italic_s ; italic_R , italic_d , italic_ρ ) be as in Section I-A2. Suppose conditions (21), (22), and (23) hold, and also assume (27). If
, 1 = (1−(1−s)2)(a−b)2+c′ 2> 1,1superscript1𝑠2superscript𝑎𝑏2superscript𝑐′21\frac{\bigl{(}1-(1-s)^{2}\bigr{)}\,\bigl{(}\sqrt{a}-\sqrt{b}\bigr{)}^{2}+c^{%
\prime}}{\,2\,}\;>\;1,divide start_ARG ( 1 - ( 1 - italic_s ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ( square-root start_ARG italic_a end_ARG - square-root start_ARG italic_b end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG start_ARG 2 end_ARG > 1 ,. , 2 = . , 3 = (28)
then there exists an estimator 𝛔^(G1,G2)^𝛔subscript𝐺1subscript𝐺2\hat{{\boldsymbol{\sigma}}}(G_{1},G_{2})over^ start_ARG bold_italic_σ end_ARG ( italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) such that 𝛔^(G1,G2)=𝛔^𝛔subscript𝐺1subscript𝐺2𝛔\hat{{\boldsymbol{\sigma}}}(G_{1},G_{2})={\boldsymbol{\sigma}}over^ start_ARG bold_italic_σ end_ARG ( italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = bold_italic_σ with high probability.
When exact matching is achievable, we can construct a new Contextual Stochastic Block Model by forming a denser union graph G1∨π∗G2subscriptsubscript𝜋subscript𝐺1subscript𝐺2G_{1}\vee_{\pi_{*}}G_{2}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∨ start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT whose edges follow
SBM(n,p(1−(1−s)2),q(1−(1−s)2))SBM𝑛𝑝1superscript1𝑠2𝑞1superscript1𝑠2\textnormal{SBM}\bigl{(}n,p(1-(1-s)^{2}),q(1-(1-s)^{2})\bigr{)}SBM ( italic_n , italic_p ( 1 - ( 1 - italic_s ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) , italic_q ( 1 - ( 1 - italic_s ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ), and by assigning to each node the averaged correlated attribute from (18). Applying the techniques of [5] to this merged structure establishes Theorem 7.
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Abbe et al. [5] showed that exact community recovery in a single CSBM G1subscript𝐺1G_{1}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is possible if
, 1 = s(a−b)2+c2>1.𝑠superscript𝑎𝑏2𝑐21\frac{s\bigl{(}\sqrt{a}-\sqrt{b}\bigr{)}^{2}+c}{2}>1.divide start_ARG italic_s ( square-root start_ARG italic_a end_ARG - square-root start_ARG italic_b end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_c end_ARG start_ARG 2 end_ARG > 1 .. , 2 =
When two correlated graphs G1subscript𝐺1G_{1}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and G2subscript𝐺2G_{2}italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT are available, an exact matching (if feasible) allows one to form a denser union graph from the correlated edges and to average the correlated attributes. This increases the effective SNR term to
, 1 = (1−(1−s)2)(a−b)2+c′2,1superscript1𝑠2superscript𝑎𝑏2superscript𝑐′2\frac{\bigl{(}1-(1-s)^{2}\bigr{)}\bigl{(}\sqrt{a}-\sqrt{b}\bigr{)}^{2}+c^{%
\prime}}{2},divide start_ARG ( 1 - ( 1 - italic_s ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ( square-root start_ARG italic_a end_ARG - square-root start_ARG italic_b end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG start_ARG 2 end_ARG ,. , 2 =
thereby relaxing the threshold to the condition in (28). A detailed proof of Theorem 7 can be found in Section XI.
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
III Correlated Contextual Stochastic Block Models
III-B Exact Community Recovery
Remark 6 (Comparison with the Single-Graph Setting).
Abbe et al. [5] showed that exact community recovery in a single CSBM G1subscript𝐺1G_{1}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is possible if
, 1 = s(a−b)2+c2>1.𝑠superscript𝑎𝑏2𝑐21\frac{s\bigl{(}\sqrt{a}-\sqrt{b}\bigr{)}^{2}+c}{2}>1.divide start_ARG italic_s ( square-root start_ARG italic_a end_ARG - square-root start_ARG italic_b end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_c end_ARG start_ARG 2 end_ARG > 1 .. , 2 =
When two correlated graphs G1subscript𝐺1G_{1}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and G2subscript𝐺2G_{2}italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT are available, an exact matching (if feasible) allows one to form a denser union graph from the correlated edges and to average the correlated attributes. This increases the effective SNR term to
, 1 = (1−(1−s)2)(a−b)2+c′2,1superscript1𝑠2superscript𝑎𝑏2superscript𝑐′2\frac{\bigl{(}1-(1-s)^{2}\bigr{)}\bigl{(}\sqrt{a}-\sqrt{b}\bigr{)}^{2}+c^{%
\prime}}{2},divide start_ARG ( 1 - ( 1 - italic_s ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ( square-root start_ARG italic_a end_ARG - square-root start_ARG italic_b end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG start_ARG 2 end_ARG ,. , 2 =
thereby relaxing the threshold to the condition in (28). A detailed proof of Theorem 7 can be found in Section XI.
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Let (G1,G2)∼CCSBMs(n,p,q,s;R,d,ρ)similar-tosubscript𝐺1subscript𝐺2CCSBMs𝑛𝑝𝑞𝑠𝑅𝑑𝜌(G_{1},G_{2})\sim\textnormal{CCSBMs}\bigl{(}n,p,q,s;R,d,\rho\bigr{)}( italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ∼ CCSBMs ( italic_n , italic_p , italic_q , italic_s ; italic_R , italic_d , italic_ρ ) be as in Section I-A2, and assume (27). If
, 1 = (1−(1−s)2)(a−b)2+c′2<1,1superscript1𝑠2superscript𝑎𝑏2superscript𝑐′21\frac{\bigl{(}1-(1-s)^{2}\bigr{)}\bigl{(}\sqrt{a}-\sqrt{b}\bigr{)}^{2}+c^{%
\prime}}{2}<1,divide start_ARG ( 1 - ( 1 - italic_s ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ( square-root start_ARG italic_a end_ARG - square-root start_ARG italic_b end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG start_ARG 2 end_ARG < 1 ,. , 2 = . , 3 = (29)
then no estimator can achieve exact community recovery with high probability.
The proof of Theorem 8 resembles the simulation argument used in Theorem 3.4 of [6], where one constructs a graph HH\mathrm{H}roman_H to mirror (G1,G2)subscript𝐺1subscript𝐺2(G_{1},G_{2})( italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ). By showing that exact community recovery in HH\mathrm{H}roman_H is impossible under (29), it follows that (G1,G2)subscript𝐺1subscript𝐺2(G_{1},G_{2})( italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) also fails to admit exact community recovery. A detailed proof is given in Section XII.
Figure 3 visualizes the parameter ranges under which community recovery is possible or impossible in correlated CSBMs. It highlights how adding correlated node attributes and edges can expand the regime where exact community detection succeeds.
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
III Correlated Contextual Stochastic Block Models
III-B Exact Community Recovery
Theorem 8 (Impossibility for Exact Community Recovery).
Let (G1,G2)∼CCSBMs(n,p,q,s;R,d,ρ)similar-tosubscript𝐺1subscript𝐺2CCSBMs𝑛𝑝𝑞𝑠𝑅𝑑𝜌(G_{1},G_{2})\sim\textnormal{CCSBMs}\bigl{(}n,p,q,s;R,d,\rho\bigr{)}( italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ∼ CCSBMs ( italic_n , italic_p , italic_q , italic_s ; italic_R , italic_d , italic_ρ ) be as in Section I-A2, and assume (27). If
, 1 = (1−(1−s)2)(a−b)2+c′2<1,1superscript1𝑠2superscript𝑎𝑏2superscript𝑐′21\frac{\bigl{(}1-(1-s)^{2}\bigr{)}\bigl{(}\sqrt{a}-\sqrt{b}\bigr{)}^{2}+c^{%
\prime}}{2}<1,divide start_ARG ( 1 - ( 1 - italic_s ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ( square-root start_ARG italic_a end_ARG - square-root start_ARG italic_b end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG start_ARG 2 end_ARG < 1 ,. , 2 = . , 3 = (29)
then no estimator can achieve exact community recovery with high probability.
The proof of Theorem 8 resembles the simulation argument used in Theorem 3.4 of [6], where one constructs a graph HH\mathrm{H}roman_H to mirror (G1,G2)subscript𝐺1subscript𝐺2(G_{1},G_{2})( italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ). By showing that exact community recovery in HH\mathrm{H}roman_H is impossible under (29), it follows that (G1,G2)subscript𝐺1subscript𝐺2(G_{1},G_{2})( italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) also fails to admit exact community recovery. A detailed proof is given in Section XII.
Figure 3 visualizes the parameter ranges under which community recovery is possible or impossible in correlated CSBMs. It highlights how adding correlated node attributes and edges can expand the regime where exact community detection succeeds.
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Suppose
, 1 = (21+ρR)221+ρR+dn=2logn⟹21+ρR≈2dnlognor 2logn,formulae-sequencesuperscript21𝜌𝑅221𝜌𝑅𝑑𝑛2𝑛⟹21𝜌𝑅2𝑑𝑛𝑛or2𝑛\frac{\left(\frac{2}{1+\rho}R\right)^{2}}{\tfrac{2}{1+\rho}R+\tfrac{d}{n}}=2\,%
\log n\quad\Longrightarrow\quad\frac{2}{1+\rho}\,R\approx\sqrt{\frac{2d}{n}\,%
\log n}\;\;\text{or}\;\;2\log n,divide start_ARG ( divide start_ARG 2 end_ARG start_ARG 1 + italic_ρ end_ARG italic_R ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG divide start_ARG 2 end_ARG start_ARG 1 + italic_ρ end_ARG italic_R + divide start_ARG italic_d end_ARG start_ARG italic_n end_ARG end_ARG = 2 roman_log italic_n ⟹ divide start_ARG 2 end_ARG start_ARG 1 + italic_ρ end_ARG italic_R ≈ square-root start_ARG divide start_ARG 2 italic_d end_ARG start_ARG italic_n end_ARG roman_log italic_n end_ARG or 2 roman_log italic_n ,. , 2 =
depending on whether d𝑑ditalic_d dominates nlogn𝑛𝑛n\log nitalic_n roman_log italic_n or not.
In the high-dimensional regime d=ω(nlogn)𝑑𝜔𝑛𝑛d=\omega(n\log n)italic_d = italic_ω ( italic_n roman_log italic_n ), we obtain
21+ρR≈2dnlogn21𝜌𝑅2𝑑𝑛𝑛\frac{2}{1+\rho}\,R\approx\sqrt{\tfrac{2d}{n}\log n}divide start_ARG 2 end_ARG start_ARG 1 + italic_ρ end_ARG italic_R ≈ square-root start_ARG divide start_ARG 2 italic_d end_ARG start_ARG italic_n end_ARG roman_log italic_n end_ARG, yielding
R2R+dn≈12(1+ρ)2lognsuperscript𝑅2𝑅𝑑𝑛12superscript1𝜌2𝑛\frac{R^{2}}{\,R+\tfrac{d}{n}\,}\approx\tfrac{1}{2}(1+\rho)^{2}\log ndivide start_ARG italic_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_R + divide start_ARG italic_d end_ARG start_ARG italic_n end_ARG end_ARG ≈ divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( 1 + italic_ρ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT roman_log italic_n.
In contrast, if d=o(nlogn)𝑑𝑜𝑛𝑛d=o(n\log n)italic_d = italic_o ( italic_n roman_log italic_n ), we have
21+ρR≈2logn21𝜌𝑅2𝑛\frac{2}{1+\rho}\,R\approx 2\log ndivide start_ARG 2 end_ARG start_ARG 1 + italic_ρ end_ARG italic_R ≈ 2 roman_log italic_n, so
R2R+dn≈(1+ρ)lognsuperscript𝑅2𝑅𝑑𝑛1𝜌𝑛\frac{R^{2}}{\,R+\tfrac{d}{n}\,}\approx(1+\rho)\log ndivide start_ARG italic_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_R + divide start_ARG italic_d end_ARG start_ARG italic_n end_ARG end_ARG ≈ ( 1 + italic_ρ ) roman_log italic_n.
Finally, in the intermediate case d=Θ(nlogn)𝑑Θ𝑛𝑛d=\Theta(n\log n)italic_d = roman_Θ ( italic_n roman_log italic_n ), the achievable SNR lies between these two extremes. These scenarios explain how the y𝑦yitalic_y-intercept in Figure 3 varies with respect to d𝑑ditalic_d.
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
III Correlated Contextual Stochastic Block Models
III-B Exact Community Recovery
Remark 7 (SNR of Node Attributes).
Suppose
, 1 = (21+ρR)221+ρR+dn=2logn⟹21+ρR≈2dnlognor 2logn,formulae-sequencesuperscript21𝜌𝑅221𝜌𝑅𝑑𝑛2𝑛⟹21𝜌𝑅2𝑑𝑛𝑛or2𝑛\frac{\left(\frac{2}{1+\rho}R\right)^{2}}{\tfrac{2}{1+\rho}R+\tfrac{d}{n}}=2\,%
\log n\quad\Longrightarrow\quad\frac{2}{1+\rho}\,R\approx\sqrt{\frac{2d}{n}\,%
\log n}\;\;\text{or}\;\;2\log n,divide start_ARG ( divide start_ARG 2 end_ARG start_ARG 1 + italic_ρ end_ARG italic_R ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG divide start_ARG 2 end_ARG start_ARG 1 + italic_ρ end_ARG italic_R + divide start_ARG italic_d end_ARG start_ARG italic_n end_ARG end_ARG = 2 roman_log italic_n ⟹ divide start_ARG 2 end_ARG start_ARG 1 + italic_ρ end_ARG italic_R ≈ square-root start_ARG divide start_ARG 2 italic_d end_ARG start_ARG italic_n end_ARG roman_log italic_n end_ARG or 2 roman_log italic_n ,. , 2 =
depending on whether d𝑑ditalic_d dominates nlogn𝑛𝑛n\log nitalic_n roman_log italic_n or not.
In the high-dimensional regime d=ω(nlogn)𝑑𝜔𝑛𝑛d=\omega(n\log n)italic_d = italic_ω ( italic_n roman_log italic_n ), we obtain
21+ρR≈2dnlogn21𝜌𝑅2𝑑𝑛𝑛\frac{2}{1+\rho}\,R\approx\sqrt{\tfrac{2d}{n}\log n}divide start_ARG 2 end_ARG start_ARG 1 + italic_ρ end_ARG italic_R ≈ square-root start_ARG divide start_ARG 2 italic_d end_ARG start_ARG italic_n end_ARG roman_log italic_n end_ARG, yielding
R2R+dn≈12(1+ρ)2lognsuperscript𝑅2𝑅𝑑𝑛12superscript1𝜌2𝑛\frac{R^{2}}{\,R+\tfrac{d}{n}\,}\approx\tfrac{1}{2}(1+\rho)^{2}\log ndivide start_ARG italic_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_R + divide start_ARG italic_d end_ARG start_ARG italic_n end_ARG end_ARG ≈ divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( 1 + italic_ρ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT roman_log italic_n.
In contrast, if d=o(nlogn)𝑑𝑜𝑛𝑛d=o(n\log n)italic_d = italic_o ( italic_n roman_log italic_n ), we have
21+ρR≈2logn21𝜌𝑅2𝑛\frac{2}{1+\rho}\,R\approx 2\log ndivide start_ARG 2 end_ARG start_ARG 1 + italic_ρ end_ARG italic_R ≈ 2 roman_log italic_n, so
R2R+dn≈(1+ρ)lognsuperscript𝑅2𝑅𝑑𝑛1𝜌𝑛\frac{R^{2}}{\,R+\tfrac{d}{n}\,}\approx(1+\rho)\log ndivide start_ARG italic_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_R + divide start_ARG italic_d end_ARG start_ARG italic_n end_ARG end_ARG ≈ ( 1 + italic_ρ ) roman_log italic_n.
Finally, in the intermediate case d=Θ(nlogn)𝑑Θ𝑛𝑛d=\Theta(n\log n)italic_d = roman_Θ ( italic_n roman_log italic_n ), the achievable SNR lies between these two extremes. These scenarios explain how the y𝑦yitalic_y-intercept in Figure 3 varies with respect to d𝑑ditalic_d.
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Without loss of generality, let π∗:[n]→[n]:subscript𝜋→delimited-[]𝑛delimited-[]𝑛\pi_{*}:[n]\to[n]italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT : [ italic_n ] → [ italic_n ] be the identity permutation. In this section, we outline the proofs of exact matching for the two proposed models.
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
IV Outline of the Proof
Without loss of generality, let π∗:[n]→[n]:subscript𝜋→delimited-[]𝑛delimited-[]𝑛\pi_{*}:[n]\to[n]italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT : [ italic_n ] → [ italic_n ] be the identity permutation. In this section, we outline the proofs of exact matching for the two proposed models.
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We begin by explaining how Theorem 1 can be established via the estimator in (6). The argument closely follows the ideas of [10], adapted to our setting.
Define
, 1 = Zij:=∥𝒙i−𝒚j∥2.assignsubscript𝑍𝑖𝑗superscriptdelimited-∥∥subscript𝒙𝑖subscript𝒚𝑗2Z_{ij}:=\lVert{\boldsymbol{x}}_{i}-{\boldsymbol{y}}_{j}\rVert^{2}.italic_Z start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT := ∥ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_italic_y start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT .. , 2 = . , 3 = (30)
Then the estimator (6) can be rewritten as
, 1 = π^=argminπ∈Sn∑i=1nZiπ(i).^𝜋subscriptargmin𝜋subscript𝑆𝑛superscriptsubscript𝑖1𝑛subscript𝑍𝑖𝜋𝑖\hat{\pi}=\operatorname*{arg\,min}_{\pi\in S_{n}}\sum\limits_{i=1}^{n}Z_{i\pi(%
i)}.over^ start_ARG italic_π end_ARG = start_OPERATOR roman_arg roman_min end_OPERATOR start_POSTSUBSCRIPT italic_π ∈ italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_Z start_POSTSUBSCRIPT italic_i italic_π ( italic_i ) end_POSTSUBSCRIPT .. , 2 = . , 3 = (31)
Let
, 1 = ℳ:={i∈[n]:π^(i)≠π∗(i)}.assignℳconditional-set𝑖delimited-[]𝑛^𝜋𝑖subscript𝜋𝑖\mathcal{M}:=\{i\in[n]:\hat{\pi}(i)\neq\pi_{*}(i)\}.caligraphic_M := { italic_i ∈ [ italic_n ] : over^ start_ARG italic_π end_ARG ( italic_i ) ≠ italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_i ) } .. , 2 = . , 3 = (32)
be the set of mismatched nodes. Our goal is to show that ℙ(|ℳ|=0)=1−o(1)ℙℳ01𝑜1\mathbb{P}\bigl{(}|\mathcal{M}|=0\bigr{)}=1-o(1)blackboard_P ( | caligraphic_M | = 0 ) = 1 - italic_o ( 1 ).
Consider the event
, 1 = ℱt={∑k=1tZikik≥∑k=1tZikik+1},subscriptℱ𝑡superscriptsubscript𝑘1𝑡subscript𝑍subscript𝑖𝑘subscript𝑖𝑘superscriptsubscript𝑘1𝑡subscript𝑍subscript𝑖𝑘subscript𝑖𝑘1\mathcal{F}_{t}=\left\{\sum_{k=1}^{t}Z_{i_{k}i_{k}}\geq\sum_{k=1}^{t}Z_{i_{k}i%
_{k+1}}\right\},caligraphic_F start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT = { ∑ start_POSTSUBSCRIPT italic_k = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT italic_Z start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ≥ ∑ start_POSTSUBSCRIPT italic_k = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT italic_Z start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT } ,. , 2 = . , 3 = (33)
where i1,…,itsubscript𝑖1…subscript𝑖𝑡i_{1},\ldots,i_{t}italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_i start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT are t𝑡titalic_t distinct nodes in [n]delimited-[]𝑛[n][ italic_n ], and it+1=i1subscript𝑖𝑡1subscript𝑖1i_{t+1}=i_{1}italic_i start_POSTSUBSCRIPT italic_t + 1 end_POSTSUBSCRIPT = italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT. We will show that the probability of ℱtsubscriptℱ𝑡\mathcal{F}_{t}caligraphic_F start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT is sufficiently small if either (7) or (8) holds. Denote this upper bound by f(t)𝑓𝑡f(t)italic_f ( italic_t ). Since there are (nt)(t−1)!binomial𝑛𝑡𝑡1{n\choose t}(t-1)!( binomial start_ARG italic_n end_ARG start_ARG italic_t end_ARG ) ( italic_t - 1 ) ! ways to choose t𝑡titalic_t distinct nodes, we obtain
, 1 = 𝔼(|ℳ|)≤∑t=2nf(t)(nt)(t−1)!𝔼ℳsuperscriptsubscript𝑡2𝑛𝑓𝑡binomial𝑛𝑡𝑡1\mathbb{E}(|\mathcal{M}|)\leq\sum\limits_{t=2}^{n}f(t){n\choose t}(t-1)!blackboard_E ( | caligraphic_M | ) ≤ ∑ start_POSTSUBSCRIPT italic_t = 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_f ( italic_t ) ( binomial start_ARG italic_n end_ARG start_ARG italic_t end_ARG ) ( italic_t - 1 ) !. , 2 = . , 3 = (34)
where t=1𝑡1t=1italic_t = 1 cannot yield a mismatch by itself. If f(t)𝑓𝑡f(t)italic_f ( italic_t ) is made sufficiently small, it follows that 𝔼(|ℳ|)=o(1)𝔼ℳ𝑜1\mathbb{E}\bigl{(}|\mathcal{M}|\bigr{)}=o(1)blackboard_E ( | caligraphic_M | ) = italic_o ( 1 ), and by Markov’s inequality, ℙ(|ℳ|=0)=1−o(1)ℙℳ01𝑜1\mathbb{P}\bigl{(}|\mathcal{M}|=0\bigr{)}=1-o(1)blackboard_P ( | caligraphic_M | = 0 ) = 1 - italic_o ( 1 ). Achieving a suitably small f(t)𝑓𝑡f(t)italic_f ( italic_t ) necessitates either (7) or (8). A rigorous proof can be found in Section VI.
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
IV Outline of the Proof
IV-A Proof Sketch of Theorem 1
We begin by explaining how Theorem 1 can be established via the estimator in (6). The argument closely follows the ideas of [10], adapted to our setting.
Define
, 1 = Zij:=∥𝒙i−𝒚j∥2.assignsubscript𝑍𝑖𝑗superscriptdelimited-∥∥subscript𝒙𝑖subscript𝒚𝑗2Z_{ij}:=\lVert{\boldsymbol{x}}_{i}-{\boldsymbol{y}}_{j}\rVert^{2}.italic_Z start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT := ∥ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_italic_y start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT .. , 2 = . , 3 = (30)
Then the estimator (6) can be rewritten as
, 1 = π^=argminπ∈Sn∑i=1nZiπ(i).^𝜋subscriptargmin𝜋subscript𝑆𝑛superscriptsubscript𝑖1𝑛subscript𝑍𝑖𝜋𝑖\hat{\pi}=\operatorname*{arg\,min}_{\pi\in S_{n}}\sum\limits_{i=1}^{n}Z_{i\pi(%
i)}.over^ start_ARG italic_π end_ARG = start_OPERATOR roman_arg roman_min end_OPERATOR start_POSTSUBSCRIPT italic_π ∈ italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_Z start_POSTSUBSCRIPT italic_i italic_π ( italic_i ) end_POSTSUBSCRIPT .. , 2 = . , 3 = (31)
Let
, 1 = ℳ:={i∈[n]:π^(i)≠π∗(i)}.assignℳconditional-set𝑖delimited-[]𝑛^𝜋𝑖subscript𝜋𝑖\mathcal{M}:=\{i\in[n]:\hat{\pi}(i)\neq\pi_{*}(i)\}.caligraphic_M := { italic_i ∈ [ italic_n ] : over^ start_ARG italic_π end_ARG ( italic_i ) ≠ italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_i ) } .. , 2 = . , 3 = (32)
be the set of mismatched nodes. Our goal is to show that ℙ(|ℳ|=0)=1−o(1)ℙℳ01𝑜1\mathbb{P}\bigl{(}|\mathcal{M}|=0\bigr{)}=1-o(1)blackboard_P ( | caligraphic_M | = 0 ) = 1 - italic_o ( 1 ).
Consider the event
, 1 = ℱt={∑k=1tZikik≥∑k=1tZikik+1},subscriptℱ𝑡superscriptsubscript𝑘1𝑡subscript𝑍subscript𝑖𝑘subscript𝑖𝑘superscriptsubscript𝑘1𝑡subscript𝑍subscript𝑖𝑘subscript𝑖𝑘1\mathcal{F}_{t}=\left\{\sum_{k=1}^{t}Z_{i_{k}i_{k}}\geq\sum_{k=1}^{t}Z_{i_{k}i%
_{k+1}}\right\},caligraphic_F start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT = { ∑ start_POSTSUBSCRIPT italic_k = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT italic_Z start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ≥ ∑ start_POSTSUBSCRIPT italic_k = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT italic_Z start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT } ,. , 2 = . , 3 = (33)
where i1,…,itsubscript𝑖1…subscript𝑖𝑡i_{1},\ldots,i_{t}italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_i start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT are t𝑡titalic_t distinct nodes in [n]delimited-[]𝑛[n][ italic_n ], and it+1=i1subscript𝑖𝑡1subscript𝑖1i_{t+1}=i_{1}italic_i start_POSTSUBSCRIPT italic_t + 1 end_POSTSUBSCRIPT = italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT. We will show that the probability of ℱtsubscriptℱ𝑡\mathcal{F}_{t}caligraphic_F start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT is sufficiently small if either (7) or (8) holds. Denote this upper bound by f(t)𝑓𝑡f(t)italic_f ( italic_t ). Since there are (nt)(t−1)!binomial𝑛𝑡𝑡1{n\choose t}(t-1)!( binomial start_ARG italic_n end_ARG start_ARG italic_t end_ARG ) ( italic_t - 1 ) ! ways to choose t𝑡titalic_t distinct nodes, we obtain
, 1 = 𝔼(|ℳ|)≤∑t=2nf(t)(nt)(t−1)!𝔼ℳsuperscriptsubscript𝑡2𝑛𝑓𝑡binomial𝑛𝑡𝑡1\mathbb{E}(|\mathcal{M}|)\leq\sum\limits_{t=2}^{n}f(t){n\choose t}(t-1)!blackboard_E ( | caligraphic_M | ) ≤ ∑ start_POSTSUBSCRIPT italic_t = 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_f ( italic_t ) ( binomial start_ARG italic_n end_ARG start_ARG italic_t end_ARG ) ( italic_t - 1 ) !. , 2 = . , 3 = (34)
where t=1𝑡1t=1italic_t = 1 cannot yield a mismatch by itself. If f(t)𝑓𝑡f(t)italic_f ( italic_t ) is made sufficiently small, it follows that 𝔼(|ℳ|)=o(1)𝔼ℳ𝑜1\mathbb{E}\bigl{(}|\mathcal{M}|\bigr{)}=o(1)blackboard_E ( | caligraphic_M | ) = italic_o ( 1 ), and by Markov’s inequality, ℙ(|ℳ|=0)=1−o(1)ℙℳ01𝑜1\mathbb{P}\bigl{(}|\mathcal{M}|=0\bigr{)}=1-o(1)blackboard_P ( | caligraphic_M | = 0 ) = 1 - italic_o ( 1 ). Achieving a suitably small f(t)𝑓𝑡f(t)italic_f ( italic_t ) necessitates either (7) or (8). A rigorous proof can be found in Section VI.
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Exact matching in correlated CSBMs is proved via a two-step algorithm:
1.
k𝑘kitalic_k-core matching using edges:
We chooseReport issue for preceding element
k=logn(loglogn)2∨nps2(lognps2)2.𝑘𝑛superscript𝑛2𝑛𝑝superscript𝑠2superscript𝑛𝑝superscript𝑠22k=\frac{\log n}{(\log\log n)^{2}}\vee\frac{nps^{2}}{(\log nps^{2})^{2}}.italic_k = divide start_ARG roman_log italic_n end_ARG start_ARG ( roman_log roman_log italic_n ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ∨ divide start_ARG italic_n italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG ( roman_log italic_n italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG .
Applying k𝑘kitalic_k-core matching with this k𝑘kitalic_k matches n−o(n)𝑛𝑜𝑛n-o(n)italic_n - italic_o ( italic_n ) nodes correctly, leaving a set F𝐹Fitalic_F of unmatched nodes satisfyingReport issue for preceding element
|F|≤n1−ns2(p+q)2logn+o(1).𝐹superscript𝑛1𝑛superscript𝑠2𝑝𝑞2𝑛𝑜1|F|\leq n^{1-\frac{ns^{2}(p+q)}{2\log n}+o(1)}.| italic_F | ≤ italic_n start_POSTSUPERSCRIPT 1 - divide start_ARG italic_n italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_p + italic_q ) end_ARG start_ARG 2 roman_log italic_n end_ARG + italic_o ( 1 ) end_POSTSUPERSCRIPT .
(35)
2.
Matching the remainder via node attributes:
The remaining unmatched nodes in F𝐹Fitalic_F form a correlated Gaussian Mixture Model. Under assumptions (22) and (23), Theorem 1 applies, ensuring that these nodes can also be matched correctly using their node attributes.Report issue for preceding element
A full proof of Theorem 5, detailing each step, can be found in Section IX.
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
IV Outline of the Proof
IV-B Proof Sketch of Theorem 5
Exact matching in correlated CSBMs is proved via a two-step algorithm:
1.
k𝑘kitalic_k-core matching using edges:
We chooseReport issue for preceding element
k=logn(loglogn)2∨nps2(lognps2)2.𝑘𝑛superscript𝑛2𝑛𝑝superscript𝑠2superscript𝑛𝑝superscript𝑠22k=\frac{\log n}{(\log\log n)^{2}}\vee\frac{nps^{2}}{(\log nps^{2})^{2}}.italic_k = divide start_ARG roman_log italic_n end_ARG start_ARG ( roman_log roman_log italic_n ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ∨ divide start_ARG italic_n italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG ( roman_log italic_n italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG .
Applying k𝑘kitalic_k-core matching with this k𝑘kitalic_k matches n−o(n)𝑛𝑜𝑛n-o(n)italic_n - italic_o ( italic_n ) nodes correctly, leaving a set F𝐹Fitalic_F of unmatched nodes satisfyingReport issue for preceding element
|F|≤n1−ns2(p+q)2logn+o(1).𝐹superscript𝑛1𝑛superscript𝑠2𝑝𝑞2𝑛𝑜1|F|\leq n^{1-\frac{ns^{2}(p+q)}{2\log n}+o(1)}.| italic_F | ≤ italic_n start_POSTSUPERSCRIPT 1 - divide start_ARG italic_n italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_p + italic_q ) end_ARG start_ARG 2 roman_log italic_n end_ARG + italic_o ( 1 ) end_POSTSUPERSCRIPT .
(35)
2.
Matching the remainder via node attributes:
The remaining unmatched nodes in F𝐹Fitalic_F form a correlated Gaussian Mixture Model. Under assumptions (22) and (23), Theorem 1 applies, ensuring that these nodes can also be matched correctly using their node attributes.Report issue for preceding element
A full proof of Theorem 5, detailing each step, can be found in Section IX.
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In this work, we studied the problem of community recovery in the presence of two correlated networks. We introduced, for the first time, Correlated Gaussian Mixture Models (CGMMs), which focus on correlated node attributes, and Correlated Contextual Stochastic Block Models (CCSBMs), which leverage both correlated node attributes and edges. We showed that, while exact community recovery may be impossible for a single network, there are parameter regimes in which having a second, correlated network serves as valuable side information, thereby making exact community recovery feasible. In particular, we found that sufficiently high-dimensional node attributes–on the order of ω(logn)𝜔𝑛\omega(\log n)italic_ω ( roman_log italic_n )–can enable effective recovery.
To obtain these results, we first identified conditions for exact matching between the two networks. Notably, exact matching is easier to achieve when both node attributes and edges are correlated, compared to scenarios where only one is available. Below, we discuss several open problems stemming from our study.
| 217
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
V Discussion and Open Problems
In this work, we studied the problem of community recovery in the presence of two correlated networks. We introduced, for the first time, Correlated Gaussian Mixture Models (CGMMs), which focus on correlated node attributes, and Correlated Contextual Stochastic Block Models (CCSBMs), which leverage both correlated node attributes and edges. We showed that, while exact community recovery may be impossible for a single network, there are parameter regimes in which having a second, correlated network serves as valuable side information, thereby making exact community recovery feasible. In particular, we found that sufficiently high-dimensional node attributes–on the order of ω(logn)𝜔𝑛\omega(\log n)italic_ω ( roman_log italic_n )–can enable effective recovery.
To obtain these results, we first identified conditions for exact matching between the two networks. Notably, exact matching is easier to achieve when both node attributes and edges are correlated, compared to scenarios where only one is available. Below, we discuss several open problems stemming from our study.
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46
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Consider first the case of Correlated Gaussian Mixture Models. In Theorem 1, either d=ω(logn)𝑑𝜔𝑛d=\omega(\log n)italic_d = italic_ω ( roman_log italic_n ) or ‖𝝁‖2≥2logn+ω(1)superscriptnorm𝝁22𝑛𝜔1\|\boldsymbol{\mu}\|^{2}\geq 2\log n+\omega(1)∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ 2 roman_log italic_n + italic_ω ( 1 ) must hold for exact matching. In contrast, in the community-free correlated Gaussian database model (i.e., 𝝁=0𝝁0\boldsymbol{\mu}=0bold_italic_μ = 0), Dai et al. [9] showed that exact matching is possible with
d4log11−ρ2≥logn+ω(1)𝑑411superscript𝜌2𝑛𝜔1\frac{d}{4}\log\frac{1}{1-\rho^{2}}\geq\log n+\omega(1)divide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≥ roman_log italic_n + italic_ω ( 1 )
without further constraints on d𝑑ditalic_d. Meanwhile, Theorem 2 leaves some gaps, especially for d=Ω(n)𝑑Ω𝑛d=\Omega(n)italic_d = roman_Ω ( italic_n ). We hypothesize that higher-dimensional attributes should make exact matching progressively easier, suggesting that
d4log11−ρ2≥(1+ϵ)logn𝑑411superscript𝜌21italic-ϵ𝑛\frac{d}{4}\log\frac{1}{1-\rho^{2}}\geq(1+\epsilon)\log ndivide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≥ ( 1 + italic_ϵ ) roman_log italic_n
may indeed be the core information-theoretic threshold, even without additional conditions on 𝝁𝝁\boldsymbol{\mu}bold_italic_μ and d𝑑ditalic_d. Resolving this would also address the same gap in the CCSBM setting, since the latter relies on CGMMs for unmatched nodes.
| 611
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
V Discussion and Open Problems
V-1 Closing the information-theoretic gap for exact matching
Consider first the case of Correlated Gaussian Mixture Models. In Theorem 1, either d=ω(logn)𝑑𝜔𝑛d=\omega(\log n)italic_d = italic_ω ( roman_log italic_n ) or ‖𝝁‖2≥2logn+ω(1)superscriptnorm𝝁22𝑛𝜔1\|\boldsymbol{\mu}\|^{2}\geq 2\log n+\omega(1)∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ 2 roman_log italic_n + italic_ω ( 1 ) must hold for exact matching. In contrast, in the community-free correlated Gaussian database model (i.e., 𝝁=0𝝁0\boldsymbol{\mu}=0bold_italic_μ = 0), Dai et al. [9] showed that exact matching is possible with
d4log11−ρ2≥logn+ω(1)𝑑411superscript𝜌2𝑛𝜔1\frac{d}{4}\log\frac{1}{1-\rho^{2}}\geq\log n+\omega(1)divide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≥ roman_log italic_n + italic_ω ( 1 )
without further constraints on d𝑑ditalic_d. Meanwhile, Theorem 2 leaves some gaps, especially for d=Ω(n)𝑑Ω𝑛d=\Omega(n)italic_d = roman_Ω ( italic_n ). We hypothesize that higher-dimensional attributes should make exact matching progressively easier, suggesting that
d4log11−ρ2≥(1+ϵ)logn𝑑411superscript𝜌21italic-ϵ𝑛\frac{d}{4}\log\frac{1}{1-\rho^{2}}\geq(1+\epsilon)\log ndivide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≥ ( 1 + italic_ϵ ) roman_log italic_n
may indeed be the core information-theoretic threshold, even without additional conditions on 𝝁𝝁\boldsymbol{\mu}bold_italic_μ and d𝑑ditalic_d. Resolving this would also address the same gap in the CCSBM setting, since the latter relies on CGMMs for unmatched nodes.
| 645
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Theorems 3 and 7 establish achievable regions for exact community recovery in CGMMs and CCSBMs under the assumption that exact matching is possible. Comparing these to the converse results (Theorems 4 and 8) reveals that the stated conditions are not necessarily tight. Thus, it is natural to ask whether exact matching is truly needed for exact community recovery.
Indeed, in correlated SBMs, Rácz and Sridhar [6] used exact matching to derive conditions for community recovery, whereas Gaudio et al. [8] refined the threshold by using partial matching (e.g., k𝑘kitalic_k-core). Similarly, one might conjecture that for correlated GMMs there exists a constant κ𝜅\kappaitalic_κ such that, even if d4log11−ρ2≥(1−κ)logn𝑑411superscript𝜌21𝜅𝑛\frac{d}{4}\log\frac{1}{1-\rho^{2}}\geq(1-\kappa)\log ndivide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≥ ( 1 - italic_κ ) roman_log italic_n
there exists an estimator 𝝈~~𝝈\tilde{{\boldsymbol{\sigma}}}over~ start_ARG bold_italic_σ end_ARG that can exactly recover 𝝈𝝈{\boldsymbol{\sigma}}bold_italic_σ even without full matching. Exploring this possibility is an intriguing open problem.
| 362
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
V Discussion and Open Problems
V-2 Closing the information-theoretic gap for exact community recovery
Theorems 3 and 7 establish achievable regions for exact community recovery in CGMMs and CCSBMs under the assumption that exact matching is possible. Comparing these to the converse results (Theorems 4 and 8) reveals that the stated conditions are not necessarily tight. Thus, it is natural to ask whether exact matching is truly needed for exact community recovery.
Indeed, in correlated SBMs, Rácz and Sridhar [6] used exact matching to derive conditions for community recovery, whereas Gaudio et al. [8] refined the threshold by using partial matching (e.g., k𝑘kitalic_k-core). Similarly, one might conjecture that for correlated GMMs there exists a constant κ𝜅\kappaitalic_κ such that, even if d4log11−ρ2≥(1−κ)logn𝑑411superscript𝜌21𝜅𝑛\frac{d}{4}\log\frac{1}{1-\rho^{2}}\geq(1-\kappa)\log ndivide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≥ ( 1 - italic_κ ) roman_log italic_n
there exists an estimator 𝝈~~𝝈\tilde{{\boldsymbol{\sigma}}}over~ start_ARG bold_italic_σ end_ARG that can exactly recover 𝝈𝝈{\boldsymbol{\sigma}}bold_italic_σ even without full matching. Exploring this possibility is an intriguing open problem.
| 397
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Our analysis has focused on the simplest case of two communities. A natural next step is to consider models with r≥2𝑟2r\geq 2italic_r ≥ 2 communities. It would be interesting to investigate whether the conditions for exact matching and exact community recovery generalize in a straightforward manner or require fundamentally new techniques.
| 68
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
V Discussion and Open Problems
V-3 Generalizing to more communities
Our analysis has focused on the simplest case of two communities. A natural next step is to consider models with r≥2𝑟2r\geq 2italic_r ≥ 2 communities. It would be interesting to investigate whether the conditions for exact matching and exact community recovery generalize in a straightforward manner or require fundamentally new techniques.
| 97
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We have considered only two correlated graphs. Another direction is to explore the setting where more than two correlated graphs are given. For example, Ameen and Hajek [37] established exact matching thresholds for r≥2𝑟2r\geq 2italic_r ≥ 2 correlated Erdős-Rényi graphs when the average degree is on the order of logn𝑛\log nroman_log italic_n. An open question is to characterize exact matching in the presence of r𝑟ritalic_r correlated graphs (possibly with node attributes) and then design optimal strategies for combining their edge and attribute information to further improve community recovery. We conjecture that taking the union of edges and averaging node attributes across multiple graphs may yield the best results.
| 158
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
V Discussion and Open Problems
V-4 Multiple correlated graphs
We have considered only two correlated graphs. Another direction is to explore the setting where more than two correlated graphs are given. For example, Ameen and Hajek [37] established exact matching thresholds for r≥2𝑟2r\geq 2italic_r ≥ 2 correlated Erdős-Rényi graphs when the average degree is on the order of logn𝑛\log nroman_log italic_n. An open question is to characterize exact matching in the presence of r𝑟ritalic_r correlated graphs (possibly with node attributes) and then design optimal strategies for combining their edge and attribute information to further improve community recovery. We conjecture that taking the union of edges and averaging node attributes across multiple graphs may yield the best results.
| 185
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From a computational viewpoint, there remain major challenges. Although [5] showed that exact community recovery can be done with spectral methods in Gaussian Mixture Models or Contextual Stochastic Block Models in O(n3)𝑂superscript𝑛3O(n^{3})italic_O ( italic_n start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) time, exact matching under CGMMs currently relies on a Hungarian algorithm to minimize pairwise distances, incurring O(n2d+n3)𝑂superscript𝑛2𝑑superscript𝑛3O(n^{2}d+n^{3})italic_O ( italic_n start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_d + italic_n start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) complexity. For CCSBMs, the situation is more severe: the k𝑘kitalic_k-core method requires examining all matchings, leading to Θ(n!)Θ𝑛\Theta(n!)roman_Θ ( italic_n ! ) complexity. Developing polynomial-time algorithms that achieve k𝑘kitalic_k-core-level performance for exact matching–and thus enable more efficient community recovery in CCSBMs–would be a significant breakthrough for large-scale network analysis.
| 267
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
V Discussion and Open Problems
V-5 Efficient algorithms
From a computational viewpoint, there remain major challenges. Although [5] showed that exact community recovery can be done with spectral methods in Gaussian Mixture Models or Contextual Stochastic Block Models in O(n3)𝑂superscript𝑛3O(n^{3})italic_O ( italic_n start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) time, exact matching under CGMMs currently relies on a Hungarian algorithm to minimize pairwise distances, incurring O(n2d+n3)𝑂superscript𝑛2𝑑superscript𝑛3O(n^{2}d+n^{3})italic_O ( italic_n start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_d + italic_n start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) complexity. For CCSBMs, the situation is more severe: the k𝑘kitalic_k-core method requires examining all matchings, leading to Θ(n!)Θ𝑛\Theta(n!)roman_Θ ( italic_n ! ) complexity. Developing polynomial-time algorithms that achieve k𝑘kitalic_k-core-level performance for exact matching–and thus enable more efficient community recovery in CCSBMs–would be a significant breakthrough for large-scale network analysis.
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We analyze the estimator (31), which finds a permutation that minimizes the sum of attribute distances, and establish the conditions under which no mismatched node pairs arise. Our proof technique builds on the approach of [10], where the estimator (31) was analyzed in the context of geometric partial matching without community structures, assuming an identical distribution for all node attribute vectors. In contrast, we analyze the correlated Gaussian Mixture Models where node attribute distributions vary with the unknown community labels. We demonstrate that the estimator (31) achieves the exact matching when conditions (7) or (8) are met.
First, for the analysis, we present a lemma that provides an upper bound for ℙ(ℱt)ℙsubscriptℱ𝑡\mathbb{P}\left(\mathcal{F}_{t}\right)blackboard_P ( caligraphic_F start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ), where ℱtsubscriptℱ𝑡\mathcal{F}_{t}caligraphic_F start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT is defined in (33). For t≥2𝑡2t\geq 2italic_t ≥ 2 and α>0𝛼0\alpha>0italic_α > 0, let us define
, 1 = S(α,t):=∑j=1t−1log(1+12α(1−cos(2πtj))),assign𝑆𝛼𝑡superscriptsubscript𝑗1𝑡1112𝛼12𝜋𝑡𝑗S\left(\alpha,t\right):=\sum_{j=1}^{t-1}\log\left(1+\frac{1}{2\alpha}\left(1-%
\cos\left(\frac{2\pi}{t}j\right)\right)\right),italic_S ( italic_α , italic_t ) := ∑ start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t - 1 end_POSTSUPERSCRIPT roman_log ( 1 + divide start_ARG 1 end_ARG start_ARG 2 italic_α end_ARG ( 1 - roman_cos ( divide start_ARG 2 italic_π end_ARG start_ARG italic_t end_ARG italic_j ) ) ) ,. , 2 = . , 3 = (36)
, 1 = I(α):=∫01log(1+12α(1−cos(2πx)))𝑑x.assign𝐼𝛼superscriptsubscript01112𝛼12𝜋𝑥differential-d𝑥I(\alpha):=\int_{0}^{1}\log\left(1+\frac{1}{2\alpha}\left(1-\cos\left(2\pi x%
\right)\right)\right)dx.italic_I ( italic_α ) := ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT roman_log ( 1 + divide start_ARG 1 end_ARG start_ARG 2 italic_α end_ARG ( 1 - roman_cos ( 2 italic_π italic_x ) ) ) italic_d italic_x .. , 2 = . , 3 = (37)
Additionally, let us define two events
, 1 = 𝒜1:={−‖𝝁‖2≤⟨𝝁,𝒛i⟩≤‖𝝁‖2 for all i∈[n]}assignsubscript𝒜1superscriptnorm𝝁2𝝁subscript𝒛𝑖superscriptnorm𝝁2 for all 𝑖delimited-[]𝑛\mathcal{A}_{1}:=\left\{-\|{\boldsymbol{\mu}}\|^{2}\leq\langle{\boldsymbol{\mu%
}},{\boldsymbol{z}}_{i}\rangle\leq\|{\boldsymbol{\mu}}\|^{2}\text{ for all }i%
\in[n]\right\}caligraphic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT := { - ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≤ ⟨ bold_italic_μ , bold_italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⟩ ≤ ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT for all italic_i ∈ [ italic_n ] }. , 2 = . , 3 = (38)
, 1 = 𝒜2:={ρ−λ2∥𝒛i−𝒛j+𝝁+1−λρ−λ𝝁∥2≥ρ−λ2(1+1−λρ−λ)2∥𝝁∥2−2(1−λ)∥𝝁∥2 and ρ−λ2∥𝒛i−𝒛j+𝝁−1−λρ−λ𝝁∥2≥ρ−λ2(1−1−λρ−λ)2∥𝝁∥2 for all distinct i,j∈[n]}.\begin{gathered}\mathcal{A}_{2}:=\left\{\frac{\rho-\lambda}{2}\left\|{%
\boldsymbol{z}}_{i}-{\boldsymbol{z}}_{j}+\boldsymbol{\mu}+\frac{1-\lambda}{%
\rho-\lambda}\boldsymbol{\mu}\right\|^{2}\geq\frac{\rho-\lambda}{2}\left(1+%
\frac{1-\lambda}{\rho-\lambda}\right)^{2}\|\boldsymbol{\mu}\|^{2}-2(1-\lambda)%
\|\boldsymbol{\mu}\|^{2}\text{ and }\right.\\
\left.\frac{\rho-\lambda}{2}\left\|{\boldsymbol{z}}_{i}-{\boldsymbol{z}}_{j}+%
\boldsymbol{\mu}-\frac{1-\lambda}{\rho-\lambda}\boldsymbol{\mu}\right\|^{2}%
\geq\frac{\rho-\lambda}{2}\left(1-\frac{1-\lambda}{\rho-\lambda}\right)^{2}\|%
\boldsymbol{\mu}\|^{2}\text{ for all distinct }i,j\in[n]\right\}.\end{gathered}start_ROW start_CELL caligraphic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT := { divide start_ARG italic_ρ - italic_λ end_ARG start_ARG 2 end_ARG ∥ bold_italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_italic_z start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT + bold_italic_μ + divide start_ARG 1 - italic_λ end_ARG start_ARG italic_ρ - italic_λ end_ARG bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ divide start_ARG italic_ρ - italic_λ end_ARG start_ARG 2 end_ARG ( 1 + divide start_ARG 1 - italic_λ end_ARG start_ARG italic_ρ - italic_λ end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 2 ( 1 - italic_λ ) ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT and end_CELL end_ROW start_ROW start_CELL divide start_ARG italic_ρ - italic_λ end_ARG start_ARG 2 end_ARG ∥ bold_italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_italic_z start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT + bold_italic_μ - divide start_ARG 1 - italic_λ end_ARG start_ARG italic_ρ - italic_λ end_ARG bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ divide start_ARG italic_ρ - italic_λ end_ARG start_ARG 2 end_ARG ( 1 - divide start_ARG 1 - italic_λ end_ARG start_ARG italic_ρ - italic_λ end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT for all distinct italic_i , italic_j ∈ [ italic_n ] } . end_CELL end_ROW. , 2 = . , 3 = (39)
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
VI Proof of Theorem 1:
Achievability of Exact Matching in Correlated Gaussian Mixture Models
We analyze the estimator (31), which finds a permutation that minimizes the sum of attribute distances, and establish the conditions under which no mismatched node pairs arise. Our proof technique builds on the approach of [10], where the estimator (31) was analyzed in the context of geometric partial matching without community structures, assuming an identical distribution for all node attribute vectors. In contrast, we analyze the correlated Gaussian Mixture Models where node attribute distributions vary with the unknown community labels. We demonstrate that the estimator (31) achieves the exact matching when conditions (7) or (8) are met.
First, for the analysis, we present a lemma that provides an upper bound for ℙ(ℱt)ℙsubscriptℱ𝑡\mathbb{P}\left(\mathcal{F}_{t}\right)blackboard_P ( caligraphic_F start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ), where ℱtsubscriptℱ𝑡\mathcal{F}_{t}caligraphic_F start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT is defined in (33). For t≥2𝑡2t\geq 2italic_t ≥ 2 and α>0𝛼0\alpha>0italic_α > 0, let us define
, 1 = S(α,t):=∑j=1t−1log(1+12α(1−cos(2πtj))),assign𝑆𝛼𝑡superscriptsubscript𝑗1𝑡1112𝛼12𝜋𝑡𝑗S\left(\alpha,t\right):=\sum_{j=1}^{t-1}\log\left(1+\frac{1}{2\alpha}\left(1-%
\cos\left(\frac{2\pi}{t}j\right)\right)\right),italic_S ( italic_α , italic_t ) := ∑ start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t - 1 end_POSTSUPERSCRIPT roman_log ( 1 + divide start_ARG 1 end_ARG start_ARG 2 italic_α end_ARG ( 1 - roman_cos ( divide start_ARG 2 italic_π end_ARG start_ARG italic_t end_ARG italic_j ) ) ) ,. , 2 = . , 3 = (36)
, 1 = I(α):=∫01log(1+12α(1−cos(2πx)))𝑑x.assign𝐼𝛼superscriptsubscript01112𝛼12𝜋𝑥differential-d𝑥I(\alpha):=\int_{0}^{1}\log\left(1+\frac{1}{2\alpha}\left(1-\cos\left(2\pi x%
\right)\right)\right)dx.italic_I ( italic_α ) := ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT roman_log ( 1 + divide start_ARG 1 end_ARG start_ARG 2 italic_α end_ARG ( 1 - roman_cos ( 2 italic_π italic_x ) ) ) italic_d italic_x .. , 2 = . , 3 = (37)
Additionally, let us define two events
, 1 = 𝒜1:={−‖𝝁‖2≤⟨𝝁,𝒛i⟩≤‖𝝁‖2 for all i∈[n]}assignsubscript𝒜1superscriptnorm𝝁2𝝁subscript𝒛𝑖superscriptnorm𝝁2 for all 𝑖delimited-[]𝑛\mathcal{A}_{1}:=\left\{-\|{\boldsymbol{\mu}}\|^{2}\leq\langle{\boldsymbol{\mu%
}},{\boldsymbol{z}}_{i}\rangle\leq\|{\boldsymbol{\mu}}\|^{2}\text{ for all }i%
\in[n]\right\}caligraphic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT := { - ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≤ ⟨ bold_italic_μ , bold_italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⟩ ≤ ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT for all italic_i ∈ [ italic_n ] }. , 2 = . , 3 = (38)
, 1 = 𝒜2:={ρ−λ2∥𝒛i−𝒛j+𝝁+1−λρ−λ𝝁∥2≥ρ−λ2(1+1−λρ−λ)2∥𝝁∥2−2(1−λ)∥𝝁∥2 and ρ−λ2∥𝒛i−𝒛j+𝝁−1−λρ−λ𝝁∥2≥ρ−λ2(1−1−λρ−λ)2∥𝝁∥2 for all distinct i,j∈[n]}.\begin{gathered}\mathcal{A}_{2}:=\left\{\frac{\rho-\lambda}{2}\left\|{%
\boldsymbol{z}}_{i}-{\boldsymbol{z}}_{j}+\boldsymbol{\mu}+\frac{1-\lambda}{%
\rho-\lambda}\boldsymbol{\mu}\right\|^{2}\geq\frac{\rho-\lambda}{2}\left(1+%
\frac{1-\lambda}{\rho-\lambda}\right)^{2}\|\boldsymbol{\mu}\|^{2}-2(1-\lambda)%
\|\boldsymbol{\mu}\|^{2}\text{ and }\right.\\
\left.\frac{\rho-\lambda}{2}\left\|{\boldsymbol{z}}_{i}-{\boldsymbol{z}}_{j}+%
\boldsymbol{\mu}-\frac{1-\lambda}{\rho-\lambda}\boldsymbol{\mu}\right\|^{2}%
\geq\frac{\rho-\lambda}{2}\left(1-\frac{1-\lambda}{\rho-\lambda}\right)^{2}\|%
\boldsymbol{\mu}\|^{2}\text{ for all distinct }i,j\in[n]\right\}.\end{gathered}start_ROW start_CELL caligraphic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT := { divide start_ARG italic_ρ - italic_λ end_ARG start_ARG 2 end_ARG ∥ bold_italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_italic_z start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT + bold_italic_μ + divide start_ARG 1 - italic_λ end_ARG start_ARG italic_ρ - italic_λ end_ARG bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ divide start_ARG italic_ρ - italic_λ end_ARG start_ARG 2 end_ARG ( 1 + divide start_ARG 1 - italic_λ end_ARG start_ARG italic_ρ - italic_λ end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 2 ( 1 - italic_λ ) ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT and end_CELL end_ROW start_ROW start_CELL divide start_ARG italic_ρ - italic_λ end_ARG start_ARG 2 end_ARG ∥ bold_italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_italic_z start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT + bold_italic_μ - divide start_ARG 1 - italic_λ end_ARG start_ARG italic_ρ - italic_λ end_ARG bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ divide start_ARG italic_ρ - italic_λ end_ARG start_ARG 2 end_ARG ( 1 - divide start_ARG 1 - italic_λ end_ARG start_ARG italic_ρ - italic_λ end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT for all distinct italic_i , italic_j ∈ [ italic_n ] } . end_CELL end_ROW. , 2 = . , 3 = (39)
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For any t𝑡titalic_t distinct integers i1,…,it∈[n]subscript𝑖1…subscript𝑖𝑡delimited-[]𝑛i_{1},...,i_{t}\in[n]italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_i start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ∈ [ italic_n ], on the event 𝒜1subscript𝒜1\mathcal{A}_{1}caligraphic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, it holds that
, 1 = ℙ(ℱt)≤exp(−d2S(1−ρ2ρ2,t));ℙsubscriptℱ𝑡𝑑2𝑆1superscript𝜌2superscript𝜌2𝑡\mathbb{P}\left(\mathcal{F}_{t}\right)\leq\exp\left(-\frac{d}{2}S\left(\frac{1%
-\rho^{2}}{\rho^{2}},t\right)\right);blackboard_P ( caligraphic_F start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) ≤ roman_exp ( - divide start_ARG italic_d end_ARG start_ARG 2 end_ARG italic_S ( divide start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG , italic_t ) ) ;. , 2 = . , 3 = (40)
and on the event 𝒜2subscript𝒜2\mathcal{A}_{2}caligraphic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT for some λ∈(0,ρ)𝜆0𝜌\lambda\in(0,\rho)italic_λ ∈ ( 0 , italic_ρ ), it holds that
, 1 = ℙ(ℱt)≤exp(−d2S(1−ρ2λ2,t)).ℙsubscriptℱ𝑡𝑑2𝑆1superscript𝜌2superscript𝜆2𝑡\mathbb{P}\left(\mathcal{F}_{t}\right)\leq\exp\left(-\frac{d}{2}S\left(\frac{1%
-\rho^{2}}{\lambda^{2}},t\right)\right).blackboard_P ( caligraphic_F start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) ≤ roman_exp ( - divide start_ARG italic_d end_ARG start_ARG 2 end_ARG italic_S ( divide start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_λ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG , italic_t ) ) .. , 2 = . , 3 = (41)
The proof of Lemma 1 can be found in Section XIII.
Furthermore, it is known that S(α,t)𝑆𝛼𝑡S(\alpha,t)italic_S ( italic_α , italic_t ) has the following lower bound:
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
VI Proof of Theorem 1:
Achievability of Exact Matching in Correlated Gaussian Mixture Models
Lemma 1.
For any t𝑡titalic_t distinct integers i1,…,it∈[n]subscript𝑖1…subscript𝑖𝑡delimited-[]𝑛i_{1},...,i_{t}\in[n]italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_i start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ∈ [ italic_n ], on the event 𝒜1subscript𝒜1\mathcal{A}_{1}caligraphic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, it holds that
, 1 = ℙ(ℱt)≤exp(−d2S(1−ρ2ρ2,t));ℙsubscriptℱ𝑡𝑑2𝑆1superscript𝜌2superscript𝜌2𝑡\mathbb{P}\left(\mathcal{F}_{t}\right)\leq\exp\left(-\frac{d}{2}S\left(\frac{1%
-\rho^{2}}{\rho^{2}},t\right)\right);blackboard_P ( caligraphic_F start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) ≤ roman_exp ( - divide start_ARG italic_d end_ARG start_ARG 2 end_ARG italic_S ( divide start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG , italic_t ) ) ;. , 2 = . , 3 = (40)
and on the event 𝒜2subscript𝒜2\mathcal{A}_{2}caligraphic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT for some λ∈(0,ρ)𝜆0𝜌\lambda\in(0,\rho)italic_λ ∈ ( 0 , italic_ρ ), it holds that
, 1 = ℙ(ℱt)≤exp(−d2S(1−ρ2λ2,t)).ℙsubscriptℱ𝑡𝑑2𝑆1superscript𝜌2superscript𝜆2𝑡\mathbb{P}\left(\mathcal{F}_{t}\right)\leq\exp\left(-\frac{d}{2}S\left(\frac{1%
-\rho^{2}}{\lambda^{2}},t\right)\right).blackboard_P ( caligraphic_F start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) ≤ roman_exp ( - divide start_ARG italic_d end_ARG start_ARG 2 end_ARG italic_S ( divide start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_λ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG , italic_t ) ) .. , 2 = . , 3 = (41)
The proof of Lemma 1 can be found in Section XIII.
Furthermore, it is known that S(α,t)𝑆𝛼𝑡S(\alpha,t)italic_S ( italic_α , italic_t ) has the following lower bound:
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For all t0≥2subscript𝑡02t_{0}\geq 2italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≥ 2 and t>t0𝑡subscript𝑡0t>t_{0}italic_t > italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, we have
, 1 = S(α,t)>S(α,t0)+(t−t0)I(α),𝑆𝛼𝑡𝑆𝛼subscript𝑡0𝑡subscript𝑡0𝐼𝛼S(\alpha,t)>S(\alpha,t_{0})+(t-t_{0})I(\alpha),italic_S ( italic_α , italic_t ) > italic_S ( italic_α , italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) + ( italic_t - italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) italic_I ( italic_α ) ,. , 2 = . , 3 = (42)
where t0I(α)>S(α,t0)subscript𝑡0𝐼𝛼𝑆𝛼subscript𝑡0t_{0}I(\alpha)>S(\alpha,t_{0})italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_I ( italic_α ) > italic_S ( italic_α , italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ).
Using the above two lemmas, we can prove Theorem 1 as follows:
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
VI Proof of Theorem 1:
Achievability of Exact Matching in Correlated Gaussian Mixture Models
Lemma 2 (Corollary 2.1 in [10]).
For all t0≥2subscript𝑡02t_{0}\geq 2italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≥ 2 and t>t0𝑡subscript𝑡0t>t_{0}italic_t > italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, we have
, 1 = S(α,t)>S(α,t0)+(t−t0)I(α),𝑆𝛼𝑡𝑆𝛼subscript𝑡0𝑡subscript𝑡0𝐼𝛼S(\alpha,t)>S(\alpha,t_{0})+(t-t_{0})I(\alpha),italic_S ( italic_α , italic_t ) > italic_S ( italic_α , italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) + ( italic_t - italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) italic_I ( italic_α ) ,. , 2 = . , 3 = (42)
where t0I(α)>S(α,t0)subscript𝑡0𝐼𝛼𝑆𝛼subscript𝑡0t_{0}I(\alpha)>S(\alpha,t_{0})italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_I ( italic_α ) > italic_S ( italic_α , italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ).
Using the above two lemmas, we can prove Theorem 1 as follows:
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Recall that the estimator we use is π^=argminπ∈Sn∑i=1nZiπ(i)^𝜋subscriptargmin𝜋subscript𝑆𝑛superscriptsubscript𝑖1𝑛subscript𝑍𝑖𝜋𝑖\hat{\pi}=\operatorname*{arg\,min}_{\pi\in S_{n}}\sum\limits_{i=1}^{n}Z_{i\pi(%
i)}over^ start_ARG italic_π end_ARG = start_OPERATOR roman_arg roman_min end_OPERATOR start_POSTSUBSCRIPT italic_π ∈ italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_Z start_POSTSUBSCRIPT italic_i italic_π ( italic_i ) end_POSTSUBSCRIPT. We define the set of mismatched nodes by ℳ:={i∈[n]:π^(i)≠π∗(i)}assignℳconditional-set𝑖delimited-[]𝑛^𝜋𝑖subscript𝜋𝑖\mathcal{M}:=\{i\in[n]:\hat{\pi}(i)\neq\pi_{*}(i)\}caligraphic_M := { italic_i ∈ [ italic_n ] : over^ start_ARG italic_π end_ARG ( italic_i ) ≠ italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_i ) }. To show that π^^𝜋\hat{\pi}over^ start_ARG italic_π end_ARG can achieve exact matching, we will show that ℙ(|ℳ|=0)=1−o(1)ℙℳ01𝑜1\mathbb{P}(|\mathcal{M}|=0)=1-o(1)blackboard_P ( | caligraphic_M | = 0 ) = 1 - italic_o ( 1 ). First, let us show that exact matching is possible when (7) holds. On the event 𝒜1subscript𝒜1\mathcal{A}_{1}caligraphic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, we can obtain
| 511
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
VI Proof of Theorem 1:
Achievability of Exact Matching in Correlated Gaussian Mixture Models
Proof:
Recall that the estimator we use is π^=argminπ∈Sn∑i=1nZiπ(i)^𝜋subscriptargmin𝜋subscript𝑆𝑛superscriptsubscript𝑖1𝑛subscript𝑍𝑖𝜋𝑖\hat{\pi}=\operatorname*{arg\,min}_{\pi\in S_{n}}\sum\limits_{i=1}^{n}Z_{i\pi(%
i)}over^ start_ARG italic_π end_ARG = start_OPERATOR roman_arg roman_min end_OPERATOR start_POSTSUBSCRIPT italic_π ∈ italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_Z start_POSTSUBSCRIPT italic_i italic_π ( italic_i ) end_POSTSUBSCRIPT. We define the set of mismatched nodes by ℳ:={i∈[n]:π^(i)≠π∗(i)}assignℳconditional-set𝑖delimited-[]𝑛^𝜋𝑖subscript𝜋𝑖\mathcal{M}:=\{i\in[n]:\hat{\pi}(i)\neq\pi_{*}(i)\}caligraphic_M := { italic_i ∈ [ italic_n ] : over^ start_ARG italic_π end_ARG ( italic_i ) ≠ italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_i ) }. To show that π^^𝜋\hat{\pi}over^ start_ARG italic_π end_ARG can achieve exact matching, we will show that ℙ(|ℳ|=0)=1−o(1)ℙℳ01𝑜1\mathbb{P}(|\mathcal{M}|=0)=1-o(1)blackboard_P ( | caligraphic_M | = 0 ) = 1 - italic_o ( 1 ). First, let us show that exact matching is possible when (7) holds. On the event 𝒜1subscript𝒜1\mathcal{A}_{1}caligraphic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, we can obtain
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, 1 = 𝔼(|ℳ|)𝔼ℳ\displaystyle\mathbb{E}(|\mathcal{M}|)blackboard_E ( | caligraphic_M | ). , 2 = =∑t=2nℙ(ℱt)(nt)(t−1)!absentsuperscriptsubscript𝑡2𝑛ℙsubscriptℱ𝑡binomial𝑛𝑡𝑡1\displaystyle=\sum\limits_{t=2}^{n}\mathbb{P}(\mathcal{F}_{t}){n\choose t}(t-1)!= ∑ start_POSTSUBSCRIPT italic_t = 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT blackboard_P ( caligraphic_F start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) ( binomial start_ARG italic_n end_ARG start_ARG italic_t end_ARG ) ( italic_t - 1 ) !. , 3 = . , 4 = (43). , 1 = . , 2 = ≤(a)∑t=2nexp(−d2S(1−ρ2ρ2,t))n(n−1)⋯(n−t+1)tsuperscript𝑎absentsuperscriptsubscript𝑡2𝑛𝑑2𝑆1superscript𝜌2superscript𝜌2𝑡𝑛𝑛1⋯𝑛𝑡1𝑡\displaystyle\stackrel{{\scriptstyle(a)}}{{\leq}}\sum\limits_{t=2}^{n}\ \exp%
\left(-\frac{d}{2}S\left(\frac{1-\rho^{2}}{\rho^{2}},t\right)\right)\frac{n(n-%
1)\cdots(n-t+1)}{t}start_RELOP SUPERSCRIPTOP start_ARG ≤ end_ARG start_ARG ( italic_a ) end_ARG end_RELOP ∑ start_POSTSUBSCRIPT italic_t = 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT roman_exp ( - divide start_ARG italic_d end_ARG start_ARG 2 end_ARG italic_S ( divide start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG , italic_t ) ) divide start_ARG italic_n ( italic_n - 1 ) ⋯ ( italic_n - italic_t + 1 ) end_ARG start_ARG italic_t end_ARG. , 3 = . , 4 = (43). , 1 = . , 2 = ≤(b)∑t=2nexp(tlogn−d2S(1−ρ2ρ2,t))superscript𝑏absentsuperscriptsubscript𝑡2𝑛𝑡𝑛𝑑2𝑆1superscript𝜌2superscript𝜌2𝑡\displaystyle\stackrel{{\scriptstyle(b)}}{{\leq}}\sum\limits_{t=2}^{n}\ \exp%
\left(t\log n-\frac{d}{2}S\left(\frac{1-\rho^{2}}{\rho^{2}},t\right)\right)start_RELOP SUPERSCRIPTOP start_ARG ≤ end_ARG start_ARG ( italic_b ) end_ARG end_RELOP ∑ start_POSTSUBSCRIPT italic_t = 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT roman_exp ( italic_t roman_log italic_n - divide start_ARG italic_d end_ARG start_ARG 2 end_ARG italic_S ( divide start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG , italic_t ) ). , 3 = . , 4 = (43). , 1 = . , 2 = ≤(c)∑t=2nexp(tlogn−d2S(1−ρ2ρ2,2)−d2(t−2)I(1−ρ2ρ2))superscript𝑐absentsuperscriptsubscript𝑡2𝑛𝑡𝑛𝑑2𝑆1superscript𝜌2superscript𝜌22𝑑2𝑡2𝐼1superscript𝜌2superscript𝜌2\displaystyle\stackrel{{\scriptstyle(c)}}{{\leq}}\sum\limits_{t=2}^{n}\ \exp%
\left(t\log n-\frac{d}{2}S\left(\frac{1-\rho^{2}}{\rho^{2}},2\right)-\frac{d}{%
2}(t-2)I\left(\frac{1-\rho^{2}}{\rho^{2}}\right)\right)start_RELOP SUPERSCRIPTOP start_ARG ≤ end_ARG start_ARG ( italic_c ) end_ARG end_RELOP ∑ start_POSTSUBSCRIPT italic_t = 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT roman_exp ( italic_t roman_log italic_n - divide start_ARG italic_d end_ARG start_ARG 2 end_ARG italic_S ( divide start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG , 2 ) - divide start_ARG italic_d end_ARG start_ARG 2 end_ARG ( italic_t - 2 ) italic_I ( divide start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) ). , 3 = . , 4 = (43). , 1 = . , 2 = ≤(d)∑t=2nexp(tlogn−dt4S(1−ρ2ρ2,2)).superscript𝑑absentsuperscriptsubscript𝑡2𝑛𝑡𝑛𝑑𝑡4𝑆1superscript𝜌2superscript𝜌22\displaystyle\stackrel{{\scriptstyle(d)}}{{\leq}}\sum\limits_{t=2}^{n}\ \exp%
\left(t\log n-\frac{dt}{4}S\left(\frac{1-\rho^{2}}{\rho^{2}},2\right)\right).start_RELOP SUPERSCRIPTOP start_ARG ≤ end_ARG start_ARG ( italic_d ) end_ARG end_RELOP ∑ start_POSTSUBSCRIPT italic_t = 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT roman_exp ( italic_t roman_log italic_n - divide start_ARG italic_d italic_t end_ARG start_ARG 4 end_ARG italic_S ( divide start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG , 2 ) ) .. , 3 = . , 4 = (43)
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
VI Proof of Theorem 1:
Achievability of Exact Matching in Correlated Gaussian Mixture Models
Proof:
, 1 = 𝔼(|ℳ|)𝔼ℳ\displaystyle\mathbb{E}(|\mathcal{M}|)blackboard_E ( | caligraphic_M | ). , 2 = =∑t=2nℙ(ℱt)(nt)(t−1)!absentsuperscriptsubscript𝑡2𝑛ℙsubscriptℱ𝑡binomial𝑛𝑡𝑡1\displaystyle=\sum\limits_{t=2}^{n}\mathbb{P}(\mathcal{F}_{t}){n\choose t}(t-1)!= ∑ start_POSTSUBSCRIPT italic_t = 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT blackboard_P ( caligraphic_F start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) ( binomial start_ARG italic_n end_ARG start_ARG italic_t end_ARG ) ( italic_t - 1 ) !. , 3 = . , 4 = (43). , 1 = . , 2 = ≤(a)∑t=2nexp(−d2S(1−ρ2ρ2,t))n(n−1)⋯(n−t+1)tsuperscript𝑎absentsuperscriptsubscript𝑡2𝑛𝑑2𝑆1superscript𝜌2superscript𝜌2𝑡𝑛𝑛1⋯𝑛𝑡1𝑡\displaystyle\stackrel{{\scriptstyle(a)}}{{\leq}}\sum\limits_{t=2}^{n}\ \exp%
\left(-\frac{d}{2}S\left(\frac{1-\rho^{2}}{\rho^{2}},t\right)\right)\frac{n(n-%
1)\cdots(n-t+1)}{t}start_RELOP SUPERSCRIPTOP start_ARG ≤ end_ARG start_ARG ( italic_a ) end_ARG end_RELOP ∑ start_POSTSUBSCRIPT italic_t = 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT roman_exp ( - divide start_ARG italic_d end_ARG start_ARG 2 end_ARG italic_S ( divide start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG , italic_t ) ) divide start_ARG italic_n ( italic_n - 1 ) ⋯ ( italic_n - italic_t + 1 ) end_ARG start_ARG italic_t end_ARG. , 3 = . , 4 = (43). , 1 = . , 2 = ≤(b)∑t=2nexp(tlogn−d2S(1−ρ2ρ2,t))superscript𝑏absentsuperscriptsubscript𝑡2𝑛𝑡𝑛𝑑2𝑆1superscript𝜌2superscript𝜌2𝑡\displaystyle\stackrel{{\scriptstyle(b)}}{{\leq}}\sum\limits_{t=2}^{n}\ \exp%
\left(t\log n-\frac{d}{2}S\left(\frac{1-\rho^{2}}{\rho^{2}},t\right)\right)start_RELOP SUPERSCRIPTOP start_ARG ≤ end_ARG start_ARG ( italic_b ) end_ARG end_RELOP ∑ start_POSTSUBSCRIPT italic_t = 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT roman_exp ( italic_t roman_log italic_n - divide start_ARG italic_d end_ARG start_ARG 2 end_ARG italic_S ( divide start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG , italic_t ) ). , 3 = . , 4 = (43). , 1 = . , 2 = ≤(c)∑t=2nexp(tlogn−d2S(1−ρ2ρ2,2)−d2(t−2)I(1−ρ2ρ2))superscript𝑐absentsuperscriptsubscript𝑡2𝑛𝑡𝑛𝑑2𝑆1superscript𝜌2superscript𝜌22𝑑2𝑡2𝐼1superscript𝜌2superscript𝜌2\displaystyle\stackrel{{\scriptstyle(c)}}{{\leq}}\sum\limits_{t=2}^{n}\ \exp%
\left(t\log n-\frac{d}{2}S\left(\frac{1-\rho^{2}}{\rho^{2}},2\right)-\frac{d}{%
2}(t-2)I\left(\frac{1-\rho^{2}}{\rho^{2}}\right)\right)start_RELOP SUPERSCRIPTOP start_ARG ≤ end_ARG start_ARG ( italic_c ) end_ARG end_RELOP ∑ start_POSTSUBSCRIPT italic_t = 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT roman_exp ( italic_t roman_log italic_n - divide start_ARG italic_d end_ARG start_ARG 2 end_ARG italic_S ( divide start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG , 2 ) - divide start_ARG italic_d end_ARG start_ARG 2 end_ARG ( italic_t - 2 ) italic_I ( divide start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) ). , 3 = . , 4 = (43). , 1 = . , 2 = ≤(d)∑t=2nexp(tlogn−dt4S(1−ρ2ρ2,2)).superscript𝑑absentsuperscriptsubscript𝑡2𝑛𝑡𝑛𝑑𝑡4𝑆1superscript𝜌2superscript𝜌22\displaystyle\stackrel{{\scriptstyle(d)}}{{\leq}}\sum\limits_{t=2}^{n}\ \exp%
\left(t\log n-\frac{dt}{4}S\left(\frac{1-\rho^{2}}{\rho^{2}},2\right)\right).start_RELOP SUPERSCRIPTOP start_ARG ≤ end_ARG start_ARG ( italic_d ) end_ARG end_RELOP ∑ start_POSTSUBSCRIPT italic_t = 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT roman_exp ( italic_t roman_log italic_n - divide start_ARG italic_d italic_t end_ARG start_ARG 4 end_ARG italic_S ( divide start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG , 2 ) ) .. , 3 = . , 4 = (43)
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The inequality (a)𝑎(a)( italic_a ) holds by the first result (40) in Lemma 1, the inequality (b)𝑏(b)( italic_b ) holds since n(n−1)⋯(n−t+1)t≤nt𝑛𝑛1⋯𝑛𝑡1𝑡superscript𝑛𝑡\frac{n(n-1)\cdots(n-t+1)}{t}\leq n^{t}divide start_ARG italic_n ( italic_n - 1 ) ⋯ ( italic_n - italic_t + 1 ) end_ARG start_ARG italic_t end_ARG ≤ italic_n start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT, the inequality (c)𝑐(c)( italic_c ) holds by Lemma 2, the inequality (d)𝑑(d)( italic_d ) holds by 12S(1−ρ2ρ2,2)<I(1−ρ2ρ2)12𝑆1superscript𝜌2superscript𝜌22𝐼1superscript𝜌2superscript𝜌2\frac{1}{2}S\left(\frac{1-\rho^{2}}{\rho^{2}},2\right)<I\left(\frac{1-\rho^{2}%
}{\rho^{2}}\right)divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_S ( divide start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG , 2 ) < italic_I ( divide start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ), which can be shown using Lemma 17. From the definition of S(α,t)𝑆𝛼𝑡S(\alpha,t)italic_S ( italic_α , italic_t ) in (36), we have S(1−ρ2ρ2,2)=log11−ρ2𝑆1superscript𝜌2superscript𝜌2211superscript𝜌2S\left(\frac{1-\rho^{2}}{\rho^{2}},2\right)=\log\frac{1}{1-\rho^{2}}italic_S ( divide start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG , 2 ) = roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG. Thus, if d4log11−ρ2≥logn+ω(1)𝑑411superscript𝜌2𝑛𝜔1\frac{d}{4}\log\frac{1}{1-\rho^{2}}\geq\log n+\omega(1)divide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≥ roman_log italic_n + italic_ω ( 1 ), then logn−d4S(1−ρ2ρ2,2)≤−ω(1)𝑛𝑑4𝑆1superscript𝜌2superscript𝜌22𝜔1\log n-\frac{d}{4}S\left(\frac{1-\rho^{2}}{\rho^{2}},2\right)\leq-\omega(1)roman_log italic_n - divide start_ARG italic_d end_ARG start_ARG 4 end_ARG italic_S ( divide start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG , 2 ) ≤ - italic_ω ( 1 ).
Therefore, if (7) holds, then we can have
, 1 = 𝔼(|ℳ|)≤∑t=2nexp(−ω(1)⋅t)=o(1).𝔼ℳsuperscriptsubscript𝑡2𝑛⋅𝜔1𝑡𝑜1\mathbb{E}(|\mathcal{M}|)\leq\sum\limits_{t=2}^{n}\exp\left(-\omega(1)\cdot t%
\right)=o(1).blackboard_E ( | caligraphic_M | ) ≤ ∑ start_POSTSUBSCRIPT italic_t = 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT roman_exp ( - italic_ω ( 1 ) ⋅ italic_t ) = italic_o ( 1 ) .. , 2 = . , 3 = (44)
Finally, by using Markov’s inequality, we can show that
, 1 = ℙ(|ℳ|≥1)≤𝔼(|ℳ|)=o(1).ℙℳ1𝔼ℳ𝑜1\mathbb{P}(|\mathcal{M}|\geq 1)\leq\mathbb{E}(|\mathcal{M}|)=o(1).blackboard_P ( | caligraphic_M | ≥ 1 ) ≤ blackboard_E ( | caligraphic_M | ) = italic_o ( 1 ) .. , 2 = . , 3 = (45)
Moreover, when ∥𝝁∥2≥2logn+ω(1)superscriptdelimited-∥∥𝝁22𝑛𝜔1\lVert\boldsymbol{\mu}\rVert^{2}\geq 2\log n+\omega(1)∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ 2 roman_log italic_n + italic_ω ( 1 ), we can have that
, 1 = ℙ(𝒜1)≥1−nℙ(⟨𝝁,𝒛1⟩≤−∥𝝁∥2 or ⟨𝝁,𝒛1⟩≥∥𝝁∥2)≥1−2ne−12∥𝝁∥2=1−o(1)ℙsubscript𝒜11𝑛ℙ𝝁subscript𝒛1superscriptdelimited-∥∥𝝁2 or 𝝁subscript𝒛1superscriptdelimited-∥∥𝝁212𝑛superscript𝑒12superscriptdelimited-∥∥𝝁21𝑜1\displaystyle\mathbb{P}(\mathcal{A}_{1})\geq 1-n\mathbb{P}(\langle\boldsymbol{%
\mu},{\boldsymbol{z}}_{1}\rangle\leq-\lVert\boldsymbol{\mu}\rVert^{2}\text{ or%
}\langle\boldsymbol{\mu},{\boldsymbol{z}}_{1}\rangle\geq\lVert\boldsymbol{\mu%
}\rVert^{2})\geq 1-2ne^{-\frac{1}{2}\lVert\boldsymbol{\mu}\rVert^{2}}=1-o(1)blackboard_P ( caligraphic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ≥ 1 - italic_n blackboard_P ( ⟨ bold_italic_μ , bold_italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⟩ ≤ - ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT or ⟨ bold_italic_μ , bold_italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⟩ ≥ ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ≥ 1 - 2 italic_n italic_e start_POSTSUPERSCRIPT - divide start_ARG 1 end_ARG start_ARG 2 end_ARG ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT = 1 - italic_o ( 1 ). , 2 = . , 3 = (46)
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
VI Proof of Theorem 1:
Achievability of Exact Matching in Correlated Gaussian Mixture Models
Proof:
The inequality (a)𝑎(a)( italic_a ) holds by the first result (40) in Lemma 1, the inequality (b)𝑏(b)( italic_b ) holds since n(n−1)⋯(n−t+1)t≤nt𝑛𝑛1⋯𝑛𝑡1𝑡superscript𝑛𝑡\frac{n(n-1)\cdots(n-t+1)}{t}\leq n^{t}divide start_ARG italic_n ( italic_n - 1 ) ⋯ ( italic_n - italic_t + 1 ) end_ARG start_ARG italic_t end_ARG ≤ italic_n start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT, the inequality (c)𝑐(c)( italic_c ) holds by Lemma 2, the inequality (d)𝑑(d)( italic_d ) holds by 12S(1−ρ2ρ2,2)<I(1−ρ2ρ2)12𝑆1superscript𝜌2superscript𝜌22𝐼1superscript𝜌2superscript𝜌2\frac{1}{2}S\left(\frac{1-\rho^{2}}{\rho^{2}},2\right)<I\left(\frac{1-\rho^{2}%
}{\rho^{2}}\right)divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_S ( divide start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG , 2 ) < italic_I ( divide start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ), which can be shown using Lemma 17. From the definition of S(α,t)𝑆𝛼𝑡S(\alpha,t)italic_S ( italic_α , italic_t ) in (36), we have S(1−ρ2ρ2,2)=log11−ρ2𝑆1superscript𝜌2superscript𝜌2211superscript𝜌2S\left(\frac{1-\rho^{2}}{\rho^{2}},2\right)=\log\frac{1}{1-\rho^{2}}italic_S ( divide start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG , 2 ) = roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG. Thus, if d4log11−ρ2≥logn+ω(1)𝑑411superscript𝜌2𝑛𝜔1\frac{d}{4}\log\frac{1}{1-\rho^{2}}\geq\log n+\omega(1)divide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≥ roman_log italic_n + italic_ω ( 1 ), then logn−d4S(1−ρ2ρ2,2)≤−ω(1)𝑛𝑑4𝑆1superscript𝜌2superscript𝜌22𝜔1\log n-\frac{d}{4}S\left(\frac{1-\rho^{2}}{\rho^{2}},2\right)\leq-\omega(1)roman_log italic_n - divide start_ARG italic_d end_ARG start_ARG 4 end_ARG italic_S ( divide start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG , 2 ) ≤ - italic_ω ( 1 ).
Therefore, if (7) holds, then we can have
, 1 = 𝔼(|ℳ|)≤∑t=2nexp(−ω(1)⋅t)=o(1).𝔼ℳsuperscriptsubscript𝑡2𝑛⋅𝜔1𝑡𝑜1\mathbb{E}(|\mathcal{M}|)\leq\sum\limits_{t=2}^{n}\exp\left(-\omega(1)\cdot t%
\right)=o(1).blackboard_E ( | caligraphic_M | ) ≤ ∑ start_POSTSUBSCRIPT italic_t = 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT roman_exp ( - italic_ω ( 1 ) ⋅ italic_t ) = italic_o ( 1 ) .. , 2 = . , 3 = (44)
Finally, by using Markov’s inequality, we can show that
, 1 = ℙ(|ℳ|≥1)≤𝔼(|ℳ|)=o(1).ℙℳ1𝔼ℳ𝑜1\mathbb{P}(|\mathcal{M}|\geq 1)\leq\mathbb{E}(|\mathcal{M}|)=o(1).blackboard_P ( | caligraphic_M | ≥ 1 ) ≤ blackboard_E ( | caligraphic_M | ) = italic_o ( 1 ) .. , 2 = . , 3 = (45)
Moreover, when ∥𝝁∥2≥2logn+ω(1)superscriptdelimited-∥∥𝝁22𝑛𝜔1\lVert\boldsymbol{\mu}\rVert^{2}\geq 2\log n+\omega(1)∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ 2 roman_log italic_n + italic_ω ( 1 ), we can have that
, 1 = ℙ(𝒜1)≥1−nℙ(⟨𝝁,𝒛1⟩≤−∥𝝁∥2 or ⟨𝝁,𝒛1⟩≥∥𝝁∥2)≥1−2ne−12∥𝝁∥2=1−o(1)ℙsubscript𝒜11𝑛ℙ𝝁subscript𝒛1superscriptdelimited-∥∥𝝁2 or 𝝁subscript𝒛1superscriptdelimited-∥∥𝝁212𝑛superscript𝑒12superscriptdelimited-∥∥𝝁21𝑜1\displaystyle\mathbb{P}(\mathcal{A}_{1})\geq 1-n\mathbb{P}(\langle\boldsymbol{%
\mu},{\boldsymbol{z}}_{1}\rangle\leq-\lVert\boldsymbol{\mu}\rVert^{2}\text{ or%
}\langle\boldsymbol{\mu},{\boldsymbol{z}}_{1}\rangle\geq\lVert\boldsymbol{\mu%
}\rVert^{2})\geq 1-2ne^{-\frac{1}{2}\lVert\boldsymbol{\mu}\rVert^{2}}=1-o(1)blackboard_P ( caligraphic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ≥ 1 - italic_n blackboard_P ( ⟨ bold_italic_μ , bold_italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⟩ ≤ - ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT or ⟨ bold_italic_μ , bold_italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⟩ ≥ ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ≥ 1 - 2 italic_n italic_e start_POSTSUPERSCRIPT - divide start_ARG 1 end_ARG start_ARG 2 end_ARG ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT = 1 - italic_o ( 1 ). , 2 = . , 3 = (46)
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by the tail bound of normal distributions (Lemma 19).
Therefore, it holds that ℙ(|ℳ|=0)=1−o(1)ℙℳ01𝑜1\mathbb{P}(|\mathcal{M}|=0)=1-o(1)blackboard_P ( | caligraphic_M | = 0 ) = 1 - italic_o ( 1 ).
We will next show that exact matching is possible when (8) holds. Suppose that ∥𝝁∥2=O(logn)superscriptdelimited-∥∥𝝁2𝑂𝑛\lVert\boldsymbol{\mu}\rVert^{2}=O(\log n)∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = italic_O ( roman_log italic_n ).
On the event 𝒜2subscript𝒜2\mathcal{A}_{2}caligraphic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, we can obtain that
, 1 = 𝔼(|ℳ|)𝔼ℳ\displaystyle\mathbb{E}(|\mathcal{M}|)blackboard_E ( | caligraphic_M | ). , 2 = =∑t=2nℙ(ℱt)(nt)(t−1)!absentsuperscriptsubscript𝑡2𝑛ℙsubscriptℱ𝑡binomial𝑛𝑡𝑡1\displaystyle=\sum\limits_{t=2}^{n}\mathbb{P}(\mathcal{F}_{t}){n\choose t}(t-1)!= ∑ start_POSTSUBSCRIPT italic_t = 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT blackboard_P ( caligraphic_F start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) ( binomial start_ARG italic_n end_ARG start_ARG italic_t end_ARG ) ( italic_t - 1 ) !. , 3 = . , 4 = (47). , 1 = . , 2 = ≤(e)∑t=2nexp(−d2S(1−ρ2λ2,t))n(n−1)⋯(n−t+1)tsuperscript𝑒absentsuperscriptsubscript𝑡2𝑛𝑑2𝑆1superscript𝜌2superscript𝜆2𝑡𝑛𝑛1⋯𝑛𝑡1𝑡\displaystyle\stackrel{{\scriptstyle(e)}}{{\leq}}\sum\limits_{t=2}^{n}\ \exp%
\left(-\frac{d}{2}S\left(\frac{1-\rho^{2}}{\lambda^{2}},t\right)\right)\frac{n%
(n-1)\cdots(n-t+1)}{t}start_RELOP SUPERSCRIPTOP start_ARG ≤ end_ARG start_ARG ( italic_e ) end_ARG end_RELOP ∑ start_POSTSUBSCRIPT italic_t = 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT roman_exp ( - divide start_ARG italic_d end_ARG start_ARG 2 end_ARG italic_S ( divide start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_λ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG , italic_t ) ) divide start_ARG italic_n ( italic_n - 1 ) ⋯ ( italic_n - italic_t + 1 ) end_ARG start_ARG italic_t end_ARG. , 3 = . , 4 = (47). , 1 = . , 2 = ≤∑t=2nexp(tlogn−d2S(1−ρ2λ2,t))absentsuperscriptsubscript𝑡2𝑛𝑡𝑛𝑑2𝑆1superscript𝜌2superscript𝜆2𝑡\displaystyle\leq\sum\limits_{t=2}^{n}\ \exp\left(t\log n-\frac{d}{2}S\left(%
\frac{1-\rho^{2}}{\lambda^{2}},t\right)\right)≤ ∑ start_POSTSUBSCRIPT italic_t = 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT roman_exp ( italic_t roman_log italic_n - divide start_ARG italic_d end_ARG start_ARG 2 end_ARG italic_S ( divide start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_λ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG , italic_t ) ). , 3 = . , 4 = (47). , 1 = . , 2 = ≤(f)∑t=2nexp(tlogn−d2S(1−ρ2λ2,2)−d2(t−2)I(1−ρ2λ2))superscript𝑓absentsuperscriptsubscript𝑡2𝑛𝑡𝑛𝑑2𝑆1superscript𝜌2superscript𝜆22𝑑2𝑡2𝐼1superscript𝜌2superscript𝜆2\displaystyle\stackrel{{\scriptstyle(f)}}{{\leq}}\sum\limits_{t=2}^{n}\ \exp%
\left(t\log n-\frac{d}{2}S\left(\frac{1-\rho^{2}}{\lambda^{2}},2\right)-\frac{%
d}{2}(t-2)I\left(\frac{1-\rho^{2}}{\lambda^{2}}\right)\right)start_RELOP SUPERSCRIPTOP start_ARG ≤ end_ARG start_ARG ( italic_f ) end_ARG end_RELOP ∑ start_POSTSUBSCRIPT italic_t = 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT roman_exp ( italic_t roman_log italic_n - divide start_ARG italic_d end_ARG start_ARG 2 end_ARG italic_S ( divide start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_λ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG , 2 ) - divide start_ARG italic_d end_ARG start_ARG 2 end_ARG ( italic_t - 2 ) italic_I ( divide start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_λ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) ). , 3 = . , 4 = (47). , 1 = . , 2 = ≤(g)∑t=2nexp(tlogn−dt4S(1−ρ2λ2,2)).superscript𝑔absentsuperscriptsubscript𝑡2𝑛𝑡𝑛𝑑𝑡4𝑆1superscript𝜌2superscript𝜆22\displaystyle\stackrel{{\scriptstyle(g)}}{{\leq}}\sum\limits_{t=2}^{n}\ \exp%
\left(t\log n-\frac{dt}{4}S\left(\frac{1-\rho^{2}}{\lambda^{2}},2\right)\right).start_RELOP SUPERSCRIPTOP start_ARG ≤ end_ARG start_ARG ( italic_g ) end_ARG end_RELOP ∑ start_POSTSUBSCRIPT italic_t = 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT roman_exp ( italic_t roman_log italic_n - divide start_ARG italic_d italic_t end_ARG start_ARG 4 end_ARG italic_S ( divide start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_λ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG , 2 ) ) .. , 3 = . , 4 = (47)
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
VI Proof of Theorem 1:
Achievability of Exact Matching in Correlated Gaussian Mixture Models
Proof:
by the tail bound of normal distributions (Lemma 19).
Therefore, it holds that ℙ(|ℳ|=0)=1−o(1)ℙℳ01𝑜1\mathbb{P}(|\mathcal{M}|=0)=1-o(1)blackboard_P ( | caligraphic_M | = 0 ) = 1 - italic_o ( 1 ).
We will next show that exact matching is possible when (8) holds. Suppose that ∥𝝁∥2=O(logn)superscriptdelimited-∥∥𝝁2𝑂𝑛\lVert\boldsymbol{\mu}\rVert^{2}=O(\log n)∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = italic_O ( roman_log italic_n ).
On the event 𝒜2subscript𝒜2\mathcal{A}_{2}caligraphic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, we can obtain that
, 1 = 𝔼(|ℳ|)𝔼ℳ\displaystyle\mathbb{E}(|\mathcal{M}|)blackboard_E ( | caligraphic_M | ). , 2 = =∑t=2nℙ(ℱt)(nt)(t−1)!absentsuperscriptsubscript𝑡2𝑛ℙsubscriptℱ𝑡binomial𝑛𝑡𝑡1\displaystyle=\sum\limits_{t=2}^{n}\mathbb{P}(\mathcal{F}_{t}){n\choose t}(t-1)!= ∑ start_POSTSUBSCRIPT italic_t = 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT blackboard_P ( caligraphic_F start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) ( binomial start_ARG italic_n end_ARG start_ARG italic_t end_ARG ) ( italic_t - 1 ) !. , 3 = . , 4 = (47). , 1 = . , 2 = ≤(e)∑t=2nexp(−d2S(1−ρ2λ2,t))n(n−1)⋯(n−t+1)tsuperscript𝑒absentsuperscriptsubscript𝑡2𝑛𝑑2𝑆1superscript𝜌2superscript𝜆2𝑡𝑛𝑛1⋯𝑛𝑡1𝑡\displaystyle\stackrel{{\scriptstyle(e)}}{{\leq}}\sum\limits_{t=2}^{n}\ \exp%
\left(-\frac{d}{2}S\left(\frac{1-\rho^{2}}{\lambda^{2}},t\right)\right)\frac{n%
(n-1)\cdots(n-t+1)}{t}start_RELOP SUPERSCRIPTOP start_ARG ≤ end_ARG start_ARG ( italic_e ) end_ARG end_RELOP ∑ start_POSTSUBSCRIPT italic_t = 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT roman_exp ( - divide start_ARG italic_d end_ARG start_ARG 2 end_ARG italic_S ( divide start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_λ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG , italic_t ) ) divide start_ARG italic_n ( italic_n - 1 ) ⋯ ( italic_n - italic_t + 1 ) end_ARG start_ARG italic_t end_ARG. , 3 = . , 4 = (47). , 1 = . , 2 = ≤∑t=2nexp(tlogn−d2S(1−ρ2λ2,t))absentsuperscriptsubscript𝑡2𝑛𝑡𝑛𝑑2𝑆1superscript𝜌2superscript𝜆2𝑡\displaystyle\leq\sum\limits_{t=2}^{n}\ \exp\left(t\log n-\frac{d}{2}S\left(%
\frac{1-\rho^{2}}{\lambda^{2}},t\right)\right)≤ ∑ start_POSTSUBSCRIPT italic_t = 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT roman_exp ( italic_t roman_log italic_n - divide start_ARG italic_d end_ARG start_ARG 2 end_ARG italic_S ( divide start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_λ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG , italic_t ) ). , 3 = . , 4 = (47). , 1 = . , 2 = ≤(f)∑t=2nexp(tlogn−d2S(1−ρ2λ2,2)−d2(t−2)I(1−ρ2λ2))superscript𝑓absentsuperscriptsubscript𝑡2𝑛𝑡𝑛𝑑2𝑆1superscript𝜌2superscript𝜆22𝑑2𝑡2𝐼1superscript𝜌2superscript𝜆2\displaystyle\stackrel{{\scriptstyle(f)}}{{\leq}}\sum\limits_{t=2}^{n}\ \exp%
\left(t\log n-\frac{d}{2}S\left(\frac{1-\rho^{2}}{\lambda^{2}},2\right)-\frac{%
d}{2}(t-2)I\left(\frac{1-\rho^{2}}{\lambda^{2}}\right)\right)start_RELOP SUPERSCRIPTOP start_ARG ≤ end_ARG start_ARG ( italic_f ) end_ARG end_RELOP ∑ start_POSTSUBSCRIPT italic_t = 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT roman_exp ( italic_t roman_log italic_n - divide start_ARG italic_d end_ARG start_ARG 2 end_ARG italic_S ( divide start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_λ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG , 2 ) - divide start_ARG italic_d end_ARG start_ARG 2 end_ARG ( italic_t - 2 ) italic_I ( divide start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_λ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) ). , 3 = . , 4 = (47). , 1 = . , 2 = ≤(g)∑t=2nexp(tlogn−dt4S(1−ρ2λ2,2)).superscript𝑔absentsuperscriptsubscript𝑡2𝑛𝑡𝑛𝑑𝑡4𝑆1superscript𝜌2superscript𝜆22\displaystyle\stackrel{{\scriptstyle(g)}}{{\leq}}\sum\limits_{t=2}^{n}\ \exp%
\left(t\log n-\frac{dt}{4}S\left(\frac{1-\rho^{2}}{\lambda^{2}},2\right)\right).start_RELOP SUPERSCRIPTOP start_ARG ≤ end_ARG start_ARG ( italic_g ) end_ARG end_RELOP ∑ start_POSTSUBSCRIPT italic_t = 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT roman_exp ( italic_t roman_log italic_n - divide start_ARG italic_d italic_t end_ARG start_ARG 4 end_ARG italic_S ( divide start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_λ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG , 2 ) ) .. , 3 = . , 4 = (47)
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The inequality (e)𝑒(e)( italic_e ) holds by (41) in Lemma 1, the inequality (f)𝑓(f)( italic_f ) holds by Lemma 2, and the inequality (g)𝑔(g)( italic_g ) holds by 12S(1−ρ2λ2,2)<I(1−ρ2λ2)12𝑆1superscript𝜌2superscript𝜆22𝐼1superscript𝜌2superscript𝜆2\frac{1}{2}S\left(\frac{1-\rho^{2}}{\lambda^{2}},2\right)<I\left(\frac{1-\rho^%
{2}}{\lambda^{2}}\right)divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_S ( divide start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_λ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG , 2 ) < italic_I ( divide start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_λ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ), which can be shown using Lemma 17. Moreover, when d4log11−ρ2≥(1+ϵ)logn𝑑411superscript𝜌21italic-ϵ𝑛\frac{d}{4}\log\frac{1}{1-\rho^{2}}\geq(1+\epsilon)\log ndivide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≥ ( 1 + italic_ϵ ) roman_log italic_n, there exists some constant r∈(0,1)𝑟01r\in(0,1)italic_r ∈ ( 0 , 1 ), depending on ϵitalic-ϵ\epsilonitalic_ϵ, such that d4log(1+r2ρ21−ρ2)≥(1+ϵ/2)logn𝑑41superscript𝑟2superscript𝜌21superscript𝜌21italic-ϵ2𝑛\frac{d}{4}\log\left(1+\frac{r^{2}\rho^{2}}{1-\rho^{2}}\right)\geq(1+\epsilon/%
2)\log ndivide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log ( 1 + divide start_ARG italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) ≥ ( 1 + italic_ϵ / 2 ) roman_log italic_n. For such a r𝑟ritalic_r, let λ=rρ𝜆𝑟𝜌\lambda=r\rhoitalic_λ = italic_r italic_ρ. From the definition of S(α,t)𝑆𝛼𝑡S(\alpha,t)italic_S ( italic_α , italic_t ) in (36), we have S(1−ρ2λ2,2)=log(1+r2ρ21−ρ2)𝑆1superscript𝜌2superscript𝜆221superscript𝑟2superscript𝜌21superscript𝜌2S\left(\frac{1-\rho^{2}}{\lambda^{2}},2\right)=\log\left(1+\frac{r^{2}\rho^{2}%
}{1-\rho^{2}}\right)italic_S ( divide start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_λ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG , 2 ) = roman_log ( 1 + divide start_ARG italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ).
Thus, we have logn−d4S(1−ρ2λ2,2)≤−ϵ2logn𝑛𝑑4𝑆1superscript𝜌2superscript𝜆22italic-ϵ2𝑛\log n-\frac{d}{4}S\left(\frac{1-\rho^{2}}{\lambda^{2}},2\right)\leq-\frac{%
\epsilon}{2}\log nroman_log italic_n - divide start_ARG italic_d end_ARG start_ARG 4 end_ARG italic_S ( divide start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_λ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG , 2 ) ≤ - divide start_ARG italic_ϵ end_ARG start_ARG 2 end_ARG roman_log italic_n. So if (8) holds, we can get
, 1 = 𝔼(|ℳ|)≤∑t=2nexp(−tϵ2logn)=o(1).𝔼ℳsuperscriptsubscript𝑡2𝑛𝑡italic-ϵ2𝑛𝑜1\mathbb{E}(|\mathcal{M}|)\leq\sum\limits_{t=2}^{n}\exp\left(-t\frac{\epsilon}{%
2}\log n\right)=o(1).blackboard_E ( | caligraphic_M | ) ≤ ∑ start_POSTSUBSCRIPT italic_t = 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT roman_exp ( - italic_t divide start_ARG italic_ϵ end_ARG start_ARG 2 end_ARG roman_log italic_n ) = italic_o ( 1 ) .. , 2 = . , 3 = (48)
Finally, by using Markov’s inequality, we can have that
, 1 = ℙ(|ℳ|≥1)≤𝔼(|ℳ|)=o(1).ℙℳ1𝔼ℳ𝑜1\mathbb{P}(|\mathcal{M}|\geq 1)\leq\mathbb{E}(|\mathcal{M}|)=o(1).blackboard_P ( | caligraphic_M | ≥ 1 ) ≤ blackboard_E ( | caligraphic_M | ) = italic_o ( 1 ) .. , 2 = . , 3 = (49)
Lastly, we will show that when d=ω(logn)𝑑𝜔𝑛d=\omega(\log n)italic_d = italic_ω ( roman_log italic_n ), ℙ(𝒜2)≥1−o(1)ℙsubscript𝒜21𝑜1\mathbb{P}(\mathcal{A}_{2})\geq 1-o(1)blackboard_P ( caligraphic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ≥ 1 - italic_o ( 1 ), and thus it holds that ℙ(|ℳ|=0)=1−o(1)ℙℳ01𝑜1\mathbb{P}(|\mathcal{M}|=0)=1-o(1)blackboard_P ( | caligraphic_M | = 0 ) = 1 - italic_o ( 1 ).
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
VI Proof of Theorem 1:
Achievability of Exact Matching in Correlated Gaussian Mixture Models
Proof:
The inequality (e)𝑒(e)( italic_e ) holds by (41) in Lemma 1, the inequality (f)𝑓(f)( italic_f ) holds by Lemma 2, and the inequality (g)𝑔(g)( italic_g ) holds by 12S(1−ρ2λ2,2)<I(1−ρ2λ2)12𝑆1superscript𝜌2superscript𝜆22𝐼1superscript𝜌2superscript𝜆2\frac{1}{2}S\left(\frac{1-\rho^{2}}{\lambda^{2}},2\right)<I\left(\frac{1-\rho^%
{2}}{\lambda^{2}}\right)divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_S ( divide start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_λ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG , 2 ) < italic_I ( divide start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_λ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ), which can be shown using Lemma 17. Moreover, when d4log11−ρ2≥(1+ϵ)logn𝑑411superscript𝜌21italic-ϵ𝑛\frac{d}{4}\log\frac{1}{1-\rho^{2}}\geq(1+\epsilon)\log ndivide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≥ ( 1 + italic_ϵ ) roman_log italic_n, there exists some constant r∈(0,1)𝑟01r\in(0,1)italic_r ∈ ( 0 , 1 ), depending on ϵitalic-ϵ\epsilonitalic_ϵ, such that d4log(1+r2ρ21−ρ2)≥(1+ϵ/2)logn𝑑41superscript𝑟2superscript𝜌21superscript𝜌21italic-ϵ2𝑛\frac{d}{4}\log\left(1+\frac{r^{2}\rho^{2}}{1-\rho^{2}}\right)\geq(1+\epsilon/%
2)\log ndivide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log ( 1 + divide start_ARG italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) ≥ ( 1 + italic_ϵ / 2 ) roman_log italic_n. For such a r𝑟ritalic_r, let λ=rρ𝜆𝑟𝜌\lambda=r\rhoitalic_λ = italic_r italic_ρ. From the definition of S(α,t)𝑆𝛼𝑡S(\alpha,t)italic_S ( italic_α , italic_t ) in (36), we have S(1−ρ2λ2,2)=log(1+r2ρ21−ρ2)𝑆1superscript𝜌2superscript𝜆221superscript𝑟2superscript𝜌21superscript𝜌2S\left(\frac{1-\rho^{2}}{\lambda^{2}},2\right)=\log\left(1+\frac{r^{2}\rho^{2}%
}{1-\rho^{2}}\right)italic_S ( divide start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_λ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG , 2 ) = roman_log ( 1 + divide start_ARG italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ).
Thus, we have logn−d4S(1−ρ2λ2,2)≤−ϵ2logn𝑛𝑑4𝑆1superscript𝜌2superscript𝜆22italic-ϵ2𝑛\log n-\frac{d}{4}S\left(\frac{1-\rho^{2}}{\lambda^{2}},2\right)\leq-\frac{%
\epsilon}{2}\log nroman_log italic_n - divide start_ARG italic_d end_ARG start_ARG 4 end_ARG italic_S ( divide start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_λ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG , 2 ) ≤ - divide start_ARG italic_ϵ end_ARG start_ARG 2 end_ARG roman_log italic_n. So if (8) holds, we can get
, 1 = 𝔼(|ℳ|)≤∑t=2nexp(−tϵ2logn)=o(1).𝔼ℳsuperscriptsubscript𝑡2𝑛𝑡italic-ϵ2𝑛𝑜1\mathbb{E}(|\mathcal{M}|)\leq\sum\limits_{t=2}^{n}\exp\left(-t\frac{\epsilon}{%
2}\log n\right)=o(1).blackboard_E ( | caligraphic_M | ) ≤ ∑ start_POSTSUBSCRIPT italic_t = 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT roman_exp ( - italic_t divide start_ARG italic_ϵ end_ARG start_ARG 2 end_ARG roman_log italic_n ) = italic_o ( 1 ) .. , 2 = . , 3 = (48)
Finally, by using Markov’s inequality, we can have that
, 1 = ℙ(|ℳ|≥1)≤𝔼(|ℳ|)=o(1).ℙℳ1𝔼ℳ𝑜1\mathbb{P}(|\mathcal{M}|\geq 1)\leq\mathbb{E}(|\mathcal{M}|)=o(1).blackboard_P ( | caligraphic_M | ≥ 1 ) ≤ blackboard_E ( | caligraphic_M | ) = italic_o ( 1 ) .. , 2 = . , 3 = (49)
Lastly, we will show that when d=ω(logn)𝑑𝜔𝑛d=\omega(\log n)italic_d = italic_ω ( roman_log italic_n ), ℙ(𝒜2)≥1−o(1)ℙsubscript𝒜21𝑜1\mathbb{P}(\mathcal{A}_{2})\geq 1-o(1)blackboard_P ( caligraphic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ≥ 1 - italic_o ( 1 ), and thus it holds that ℙ(|ℳ|=0)=1−o(1)ℙℳ01𝑜1\mathbb{P}(|\mathcal{M}|=0)=1-o(1)blackboard_P ( | caligraphic_M | = 0 ) = 1 - italic_o ( 1 ).
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For 𝒛1,𝒛2∼𝒩(0,𝑰d)similar-tosubscript𝒛1subscript𝒛2𝒩0subscript𝑰𝑑{\boldsymbol{z}}_{1},{\boldsymbol{z}}_{2}\sim\mathcal{N}(0,{\boldsymbol{I}}_{d})bold_italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , bold_italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∼ caligraphic_N ( 0 , bold_italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ), let 𝒛=𝒛1−𝒛22∼𝒩(0,𝑰d)𝒛subscript𝒛1subscript𝒛22similar-to𝒩0subscript𝑰𝑑{\boldsymbol{z}}=\frac{{\boldsymbol{z}}_{1}-{\boldsymbol{z}}_{2}}{\sqrt{2}}%
\sim\mathcal{N}(0,{\boldsymbol{I}}_{d})bold_italic_z = divide start_ARG bold_italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - bold_italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG start_ARG square-root start_ARG 2 end_ARG end_ARG ∼ caligraphic_N ( 0 , bold_italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ).
We can have that
, 1 = . , 2 = ℙ(ρ−λ2‖𝒛1−𝒛2+𝝁+1−λρ−λ𝝁‖2≤ρ−λ2(1+1−λρ−λ)2‖𝝁‖2−2(1−λ)‖𝝁‖2)ℙ𝜌𝜆2superscriptnormsubscript𝒛1subscript𝒛2𝝁1𝜆𝜌𝜆𝝁2𝜌𝜆2superscript11𝜆𝜌𝜆2superscriptnorm𝝁221𝜆superscriptnorm𝝁2\displaystyle\mathbb{P}\left(\frac{\rho-\lambda}{2}\left\|{\boldsymbol{z}}_{1}%
-{\boldsymbol{z}}_{2}+\boldsymbol{\mu}+\frac{1-\lambda}{\rho-\lambda}%
\boldsymbol{\mu}\right\|^{2}\leq\frac{\rho-\lambda}{2}\left(1+\frac{1-\lambda}%
{\rho-\lambda}\right)^{2}\|\boldsymbol{\mu}\|^{2}-2(1-\lambda)\|\boldsymbol{%
\mu}\|^{2}\right)blackboard_P ( divide start_ARG italic_ρ - italic_λ end_ARG start_ARG 2 end_ARG ∥ bold_italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - bold_italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + bold_italic_μ + divide start_ARG 1 - italic_λ end_ARG start_ARG italic_ρ - italic_λ end_ARG bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≤ divide start_ARG italic_ρ - italic_λ end_ARG start_ARG 2 end_ARG ( 1 + divide start_ARG 1 - italic_λ end_ARG start_ARG italic_ρ - italic_λ end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 2 ( 1 - italic_λ ) ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ). , 3 = . , 4 = (50). , 1 = . , 2 = =ℙ(‖𝒛+ρ−2λ+12(ρ−λ)𝝁‖2≤12(ρ−2λ+1ρ−λ)2‖𝝁‖2−2(1−λ)ρ−λ‖𝝁‖2).absentℙsuperscriptnorm𝒛𝜌2𝜆12𝜌𝜆𝝁212superscript𝜌2𝜆1𝜌𝜆2superscriptnorm𝝁221𝜆𝜌𝜆superscriptnorm𝝁2\displaystyle=\mathbb{P}\left(\left\|{\boldsymbol{z}}+\frac{\rho-2\lambda+1}{%
\sqrt{2}(\rho-\lambda)}\boldsymbol{\mu}\right\|^{2}\leq\frac{1}{2}\left(\frac{%
\rho-2\lambda+1}{\rho-\lambda}\right)^{2}\|\boldsymbol{\mu}\|^{2}-\frac{2(1-%
\lambda)}{\rho-\lambda}\|\boldsymbol{\mu}\|^{2}\right).= blackboard_P ( ∥ bold_italic_z + divide start_ARG italic_ρ - 2 italic_λ + 1 end_ARG start_ARG square-root start_ARG 2 end_ARG ( italic_ρ - italic_λ ) end_ARG bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≤ divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( divide start_ARG italic_ρ - 2 italic_λ + 1 end_ARG start_ARG italic_ρ - italic_λ end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - divide start_ARG 2 ( 1 - italic_λ ) end_ARG start_ARG italic_ρ - italic_λ end_ARG ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) .. , 3 = . , 4 = (50)
| 1,357
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
VI Proof of Theorem 1:
Achievability of Exact Matching in Correlated Gaussian Mixture Models
Proof:
For 𝒛1,𝒛2∼𝒩(0,𝑰d)similar-tosubscript𝒛1subscript𝒛2𝒩0subscript𝑰𝑑{\boldsymbol{z}}_{1},{\boldsymbol{z}}_{2}\sim\mathcal{N}(0,{\boldsymbol{I}}_{d})bold_italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , bold_italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∼ caligraphic_N ( 0 , bold_italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ), let 𝒛=𝒛1−𝒛22∼𝒩(0,𝑰d)𝒛subscript𝒛1subscript𝒛22similar-to𝒩0subscript𝑰𝑑{\boldsymbol{z}}=\frac{{\boldsymbol{z}}_{1}-{\boldsymbol{z}}_{2}}{\sqrt{2}}%
\sim\mathcal{N}(0,{\boldsymbol{I}}_{d})bold_italic_z = divide start_ARG bold_italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - bold_italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG start_ARG square-root start_ARG 2 end_ARG end_ARG ∼ caligraphic_N ( 0 , bold_italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ).
We can have that
, 1 = . , 2 = ℙ(ρ−λ2‖𝒛1−𝒛2+𝝁+1−λρ−λ𝝁‖2≤ρ−λ2(1+1−λρ−λ)2‖𝝁‖2−2(1−λ)‖𝝁‖2)ℙ𝜌𝜆2superscriptnormsubscript𝒛1subscript𝒛2𝝁1𝜆𝜌𝜆𝝁2𝜌𝜆2superscript11𝜆𝜌𝜆2superscriptnorm𝝁221𝜆superscriptnorm𝝁2\displaystyle\mathbb{P}\left(\frac{\rho-\lambda}{2}\left\|{\boldsymbol{z}}_{1}%
-{\boldsymbol{z}}_{2}+\boldsymbol{\mu}+\frac{1-\lambda}{\rho-\lambda}%
\boldsymbol{\mu}\right\|^{2}\leq\frac{\rho-\lambda}{2}\left(1+\frac{1-\lambda}%
{\rho-\lambda}\right)^{2}\|\boldsymbol{\mu}\|^{2}-2(1-\lambda)\|\boldsymbol{%
\mu}\|^{2}\right)blackboard_P ( divide start_ARG italic_ρ - italic_λ end_ARG start_ARG 2 end_ARG ∥ bold_italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - bold_italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + bold_italic_μ + divide start_ARG 1 - italic_λ end_ARG start_ARG italic_ρ - italic_λ end_ARG bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≤ divide start_ARG italic_ρ - italic_λ end_ARG start_ARG 2 end_ARG ( 1 + divide start_ARG 1 - italic_λ end_ARG start_ARG italic_ρ - italic_λ end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 2 ( 1 - italic_λ ) ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ). , 3 = . , 4 = (50). , 1 = . , 2 = =ℙ(‖𝒛+ρ−2λ+12(ρ−λ)𝝁‖2≤12(ρ−2λ+1ρ−λ)2‖𝝁‖2−2(1−λ)ρ−λ‖𝝁‖2).absentℙsuperscriptnorm𝒛𝜌2𝜆12𝜌𝜆𝝁212superscript𝜌2𝜆1𝜌𝜆2superscriptnorm𝝁221𝜆𝜌𝜆superscriptnorm𝝁2\displaystyle=\mathbb{P}\left(\left\|{\boldsymbol{z}}+\frac{\rho-2\lambda+1}{%
\sqrt{2}(\rho-\lambda)}\boldsymbol{\mu}\right\|^{2}\leq\frac{1}{2}\left(\frac{%
\rho-2\lambda+1}{\rho-\lambda}\right)^{2}\|\boldsymbol{\mu}\|^{2}-\frac{2(1-%
\lambda)}{\rho-\lambda}\|\boldsymbol{\mu}\|^{2}\right).= blackboard_P ( ∥ bold_italic_z + divide start_ARG italic_ρ - 2 italic_λ + 1 end_ARG start_ARG square-root start_ARG 2 end_ARG ( italic_ρ - italic_λ ) end_ARG bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≤ divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( divide start_ARG italic_ρ - 2 italic_λ + 1 end_ARG start_ARG italic_ρ - italic_λ end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - divide start_ARG 2 ( 1 - italic_λ ) end_ARG start_ARG italic_ρ - italic_λ end_ARG ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) .. , 3 = . , 4 = (50)
| 1,396
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60
|
Then, ‖𝒛+ρ−2λ+12(ρ−λ)𝝁‖2superscriptnorm𝒛𝜌2𝜆12𝜌𝜆𝝁2\left\|{\boldsymbol{z}}+\frac{\rho-2\lambda+1}{\sqrt{2}(\rho-\lambda)}{%
\boldsymbol{\mu}}\right\|^{2}∥ bold_italic_z + divide start_ARG italic_ρ - 2 italic_λ + 1 end_ARG start_ARG square-root start_ARG 2 end_ARG ( italic_ρ - italic_λ ) end_ARG bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT follows a noncentral chi-squared distribution (χ2superscript𝜒2\chi^{2}italic_χ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT distribution) with d𝑑ditalic_d degrees of freedom and noncentrality parameter (ρ−2λ+1)22(ρ−λ)2‖𝝁‖2superscript𝜌2𝜆122superscript𝜌𝜆2superscriptnorm𝝁2\frac{(\rho-2\lambda+1)^{2}}{2(\rho-\lambda)^{2}}\|\boldsymbol{\mu}\|^{2}divide start_ARG ( italic_ρ - 2 italic_λ + 1 ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 ( italic_ρ - italic_λ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT. By the tail bound for noncentral chi-squared distribution (Lemma 22), if (d+2−2λρ−λ∥𝝁∥2)2d+(ρ−2λ+1ρ−λ)2∥𝝁∥2≥8logn+ω(1)superscript𝑑22𝜆𝜌𝜆superscriptdelimited-∥∥𝝁22𝑑superscript𝜌2𝜆1𝜌𝜆2superscriptdelimited-∥∥𝝁28𝑛𝜔1\frac{\left(d+\frac{2-2\lambda}{\rho-\lambda}\lVert\boldsymbol{\mu}\rVert^{2}%
\right)^{2}}{d+\left(\frac{\rho-2\lambda+1}{\rho-\lambda}\right)^{2}\lVert%
\boldsymbol{\mu}\rVert^{2}}\geq 8\log n+\omega(1)divide start_ARG ( italic_d + divide start_ARG 2 - 2 italic_λ end_ARG start_ARG italic_ρ - italic_λ end_ARG ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_d + ( divide start_ARG italic_ρ - 2 italic_λ + 1 end_ARG start_ARG italic_ρ - italic_λ end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≥ 8 roman_log italic_n + italic_ω ( 1 ), then
we have
, 1 = ℙ(‖𝒛+ρ−2λ+12(ρ−λ)𝝁‖2≤12(ρ−2λ+1ρ−λ)2‖𝝁‖2−2(1−λ)ρ−λ‖𝝁‖2)≤exp(−2logn−ω(1)).ℙsuperscriptnorm𝒛𝜌2𝜆12𝜌𝜆𝝁212superscript𝜌2𝜆1𝜌𝜆2superscriptnorm𝝁221𝜆𝜌𝜆superscriptnorm𝝁22𝑛𝜔1\mathbb{P}\left(\left\|{\boldsymbol{z}}+\frac{\rho-2\lambda+1}{\sqrt{2}(\rho-%
\lambda)}\boldsymbol{\mu}\right\|^{2}\leq\frac{1}{2}\left(\frac{\rho-2\lambda+%
1}{\rho-\lambda}\right)^{2}\|\boldsymbol{\mu}\|^{2}-\frac{2(1-\lambda)}{\rho-%
\lambda}\|\boldsymbol{\mu}\|^{2}\right)\leq\exp\left(-2\log n-\omega(1)\right).blackboard_P ( ∥ bold_italic_z + divide start_ARG italic_ρ - 2 italic_λ + 1 end_ARG start_ARG square-root start_ARG 2 end_ARG ( italic_ρ - italic_λ ) end_ARG bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≤ divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( divide start_ARG italic_ρ - 2 italic_λ + 1 end_ARG start_ARG italic_ρ - italic_λ end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - divide start_ARG 2 ( 1 - italic_λ ) end_ARG start_ARG italic_ρ - italic_λ end_ARG ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ≤ roman_exp ( - 2 roman_log italic_n - italic_ω ( 1 ) ) .. , 2 = . , 3 = (51)
Similarly, we can have that
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
VI Proof of Theorem 1:
Achievability of Exact Matching in Correlated Gaussian Mixture Models
Proof:
Then, ‖𝒛+ρ−2λ+12(ρ−λ)𝝁‖2superscriptnorm𝒛𝜌2𝜆12𝜌𝜆𝝁2\left\|{\boldsymbol{z}}+\frac{\rho-2\lambda+1}{\sqrt{2}(\rho-\lambda)}{%
\boldsymbol{\mu}}\right\|^{2}∥ bold_italic_z + divide start_ARG italic_ρ - 2 italic_λ + 1 end_ARG start_ARG square-root start_ARG 2 end_ARG ( italic_ρ - italic_λ ) end_ARG bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT follows a noncentral chi-squared distribution (χ2superscript𝜒2\chi^{2}italic_χ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT distribution) with d𝑑ditalic_d degrees of freedom and noncentrality parameter (ρ−2λ+1)22(ρ−λ)2‖𝝁‖2superscript𝜌2𝜆122superscript𝜌𝜆2superscriptnorm𝝁2\frac{(\rho-2\lambda+1)^{2}}{2(\rho-\lambda)^{2}}\|\boldsymbol{\mu}\|^{2}divide start_ARG ( italic_ρ - 2 italic_λ + 1 ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 ( italic_ρ - italic_λ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT. By the tail bound for noncentral chi-squared distribution (Lemma 22), if (d+2−2λρ−λ∥𝝁∥2)2d+(ρ−2λ+1ρ−λ)2∥𝝁∥2≥8logn+ω(1)superscript𝑑22𝜆𝜌𝜆superscriptdelimited-∥∥𝝁22𝑑superscript𝜌2𝜆1𝜌𝜆2superscriptdelimited-∥∥𝝁28𝑛𝜔1\frac{\left(d+\frac{2-2\lambda}{\rho-\lambda}\lVert\boldsymbol{\mu}\rVert^{2}%
\right)^{2}}{d+\left(\frac{\rho-2\lambda+1}{\rho-\lambda}\right)^{2}\lVert%
\boldsymbol{\mu}\rVert^{2}}\geq 8\log n+\omega(1)divide start_ARG ( italic_d + divide start_ARG 2 - 2 italic_λ end_ARG start_ARG italic_ρ - italic_λ end_ARG ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_d + ( divide start_ARG italic_ρ - 2 italic_λ + 1 end_ARG start_ARG italic_ρ - italic_λ end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≥ 8 roman_log italic_n + italic_ω ( 1 ), then
we have
, 1 = ℙ(‖𝒛+ρ−2λ+12(ρ−λ)𝝁‖2≤12(ρ−2λ+1ρ−λ)2‖𝝁‖2−2(1−λ)ρ−λ‖𝝁‖2)≤exp(−2logn−ω(1)).ℙsuperscriptnorm𝒛𝜌2𝜆12𝜌𝜆𝝁212superscript𝜌2𝜆1𝜌𝜆2superscriptnorm𝝁221𝜆𝜌𝜆superscriptnorm𝝁22𝑛𝜔1\mathbb{P}\left(\left\|{\boldsymbol{z}}+\frac{\rho-2\lambda+1}{\sqrt{2}(\rho-%
\lambda)}\boldsymbol{\mu}\right\|^{2}\leq\frac{1}{2}\left(\frac{\rho-2\lambda+%
1}{\rho-\lambda}\right)^{2}\|\boldsymbol{\mu}\|^{2}-\frac{2(1-\lambda)}{\rho-%
\lambda}\|\boldsymbol{\mu}\|^{2}\right)\leq\exp\left(-2\log n-\omega(1)\right).blackboard_P ( ∥ bold_italic_z + divide start_ARG italic_ρ - 2 italic_λ + 1 end_ARG start_ARG square-root start_ARG 2 end_ARG ( italic_ρ - italic_λ ) end_ARG bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≤ divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( divide start_ARG italic_ρ - 2 italic_λ + 1 end_ARG start_ARG italic_ρ - italic_λ end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - divide start_ARG 2 ( 1 - italic_λ ) end_ARG start_ARG italic_ρ - italic_λ end_ARG ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ≤ roman_exp ( - 2 roman_log italic_n - italic_ω ( 1 ) ) .. , 2 = . , 3 = (51)
Similarly, we can have that
| 1,370
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61
|
, 1 = . , 2 = ℙ(ρ−λ2‖𝒛1−𝒛2+𝝁−1−λρ−λ𝝁‖2≤ρ−λ2(1−1−λρ−λ)2‖𝝁‖2)ℙ𝜌𝜆2superscriptnormsubscript𝒛1subscript𝒛2𝝁1𝜆𝜌𝜆𝝁2𝜌𝜆2superscript11𝜆𝜌𝜆2superscriptnorm𝝁2\displaystyle\mathbb{P}\left(\frac{\rho-\lambda}{2}\left\|{\boldsymbol{z}}_{1}%
-{\boldsymbol{z}}_{2}+\boldsymbol{\mu}-\frac{1-\lambda}{\rho-\lambda}%
\boldsymbol{\mu}\right\|^{2}\leq\frac{\rho-\lambda}{2}\left(1-\frac{1-\lambda}%
{\rho-\lambda}\right)^{2}\|\boldsymbol{\mu}\|^{2}\right)blackboard_P ( divide start_ARG italic_ρ - italic_λ end_ARG start_ARG 2 end_ARG ∥ bold_italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - bold_italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + bold_italic_μ - divide start_ARG 1 - italic_λ end_ARG start_ARG italic_ρ - italic_λ end_ARG bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≤ divide start_ARG italic_ρ - italic_λ end_ARG start_ARG 2 end_ARG ( 1 - divide start_ARG 1 - italic_λ end_ARG start_ARG italic_ρ - italic_λ end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ). , 3 = . , 4 = (52). , 1 = . , 2 = =ℙ(‖𝒛+ρ−12(ρ−λ)𝝁‖2≤12(ρ−1ρ−λ)2‖𝝁‖2).absentℙsuperscriptnorm𝒛𝜌12𝜌𝜆𝝁212superscript𝜌1𝜌𝜆2superscriptnorm𝝁2\displaystyle=\mathbb{P}\left(\left\|{\boldsymbol{z}}+\frac{\rho-1}{\sqrt{2}(%
\rho-\lambda)}\boldsymbol{\mu}\right\|^{2}\leq\frac{1}{2}\left(\frac{\rho-1}{%
\rho-\lambda}\right)^{2}\|\boldsymbol{\mu}\|^{2}\right).= blackboard_P ( ∥ bold_italic_z + divide start_ARG italic_ρ - 1 end_ARG start_ARG square-root start_ARG 2 end_ARG ( italic_ρ - italic_λ ) end_ARG bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≤ divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( divide start_ARG italic_ρ - 1 end_ARG start_ARG italic_ρ - italic_λ end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) .. , 3 = . , 4 = (52)
Then, ‖𝒛+ρ−12(ρ−λ)𝝁‖2superscriptnorm𝒛𝜌12𝜌𝜆𝝁2\left\|{\boldsymbol{z}}+\frac{\rho-1}{\sqrt{2}(\rho-\lambda)}\boldsymbol{\mu}%
\right\|^{2}∥ bold_italic_z + divide start_ARG italic_ρ - 1 end_ARG start_ARG square-root start_ARG 2 end_ARG ( italic_ρ - italic_λ ) end_ARG bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT follows a noncentral chi-squared distribution (χ2superscript𝜒2\chi^{2}italic_χ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT distribution) with d𝑑ditalic_d degrees of freedom and noncentrality parameter (ρ−1)22(ρ−λ)2‖𝝁‖2superscript𝜌122superscript𝜌𝜆2superscriptnorm𝝁2\frac{(\rho-1)^{2}}{2(\rho-\lambda)^{2}}\|\boldsymbol{\mu}\|^{2}divide start_ARG ( italic_ρ - 1 ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 ( italic_ρ - italic_λ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT. Again, by the tail bound for noncentral chi-squared distribution (Lemma 22), if d2d+(ρ−1ρ−λ)2∥𝝁∥2≥8logn+ω(1)superscript𝑑2𝑑superscript𝜌1𝜌𝜆2superscriptdelimited-∥∥𝝁28𝑛𝜔1\frac{d^{2}}{d+\left(\frac{\rho-1}{\rho-\lambda}\right)^{2}\lVert\boldsymbol{%
\mu}\rVert^{2}}\geq 8\log n+\omega(1)divide start_ARG italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_d + ( divide start_ARG italic_ρ - 1 end_ARG start_ARG italic_ρ - italic_λ end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≥ 8 roman_log italic_n + italic_ω ( 1 ), then
we have
, 1 = ℙ(‖𝒛+ρ−12(ρ−λ)𝝁‖2≤12(ρ−1ρ−λ)2‖𝝁‖2)≤exp(−2logn−ω(1)).ℙsuperscriptnorm𝒛𝜌12𝜌𝜆𝝁212superscript𝜌1𝜌𝜆2superscriptnorm𝝁22𝑛𝜔1\mathbb{P}\left(\left\|{\boldsymbol{z}}+\frac{\rho-1}{\sqrt{2}(\rho-\lambda)}%
\boldsymbol{\mu}\right\|^{2}\leq\frac{1}{2}\left(\frac{\rho-1}{\rho-\lambda}%
\right)^{2}\|\boldsymbol{\mu}\|^{2}\right)\leq\exp\left(-2\log n-\omega(1)%
\right).blackboard_P ( ∥ bold_italic_z + divide start_ARG italic_ρ - 1 end_ARG start_ARG square-root start_ARG 2 end_ARG ( italic_ρ - italic_λ ) end_ARG bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≤ divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( divide start_ARG italic_ρ - 1 end_ARG start_ARG italic_ρ - italic_λ end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ≤ roman_exp ( - 2 roman_log italic_n - italic_ω ( 1 ) ) .. , 2 = . , 3 = (53)
| 1,796
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
VI Proof of Theorem 1:
Achievability of Exact Matching in Correlated Gaussian Mixture Models
Proof:
, 1 = . , 2 = ℙ(ρ−λ2‖𝒛1−𝒛2+𝝁−1−λρ−λ𝝁‖2≤ρ−λ2(1−1−λρ−λ)2‖𝝁‖2)ℙ𝜌𝜆2superscriptnormsubscript𝒛1subscript𝒛2𝝁1𝜆𝜌𝜆𝝁2𝜌𝜆2superscript11𝜆𝜌𝜆2superscriptnorm𝝁2\displaystyle\mathbb{P}\left(\frac{\rho-\lambda}{2}\left\|{\boldsymbol{z}}_{1}%
-{\boldsymbol{z}}_{2}+\boldsymbol{\mu}-\frac{1-\lambda}{\rho-\lambda}%
\boldsymbol{\mu}\right\|^{2}\leq\frac{\rho-\lambda}{2}\left(1-\frac{1-\lambda}%
{\rho-\lambda}\right)^{2}\|\boldsymbol{\mu}\|^{2}\right)blackboard_P ( divide start_ARG italic_ρ - italic_λ end_ARG start_ARG 2 end_ARG ∥ bold_italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - bold_italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + bold_italic_μ - divide start_ARG 1 - italic_λ end_ARG start_ARG italic_ρ - italic_λ end_ARG bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≤ divide start_ARG italic_ρ - italic_λ end_ARG start_ARG 2 end_ARG ( 1 - divide start_ARG 1 - italic_λ end_ARG start_ARG italic_ρ - italic_λ end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ). , 3 = . , 4 = (52). , 1 = . , 2 = =ℙ(‖𝒛+ρ−12(ρ−λ)𝝁‖2≤12(ρ−1ρ−λ)2‖𝝁‖2).absentℙsuperscriptnorm𝒛𝜌12𝜌𝜆𝝁212superscript𝜌1𝜌𝜆2superscriptnorm𝝁2\displaystyle=\mathbb{P}\left(\left\|{\boldsymbol{z}}+\frac{\rho-1}{\sqrt{2}(%
\rho-\lambda)}\boldsymbol{\mu}\right\|^{2}\leq\frac{1}{2}\left(\frac{\rho-1}{%
\rho-\lambda}\right)^{2}\|\boldsymbol{\mu}\|^{2}\right).= blackboard_P ( ∥ bold_italic_z + divide start_ARG italic_ρ - 1 end_ARG start_ARG square-root start_ARG 2 end_ARG ( italic_ρ - italic_λ ) end_ARG bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≤ divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( divide start_ARG italic_ρ - 1 end_ARG start_ARG italic_ρ - italic_λ end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) .. , 3 = . , 4 = (52)
Then, ‖𝒛+ρ−12(ρ−λ)𝝁‖2superscriptnorm𝒛𝜌12𝜌𝜆𝝁2\left\|{\boldsymbol{z}}+\frac{\rho-1}{\sqrt{2}(\rho-\lambda)}\boldsymbol{\mu}%
\right\|^{2}∥ bold_italic_z + divide start_ARG italic_ρ - 1 end_ARG start_ARG square-root start_ARG 2 end_ARG ( italic_ρ - italic_λ ) end_ARG bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT follows a noncentral chi-squared distribution (χ2superscript𝜒2\chi^{2}italic_χ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT distribution) with d𝑑ditalic_d degrees of freedom and noncentrality parameter (ρ−1)22(ρ−λ)2‖𝝁‖2superscript𝜌122superscript𝜌𝜆2superscriptnorm𝝁2\frac{(\rho-1)^{2}}{2(\rho-\lambda)^{2}}\|\boldsymbol{\mu}\|^{2}divide start_ARG ( italic_ρ - 1 ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 ( italic_ρ - italic_λ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT. Again, by the tail bound for noncentral chi-squared distribution (Lemma 22), if d2d+(ρ−1ρ−λ)2∥𝝁∥2≥8logn+ω(1)superscript𝑑2𝑑superscript𝜌1𝜌𝜆2superscriptdelimited-∥∥𝝁28𝑛𝜔1\frac{d^{2}}{d+\left(\frac{\rho-1}{\rho-\lambda}\right)^{2}\lVert\boldsymbol{%
\mu}\rVert^{2}}\geq 8\log n+\omega(1)divide start_ARG italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_d + ( divide start_ARG italic_ρ - 1 end_ARG start_ARG italic_ρ - italic_λ end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≥ 8 roman_log italic_n + italic_ω ( 1 ), then
we have
, 1 = ℙ(‖𝒛+ρ−12(ρ−λ)𝝁‖2≤12(ρ−1ρ−λ)2‖𝝁‖2)≤exp(−2logn−ω(1)).ℙsuperscriptnorm𝒛𝜌12𝜌𝜆𝝁212superscript𝜌1𝜌𝜆2superscriptnorm𝝁22𝑛𝜔1\mathbb{P}\left(\left\|{\boldsymbol{z}}+\frac{\rho-1}{\sqrt{2}(\rho-\lambda)}%
\boldsymbol{\mu}\right\|^{2}\leq\frac{1}{2}\left(\frac{\rho-1}{\rho-\lambda}%
\right)^{2}\|\boldsymbol{\mu}\|^{2}\right)\leq\exp\left(-2\log n-\omega(1)%
\right).blackboard_P ( ∥ bold_italic_z + divide start_ARG italic_ρ - 1 end_ARG start_ARG square-root start_ARG 2 end_ARG ( italic_ρ - italic_λ ) end_ARG bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≤ divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( divide start_ARG italic_ρ - 1 end_ARG start_ARG italic_ρ - italic_λ end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ≤ roman_exp ( - 2 roman_log italic_n - italic_ω ( 1 ) ) .. , 2 = . , 3 = (53)
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Under the assumption d=ω(logn)𝑑𝜔𝑛d=\omega(\log n)italic_d = italic_ω ( roman_log italic_n ) in (8), for any constant r∈(0,1)𝑟01r\in(0,1)italic_r ∈ ( 0 , 1 ) and the associated λ=rρ𝜆𝑟𝜌\lambda=r\rhoitalic_λ = italic_r italic_ρ, we have (d+2−2λρ−λ∥𝝁∥2)2d+(ρ−2λ+1ρ−λ)2∥𝝁∥2≥8logn+ω(1)superscript𝑑22𝜆𝜌𝜆superscriptdelimited-∥∥𝝁22𝑑superscript𝜌2𝜆1𝜌𝜆2superscriptdelimited-∥∥𝝁28𝑛𝜔1\frac{\left(d+\frac{2-2\lambda}{\rho-\lambda}\lVert\boldsymbol{\mu}\rVert^{2}%
\right)^{2}}{d+\left(\frac{\rho-2\lambda+1}{\rho-\lambda}\right)^{2}\lVert%
\boldsymbol{\mu}\rVert^{2}}\geq 8\log n+\omega(1)divide start_ARG ( italic_d + divide start_ARG 2 - 2 italic_λ end_ARG start_ARG italic_ρ - italic_λ end_ARG ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_d + ( divide start_ARG italic_ρ - 2 italic_λ + 1 end_ARG start_ARG italic_ρ - italic_λ end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≥ 8 roman_log italic_n + italic_ω ( 1 ) and d2d+(ρ−1ρ−λ)2∥𝝁∥2≥8logn+ω(1)superscript𝑑2𝑑superscript𝜌1𝜌𝜆2superscriptdelimited-∥∥𝝁28𝑛𝜔1\frac{d^{2}}{d+\left(\frac{\rho-1}{\rho-\lambda}\right)^{2}\lVert\boldsymbol{%
\mu}\rVert^{2}}\geq 8\log n+\omega(1)divide start_ARG italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_d + ( divide start_ARG italic_ρ - 1 end_ARG start_ARG italic_ρ - italic_λ end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≥ 8 roman_log italic_n + italic_ω ( 1 ). Therefore, combining (51) and (53) and taking a union bound over all distinct i,j∈[n]𝑖𝑗delimited-[]𝑛i,j\in[n]italic_i , italic_j ∈ [ italic_n ], we obtain that
, 1 = ℙ(𝒜2)≥1−o(1).ℙsubscript𝒜21𝑜1\mathbb{P}(\mathcal{A}_{2})\geq 1-o(1).blackboard_P ( caligraphic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ≥ 1 - italic_o ( 1 ) .. , 2 = . , 3 = (54)
Therefore, it holds that ℙ(|ℳ|=0)=1−o(1)ℙℳ01𝑜1\mathbb{P}(|\mathcal{M}|=0)=1-o(1)blackboard_P ( | caligraphic_M | = 0 ) = 1 - italic_o ( 1 ).
∎
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VI Proof of Theorem 1:
Achievability of Exact Matching in Correlated Gaussian Mixture Models
Proof:
Under the assumption d=ω(logn)𝑑𝜔𝑛d=\omega(\log n)italic_d = italic_ω ( roman_log italic_n ) in (8), for any constant r∈(0,1)𝑟01r\in(0,1)italic_r ∈ ( 0 , 1 ) and the associated λ=rρ𝜆𝑟𝜌\lambda=r\rhoitalic_λ = italic_r italic_ρ, we have (d+2−2λρ−λ∥𝝁∥2)2d+(ρ−2λ+1ρ−λ)2∥𝝁∥2≥8logn+ω(1)superscript𝑑22𝜆𝜌𝜆superscriptdelimited-∥∥𝝁22𝑑superscript𝜌2𝜆1𝜌𝜆2superscriptdelimited-∥∥𝝁28𝑛𝜔1\frac{\left(d+\frac{2-2\lambda}{\rho-\lambda}\lVert\boldsymbol{\mu}\rVert^{2}%
\right)^{2}}{d+\left(\frac{\rho-2\lambda+1}{\rho-\lambda}\right)^{2}\lVert%
\boldsymbol{\mu}\rVert^{2}}\geq 8\log n+\omega(1)divide start_ARG ( italic_d + divide start_ARG 2 - 2 italic_λ end_ARG start_ARG italic_ρ - italic_λ end_ARG ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_d + ( divide start_ARG italic_ρ - 2 italic_λ + 1 end_ARG start_ARG italic_ρ - italic_λ end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≥ 8 roman_log italic_n + italic_ω ( 1 ) and d2d+(ρ−1ρ−λ)2∥𝝁∥2≥8logn+ω(1)superscript𝑑2𝑑superscript𝜌1𝜌𝜆2superscriptdelimited-∥∥𝝁28𝑛𝜔1\frac{d^{2}}{d+\left(\frac{\rho-1}{\rho-\lambda}\right)^{2}\lVert\boldsymbol{%
\mu}\rVert^{2}}\geq 8\log n+\omega(1)divide start_ARG italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_d + ( divide start_ARG italic_ρ - 1 end_ARG start_ARG italic_ρ - italic_λ end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≥ 8 roman_log italic_n + italic_ω ( 1 ). Therefore, combining (51) and (53) and taking a union bound over all distinct i,j∈[n]𝑖𝑗delimited-[]𝑛i,j\in[n]italic_i , italic_j ∈ [ italic_n ], we obtain that
, 1 = ℙ(𝒜2)≥1−o(1).ℙsubscript𝒜21𝑜1\mathbb{P}(\mathcal{A}_{2})\geq 1-o(1).blackboard_P ( caligraphic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ≥ 1 - italic_o ( 1 ) .. , 2 = . , 3 = (54)
Therefore, it holds that ℙ(|ℳ|=0)=1−o(1)ℙℳ01𝑜1\mathbb{P}(|\mathcal{M}|=0)=1-o(1)blackboard_P ( | caligraphic_M | = 0 ) = 1 - italic_o ( 1 ).
∎
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Assume that the community labels 𝝈1superscript𝝈1{\boldsymbol{\sigma}}^{1}bold_italic_σ start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT and 𝝈2superscript𝝈2{\boldsymbol{\sigma}}^{2}bold_italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT are given. Without loss of generality, assume that at least half of the nodes, i.e., n/2𝑛2n/2italic_n / 2 more, are assigned +++ label. Let this set of nodes be denoted as V′superscript𝑉′V^{\prime}italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT. Additionally, assume that the π∗{V\V′}subscript𝜋\𝑉superscript𝑉′\pi_{*}\left\{V\backslash V^{\prime}\right\}italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT { italic_V \ italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT } and the mean vector 𝝁𝝁\boldsymbol{\mu}bold_italic_μ are also given.
Then, we can consider that we are solving the database alignment problem for |V′|superscript𝑉′|V^{\prime}|| italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT | nodes in the given correlated Gaussian databases.
Dai et al. [9] identified the conditions under which exact matching is impossible, and the results are as follows.
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VII Proof of Theorem 2:
Impossibility of Exact Matching in Correlated Gaussian Mixture Models
Assume that the community labels 𝝈1superscript𝝈1{\boldsymbol{\sigma}}^{1}bold_italic_σ start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT and 𝝈2superscript𝝈2{\boldsymbol{\sigma}}^{2}bold_italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT are given. Without loss of generality, assume that at least half of the nodes, i.e., n/2𝑛2n/2italic_n / 2 more, are assigned +++ label. Let this set of nodes be denoted as V′superscript𝑉′V^{\prime}italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT. Additionally, assume that the π∗{V\V′}subscript𝜋\𝑉superscript𝑉′\pi_{*}\left\{V\backslash V^{\prime}\right\}italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT { italic_V \ italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT } and the mean vector 𝝁𝝁\boldsymbol{\mu}bold_italic_μ are also given.
Then, we can consider that we are solving the database alignment problem for |V′|superscript𝑉′|V^{\prime}|| italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT | nodes in the given correlated Gaussian databases.
Dai et al. [9] identified the conditions under which exact matching is impossible, and the results are as follows.
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Consider the correlated Gaussian databases X,Y∈ℝn×d𝑋𝑌superscriptℝ𝑛𝑑X,Y\in\mathbb{R}^{n\times d}italic_X , italic_Y ∈ blackboard_R start_POSTSUPERSCRIPT italic_n × italic_d end_POSTSUPERSCRIPT. Suppose that 1≪d=O(logn)much-less-than1𝑑𝑂𝑛1\ll d=O(\log n)1 ≪ italic_d = italic_O ( roman_log italic_n ) and
, 1 = d4log11−ρ2≤(1−ϵ)logn𝑑411superscript𝜌21italic-ϵ𝑛\frac{d}{4}\log\frac{1}{1-\rho^{2}}\leq(1-\epsilon)\log ndivide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≤ ( 1 - italic_ϵ ) roman_log italic_n. , 2 = . , 3 = (55)
for an arbitrary small constant ϵ>0italic-ϵ0\epsilon>0italic_ϵ > 0. Then, there is no estimator that can exactly recover π∗subscript𝜋\pi_{*}italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT with probability 1−o(1).1𝑜11-o(1).1 - italic_o ( 1 ) .
Assume that 1≪d=O(logn)much-less-than1𝑑𝑂𝑛1\ll d=O(\log n)1 ≪ italic_d = italic_O ( roman_log italic_n ). Then, by applying Theorem 9, we can conclude that exact matching is impossible if d4log11−ρ2≤(1−ϵ)log|V′|≤(1−ϵ)logn𝑑411superscript𝜌21italic-ϵsuperscript𝑉′1italic-ϵ𝑛\frac{d}{4}\log\frac{1}{1-\rho^{2}}\leq(1-\epsilon)\log|V^{\prime}|\leq(1-%
\epsilon)\log ndivide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≤ ( 1 - italic_ϵ ) roman_log | italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT | ≤ ( 1 - italic_ϵ ) roman_log italic_n.
Wang et al. [36] also idenitfied the conditions under which exact matching is not possible, and the results are as follows.
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VII Proof of Theorem 2:
Impossibility of Exact Matching in Correlated Gaussian Mixture Models
Theorem 9 (Theorem 2 in [9]).
Consider the correlated Gaussian databases X,Y∈ℝn×d𝑋𝑌superscriptℝ𝑛𝑑X,Y\in\mathbb{R}^{n\times d}italic_X , italic_Y ∈ blackboard_R start_POSTSUPERSCRIPT italic_n × italic_d end_POSTSUPERSCRIPT. Suppose that 1≪d=O(logn)much-less-than1𝑑𝑂𝑛1\ll d=O(\log n)1 ≪ italic_d = italic_O ( roman_log italic_n ) and
, 1 = d4log11−ρ2≤(1−ϵ)logn𝑑411superscript𝜌21italic-ϵ𝑛\frac{d}{4}\log\frac{1}{1-\rho^{2}}\leq(1-\epsilon)\log ndivide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≤ ( 1 - italic_ϵ ) roman_log italic_n. , 2 = . , 3 = (55)
for an arbitrary small constant ϵ>0italic-ϵ0\epsilon>0italic_ϵ > 0. Then, there is no estimator that can exactly recover π∗subscript𝜋\pi_{*}italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT with probability 1−o(1).1𝑜11-o(1).1 - italic_o ( 1 ) .
Assume that 1≪d=O(logn)much-less-than1𝑑𝑂𝑛1\ll d=O(\log n)1 ≪ italic_d = italic_O ( roman_log italic_n ). Then, by applying Theorem 9, we can conclude that exact matching is impossible if d4log11−ρ2≤(1−ϵ)log|V′|≤(1−ϵ)logn𝑑411superscript𝜌21italic-ϵsuperscript𝑉′1italic-ϵ𝑛\frac{d}{4}\log\frac{1}{1-\rho^{2}}\leq(1-\epsilon)\log|V^{\prime}|\leq(1-%
\epsilon)\log ndivide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≤ ( 1 - italic_ϵ ) roman_log | italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT | ≤ ( 1 - italic_ϵ ) roman_log italic_n.
Wang et al. [36] also idenitfied the conditions under which exact matching is not possible, and the results are as follows.
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Consider the correlated Gaussian databases X,Y∈ℝn×d𝑋𝑌superscriptℝ𝑛𝑑X,Y\in\mathbb{R}^{n\times d}italic_X , italic_Y ∈ blackboard_R start_POSTSUPERSCRIPT italic_n × italic_d end_POSTSUPERSCRIPT. Suppose that 1ρ2−1≤d401superscript𝜌21𝑑40\frac{1}{\rho^{2}}-1\leq\frac{d}{40}divide start_ARG 1 end_ARG start_ARG italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG - 1 ≤ divide start_ARG italic_d end_ARG start_ARG 40 end_ARG and
, 1 = d4log11−ρ2≤logn−logd+C,𝑑411superscript𝜌2𝑛𝑑𝐶\frac{d}{4}\log\frac{1}{1-\rho^{2}}\leq\log n-\log d+C,divide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≤ roman_log italic_n - roman_log italic_d + italic_C ,. , 2 = . , 3 = (56)
for a constant C>0𝐶0C>0italic_C > 0. Then, there is no estimator that can exactly recover π∗subscript𝜋\pi_{*}italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT with probability 1−o(1).1𝑜11-o(1).1 - italic_o ( 1 ) .
Assume that 1ρ2−1≤d401superscript𝜌21𝑑40\frac{1}{\rho^{2}}-1\leq\frac{d}{40}divide start_ARG 1 end_ARG start_ARG italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG - 1 ≤ divide start_ARG italic_d end_ARG start_ARG 40 end_ARG. Then, by applying Theorem 10, we can conclude that exact matching is impossible if d4log11−ρ2≤log|V′|−logd+C≤logn−logd+C𝑑411superscript𝜌2superscript𝑉′𝑑𝐶𝑛𝑑𝐶\frac{d}{4}\log\frac{1}{1-\rho^{2}}\leq\log|V^{\prime}|-\log d+C\leq\log n-%
\log d+Cdivide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≤ roman_log | italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT | - roman_log italic_d + italic_C ≤ roman_log italic_n - roman_log italic_d + italic_C.
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VII Proof of Theorem 2:
Impossibility of Exact Matching in Correlated Gaussian Mixture Models
Theorem 10 (Theorem 19 in [36]).
Consider the correlated Gaussian databases X,Y∈ℝn×d𝑋𝑌superscriptℝ𝑛𝑑X,Y\in\mathbb{R}^{n\times d}italic_X , italic_Y ∈ blackboard_R start_POSTSUPERSCRIPT italic_n × italic_d end_POSTSUPERSCRIPT. Suppose that 1ρ2−1≤d401superscript𝜌21𝑑40\frac{1}{\rho^{2}}-1\leq\frac{d}{40}divide start_ARG 1 end_ARG start_ARG italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG - 1 ≤ divide start_ARG italic_d end_ARG start_ARG 40 end_ARG and
, 1 = d4log11−ρ2≤logn−logd+C,𝑑411superscript𝜌2𝑛𝑑𝐶\frac{d}{4}\log\frac{1}{1-\rho^{2}}\leq\log n-\log d+C,divide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≤ roman_log italic_n - roman_log italic_d + italic_C ,. , 2 = . , 3 = (56)
for a constant C>0𝐶0C>0italic_C > 0. Then, there is no estimator that can exactly recover π∗subscript𝜋\pi_{*}italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT with probability 1−o(1).1𝑜11-o(1).1 - italic_o ( 1 ) .
Assume that 1ρ2−1≤d401superscript𝜌21𝑑40\frac{1}{\rho^{2}}-1\leq\frac{d}{40}divide start_ARG 1 end_ARG start_ARG italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG - 1 ≤ divide start_ARG italic_d end_ARG start_ARG 40 end_ARG. Then, by applying Theorem 10, we can conclude that exact matching is impossible if d4log11−ρ2≤log|V′|−logd+C≤logn−logd+C𝑑411superscript𝜌2superscript𝑉′𝑑𝐶𝑛𝑑𝐶\frac{d}{4}\log\frac{1}{1-\rho^{2}}\leq\log|V^{\prime}|-\log d+C\leq\log n-%
\log d+Cdivide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≤ roman_log | italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT | - roman_log italic_d + italic_C ≤ roman_log italic_n - roman_log italic_d + italic_C.
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Given a permutation π:[n]→[n]:𝜋→delimited-[]𝑛delimited-[]𝑛\pi:[n]\to[n]italic_π : [ italic_n ] → [ italic_n ], let X+πYsubscript𝜋𝑋𝑌X+_{\pi}Yitalic_X + start_POSTSUBSCRIPT italic_π end_POSTSUBSCRIPT italic_Y represent the database where each node i𝑖iitalic_i is assigned the vector 𝒙i+𝒚π(i)2subscript𝒙𝑖subscript𝒚𝜋𝑖2\frac{{\boldsymbol{x}}_{i}+{\boldsymbol{y}}_{\pi(i)}}{2}divide start_ARG bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + bold_italic_y start_POSTSUBSCRIPT italic_π ( italic_i ) end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG. Recall that for two functions f,g𝑓𝑔f,gitalic_f , italic_g, 𝐨𝐯(f,g):=∑i=1n𝟙(f(i)=g(i))nassign𝐨𝐯𝑓𝑔superscriptsubscript𝑖1𝑛1𝑓𝑖𝑔𝑖𝑛\mathbf{ov}(f,g):=\sum\limits_{i=1}^{n}\frac{\mathds{1}\left(f(i)=g(i)\right)}%
{n}bold_ov ( italic_f , italic_g ) := ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT divide start_ARG blackboard_1 ( italic_f ( italic_i ) = italic_g ( italic_i ) ) end_ARG start_ARG italic_n end_ARG.
Ndaoud [14] found the conditions under which there exists an estimator 𝝈^^𝝈\hat{{\boldsymbol{\sigma}}}over^ start_ARG bold_italic_σ end_ARG, which is based on a variant of Lloyd’s iteration initialized by a spectral method, that can exactly recover 𝝈𝝈{\boldsymbol{\sigma}}bold_italic_σ from a Gaussian mixture model X𝑋Xitalic_X.
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
VIII Proof of Theorem 3:
Achievability of Exact Community Recovery in Correlated Gaussian Mixture Models
Given a permutation π:[n]→[n]:𝜋→delimited-[]𝑛delimited-[]𝑛\pi:[n]\to[n]italic_π : [ italic_n ] → [ italic_n ], let X+πYsubscript𝜋𝑋𝑌X+_{\pi}Yitalic_X + start_POSTSUBSCRIPT italic_π end_POSTSUBSCRIPT italic_Y represent the database where each node i𝑖iitalic_i is assigned the vector 𝒙i+𝒚π(i)2subscript𝒙𝑖subscript𝒚𝜋𝑖2\frac{{\boldsymbol{x}}_{i}+{\boldsymbol{y}}_{\pi(i)}}{2}divide start_ARG bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + bold_italic_y start_POSTSUBSCRIPT italic_π ( italic_i ) end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG. Recall that for two functions f,g𝑓𝑔f,gitalic_f , italic_g, 𝐨𝐯(f,g):=∑i=1n𝟙(f(i)=g(i))nassign𝐨𝐯𝑓𝑔superscriptsubscript𝑖1𝑛1𝑓𝑖𝑔𝑖𝑛\mathbf{ov}(f,g):=\sum\limits_{i=1}^{n}\frac{\mathds{1}\left(f(i)=g(i)\right)}%
{n}bold_ov ( italic_f , italic_g ) := ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT divide start_ARG blackboard_1 ( italic_f ( italic_i ) = italic_g ( italic_i ) ) end_ARG start_ARG italic_n end_ARG.
Ndaoud [14] found the conditions under which there exists an estimator 𝝈^^𝝈\hat{{\boldsymbol{\sigma}}}over^ start_ARG bold_italic_σ end_ARG, which is based on a variant of Lloyd’s iteration initialized by a spectral method, that can exactly recover 𝝈𝝈{\boldsymbol{\sigma}}bold_italic_σ from a Gaussian mixture model X𝑋Xitalic_X.
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For k>0𝑘0k>0italic_k > 0 and 𝐳i∼𝒩(0,𝐈d)similar-tosubscript𝐳𝑖𝒩0subscript𝐈𝑑{\boldsymbol{z}}_{i}\sim\mathcal{N}(0,{\boldsymbol{I}}_{d})bold_italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∼ caligraphic_N ( 0 , bold_italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ), let X:={𝐱i}i=1n∼GMM(𝛍,𝛔)assign𝑋superscriptsubscriptsubscript𝐱𝑖𝑖1𝑛similar-toGMM𝛍𝛔X:=\left\{{\boldsymbol{x}}_{i}\right\}_{i=1}^{n}\sim\operatorname{GMM}(%
\boldsymbol{\mu},{\boldsymbol{\sigma}})italic_X := { bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ∼ roman_GMM ( bold_italic_μ , bold_italic_σ ), where 𝛍∈ℝd𝛍superscriptℝ𝑑\boldsymbol{\mu}\in\mathbb{R}^{d}bold_italic_μ ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT and 𝐱i:=𝛍σi+k𝐳iassignsubscript𝐱𝑖𝛍subscript𝜎𝑖𝑘subscript𝐳𝑖{\boldsymbol{x}}_{i}:=\boldsymbol{\mu}\sigma_{i}+k{\boldsymbol{z}}_{i}bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT := bold_italic_μ italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + italic_k bold_italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT. If
, 1 = ∥𝝁∥2≥k2(1+ϵ)(1+1+2dnlogn)lognsuperscriptdelimited-∥∥𝝁2superscript𝑘21italic-ϵ112𝑑𝑛𝑛𝑛\lVert\boldsymbol{\mu}\rVert^{2}\geq k^{2}(1+\epsilon)\left(1+\sqrt{1+\frac{2d%
}{n\log n}}\right)\log n∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( 1 + italic_ϵ ) ( 1 + square-root start_ARG 1 + divide start_ARG 2 italic_d end_ARG start_ARG italic_n roman_log italic_n end_ARG end_ARG ) roman_log italic_n. , 2 = . , 3 = (57)
for an arbitrary small constant ϵ>0italic-ϵ0\epsilon>0italic_ϵ > 0, then there exists an estimator 𝛔^^𝛔\hat{{\boldsymbol{\sigma}}}over^ start_ARG bold_italic_σ end_ARG achieving ℙ(𝐨𝐯(𝛔^,𝛔)=1)=1−o(1)ℙ𝐨𝐯^𝛔𝛔11𝑜1\mathbb{P}(\mathbf{ov}(\hat{{\boldsymbol{\sigma}}},{\boldsymbol{\sigma}})=1)=1%
-o(1)blackboard_P ( bold_ov ( over^ start_ARG bold_italic_σ end_ARG , bold_italic_σ ) = 1 ) = 1 - italic_o ( 1 ).
When applying this estimator 𝝈^^𝝈\hat{{\boldsymbol{\sigma}}}over^ start_ARG bold_italic_σ end_ARG to X+π^Ysubscript^𝜋𝑋𝑌X+_{\hat{\pi}}Yitalic_X + start_POSTSUBSCRIPT over^ start_ARG italic_π end_ARG end_POSTSUBSCRIPT italic_Y for the estimator π^^𝜋\hat{\pi}over^ start_ARG italic_π end_ARG defined in (31), we can have
, 1 = ℙ(𝐨𝐯(𝝈^(X+π^Y),𝝈)≠1)ℙ𝐨𝐯^𝝈subscript^𝜋𝑋𝑌𝝈1\displaystyle\mathbb{P}\left(\mathbf{ov}(\hat{{\boldsymbol{\sigma}}}(X+_{\hat{%
\pi}}Y),{\boldsymbol{\sigma}})\neq 1\right)blackboard_P ( bold_ov ( over^ start_ARG bold_italic_σ end_ARG ( italic_X + start_POSTSUBSCRIPT over^ start_ARG italic_π end_ARG end_POSTSUBSCRIPT italic_Y ) , bold_italic_σ ) ≠ 1 ). , 2 = ≤ℙ({𝐨𝐯(𝝈^(X+π^Y),𝝈)≠1}∩{X+π^Y=X+π∗Y})+ℙ(X+π^Y≠X+π∗Y)absentℙ𝐨𝐯^𝝈subscript^𝜋𝑋𝑌𝝈1subscript^𝜋𝑋𝑌subscriptsubscript𝜋𝑋𝑌ℙsubscript^𝜋𝑋𝑌subscriptsubscript𝜋𝑋𝑌\displaystyle\leq\mathbb{P}\left(\{\mathbf{ov}(\hat{{\boldsymbol{\sigma}}}(X+_%
{\hat{\pi}}Y),{\boldsymbol{\sigma}})\neq 1\}\cap\{X+_{\hat{\pi}}Y=X+_{\pi_{*}}%
Y\}\right)+\mathbb{P}(X+_{\hat{\pi}}Y\neq X+_{\pi_{*}}Y)≤ blackboard_P ( { bold_ov ( over^ start_ARG bold_italic_σ end_ARG ( italic_X + start_POSTSUBSCRIPT over^ start_ARG italic_π end_ARG end_POSTSUBSCRIPT italic_Y ) , bold_italic_σ ) ≠ 1 } ∩ { italic_X + start_POSTSUBSCRIPT over^ start_ARG italic_π end_ARG end_POSTSUBSCRIPT italic_Y = italic_X + start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_Y } ) + blackboard_P ( italic_X + start_POSTSUBSCRIPT over^ start_ARG italic_π end_ARG end_POSTSUBSCRIPT italic_Y ≠ italic_X + start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_Y ). , 3 = . , 4 = (58). , 1 = . , 2 = ≤ℙ(𝐨𝐯(𝝈^(X+π∗Y),𝝈)≠1)+ℙ(π^≠π∗).absentℙ𝐨𝐯^𝝈subscriptsubscript𝜋𝑋𝑌𝝈1ℙ^𝜋subscript𝜋\displaystyle\leq\mathbb{P}\left(\mathbf{ov}(\hat{{\boldsymbol{\sigma}}}(X+_{%
\pi_{*}}Y),{\boldsymbol{\sigma}})\neq 1\right)+\mathbb{P}(\hat{\pi}\neq\pi_{*}).≤ blackboard_P ( bold_ov ( over^ start_ARG bold_italic_σ end_ARG ( italic_X + start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_Y ) , bold_italic_σ ) ≠ 1 ) + blackboard_P ( over^ start_ARG italic_π end_ARG ≠ italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ) .. , 3 = . , 4 = (58)
If (7) or (8) holds, then by Theorem 1, we can obtain that
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
VIII Proof of Theorem 3:
Achievability of Exact Community Recovery in Correlated Gaussian Mixture Models
Theorem 11 (Theorem 8 in [14]).
For k>0𝑘0k>0italic_k > 0 and 𝐳i∼𝒩(0,𝐈d)similar-tosubscript𝐳𝑖𝒩0subscript𝐈𝑑{\boldsymbol{z}}_{i}\sim\mathcal{N}(0,{\boldsymbol{I}}_{d})bold_italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∼ caligraphic_N ( 0 , bold_italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ), let X:={𝐱i}i=1n∼GMM(𝛍,𝛔)assign𝑋superscriptsubscriptsubscript𝐱𝑖𝑖1𝑛similar-toGMM𝛍𝛔X:=\left\{{\boldsymbol{x}}_{i}\right\}_{i=1}^{n}\sim\operatorname{GMM}(%
\boldsymbol{\mu},{\boldsymbol{\sigma}})italic_X := { bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ∼ roman_GMM ( bold_italic_μ , bold_italic_σ ), where 𝛍∈ℝd𝛍superscriptℝ𝑑\boldsymbol{\mu}\in\mathbb{R}^{d}bold_italic_μ ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT and 𝐱i:=𝛍σi+k𝐳iassignsubscript𝐱𝑖𝛍subscript𝜎𝑖𝑘subscript𝐳𝑖{\boldsymbol{x}}_{i}:=\boldsymbol{\mu}\sigma_{i}+k{\boldsymbol{z}}_{i}bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT := bold_italic_μ italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + italic_k bold_italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT. If
, 1 = ∥𝝁∥2≥k2(1+ϵ)(1+1+2dnlogn)lognsuperscriptdelimited-∥∥𝝁2superscript𝑘21italic-ϵ112𝑑𝑛𝑛𝑛\lVert\boldsymbol{\mu}\rVert^{2}\geq k^{2}(1+\epsilon)\left(1+\sqrt{1+\frac{2d%
}{n\log n}}\right)\log n∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( 1 + italic_ϵ ) ( 1 + square-root start_ARG 1 + divide start_ARG 2 italic_d end_ARG start_ARG italic_n roman_log italic_n end_ARG end_ARG ) roman_log italic_n. , 2 = . , 3 = (57)
for an arbitrary small constant ϵ>0italic-ϵ0\epsilon>0italic_ϵ > 0, then there exists an estimator 𝛔^^𝛔\hat{{\boldsymbol{\sigma}}}over^ start_ARG bold_italic_σ end_ARG achieving ℙ(𝐨𝐯(𝛔^,𝛔)=1)=1−o(1)ℙ𝐨𝐯^𝛔𝛔11𝑜1\mathbb{P}(\mathbf{ov}(\hat{{\boldsymbol{\sigma}}},{\boldsymbol{\sigma}})=1)=1%
-o(1)blackboard_P ( bold_ov ( over^ start_ARG bold_italic_σ end_ARG , bold_italic_σ ) = 1 ) = 1 - italic_o ( 1 ).
When applying this estimator 𝝈^^𝝈\hat{{\boldsymbol{\sigma}}}over^ start_ARG bold_italic_σ end_ARG to X+π^Ysubscript^𝜋𝑋𝑌X+_{\hat{\pi}}Yitalic_X + start_POSTSUBSCRIPT over^ start_ARG italic_π end_ARG end_POSTSUBSCRIPT italic_Y for the estimator π^^𝜋\hat{\pi}over^ start_ARG italic_π end_ARG defined in (31), we can have
, 1 = ℙ(𝐨𝐯(𝝈^(X+π^Y),𝝈)≠1)ℙ𝐨𝐯^𝝈subscript^𝜋𝑋𝑌𝝈1\displaystyle\mathbb{P}\left(\mathbf{ov}(\hat{{\boldsymbol{\sigma}}}(X+_{\hat{%
\pi}}Y),{\boldsymbol{\sigma}})\neq 1\right)blackboard_P ( bold_ov ( over^ start_ARG bold_italic_σ end_ARG ( italic_X + start_POSTSUBSCRIPT over^ start_ARG italic_π end_ARG end_POSTSUBSCRIPT italic_Y ) , bold_italic_σ ) ≠ 1 ). , 2 = ≤ℙ({𝐨𝐯(𝝈^(X+π^Y),𝝈)≠1}∩{X+π^Y=X+π∗Y})+ℙ(X+π^Y≠X+π∗Y)absentℙ𝐨𝐯^𝝈subscript^𝜋𝑋𝑌𝝈1subscript^𝜋𝑋𝑌subscriptsubscript𝜋𝑋𝑌ℙsubscript^𝜋𝑋𝑌subscriptsubscript𝜋𝑋𝑌\displaystyle\leq\mathbb{P}\left(\{\mathbf{ov}(\hat{{\boldsymbol{\sigma}}}(X+_%
{\hat{\pi}}Y),{\boldsymbol{\sigma}})\neq 1\}\cap\{X+_{\hat{\pi}}Y=X+_{\pi_{*}}%
Y\}\right)+\mathbb{P}(X+_{\hat{\pi}}Y\neq X+_{\pi_{*}}Y)≤ blackboard_P ( { bold_ov ( over^ start_ARG bold_italic_σ end_ARG ( italic_X + start_POSTSUBSCRIPT over^ start_ARG italic_π end_ARG end_POSTSUBSCRIPT italic_Y ) , bold_italic_σ ) ≠ 1 } ∩ { italic_X + start_POSTSUBSCRIPT over^ start_ARG italic_π end_ARG end_POSTSUBSCRIPT italic_Y = italic_X + start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_Y } ) + blackboard_P ( italic_X + start_POSTSUBSCRIPT over^ start_ARG italic_π end_ARG end_POSTSUBSCRIPT italic_Y ≠ italic_X + start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_Y ). , 3 = . , 4 = (58). , 1 = . , 2 = ≤ℙ(𝐨𝐯(𝝈^(X+π∗Y),𝝈)≠1)+ℙ(π^≠π∗).absentℙ𝐨𝐯^𝝈subscriptsubscript𝜋𝑋𝑌𝝈1ℙ^𝜋subscript𝜋\displaystyle\leq\mathbb{P}\left(\mathbf{ov}(\hat{{\boldsymbol{\sigma}}}(X+_{%
\pi_{*}}Y),{\boldsymbol{\sigma}})\neq 1\right)+\mathbb{P}(\hat{\pi}\neq\pi_{*}).≤ blackboard_P ( bold_ov ( over^ start_ARG bold_italic_σ end_ARG ( italic_X + start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_Y ) , bold_italic_σ ) ≠ 1 ) + blackboard_P ( over^ start_ARG italic_π end_ARG ≠ italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ) .. , 3 = . , 4 = (58)
If (7) or (8) holds, then by Theorem 1, we can obtain that
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, 1 = ℙ(π^≠π∗)=o(1).ℙ^𝜋subscript𝜋𝑜1\mathbb{P}(\hat{\pi}\neq\pi_{*})=o(1).blackboard_P ( over^ start_ARG italic_π end_ARG ≠ italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ) = italic_o ( 1 ) .. , 2 = . , 3 = (59)
Moreover, we have 𝒙i+𝒚π∗(i)2=𝝁𝝈i+(1+ρ)𝒛i+1−ρ2𝒘i2∼𝝁𝝈i+1+ρ2oisubscript𝒙𝑖subscript𝒚subscript𝜋𝑖2𝝁subscript𝝈𝑖1𝜌subscript𝒛𝑖1superscript𝜌2subscript𝒘𝑖2similar-to𝝁subscript𝝈𝑖1𝜌2subscript𝑜𝑖\frac{{\boldsymbol{x}}_{i}+{\boldsymbol{y}}_{\pi_{*}}(i)}{2}=\boldsymbol{\mu}{%
\boldsymbol{\sigma}}_{i}+\frac{(1+\rho){\boldsymbol{z}}_{i}+\sqrt{1-\rho^{2}}{%
\boldsymbol{w}}_{i}}{2}\sim\boldsymbol{\mu}{\boldsymbol{\sigma}}_{i}+\sqrt{%
\frac{1+\rho}{2}}o_{i}divide start_ARG bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + bold_italic_y start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_i ) end_ARG start_ARG 2 end_ARG = bold_italic_μ bold_italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + divide start_ARG ( 1 + italic_ρ ) bold_italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + square-root start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG bold_italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG ∼ bold_italic_μ bold_italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + square-root start_ARG divide start_ARG 1 + italic_ρ end_ARG start_ARG 2 end_ARG end_ARG italic_o start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT, where 𝒛i,𝒘i,𝒐i∼𝒩(0,𝑰d)similar-tosubscript𝒛𝑖subscript𝒘𝑖subscript𝒐𝑖𝒩0subscript𝑰𝑑{\boldsymbol{z}}_{i},{\boldsymbol{w}}_{i},{\boldsymbol{o}}_{i}\sim\mathcal{N}(%
0,{\boldsymbol{I}}_{d})bold_italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_italic_o start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∼ caligraphic_N ( 0 , bold_italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ). Therefore, by Theorem 11, if ∥𝝁∥2≥(1+ϵ)1+ρ2(1+1+2dnlogn)lognsuperscriptdelimited-∥∥𝝁21italic-ϵ1𝜌2112𝑑𝑛𝑛𝑛\lVert\boldsymbol{\mu}\rVert^{2}\geq(1+\epsilon)\frac{1+\rho}{2}\left(1+\sqrt{%
1+\frac{2d}{n\log n}}\right)\log n∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ ( 1 + italic_ϵ ) divide start_ARG 1 + italic_ρ end_ARG start_ARG 2 end_ARG ( 1 + square-root start_ARG 1 + divide start_ARG 2 italic_d end_ARG start_ARG italic_n roman_log italic_n end_ARG end_ARG ) roman_log italic_n, then it holds that
, 1 = ℙ(𝐨𝐯(𝝈^(X+π∗Y),𝝈)≠1)=o(1).ℙ𝐨𝐯^𝝈subscriptsubscript𝜋𝑋𝑌𝝈1𝑜1\mathbb{P}\left(\mathbf{ov}(\hat{{\boldsymbol{\sigma}}}(X+_{\pi_{*}}Y),{%
\boldsymbol{\sigma}})\neq 1\right)=o(1).blackboard_P ( bold_ov ( over^ start_ARG bold_italic_σ end_ARG ( italic_X + start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_Y ) , bold_italic_σ ) ≠ 1 ) = italic_o ( 1 ) .. , 2 = . , 3 = (60)
By combining the results from (58), (59) and (60), the proof is complete.
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
VIII Proof of Theorem 3:
Achievability of Exact Community Recovery in Correlated Gaussian Mixture Models
Theorem 11 (Theorem 8 in [14]).
, 1 = ℙ(π^≠π∗)=o(1).ℙ^𝜋subscript𝜋𝑜1\mathbb{P}(\hat{\pi}\neq\pi_{*})=o(1).blackboard_P ( over^ start_ARG italic_π end_ARG ≠ italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ) = italic_o ( 1 ) .. , 2 = . , 3 = (59)
Moreover, we have 𝒙i+𝒚π∗(i)2=𝝁𝝈i+(1+ρ)𝒛i+1−ρ2𝒘i2∼𝝁𝝈i+1+ρ2oisubscript𝒙𝑖subscript𝒚subscript𝜋𝑖2𝝁subscript𝝈𝑖1𝜌subscript𝒛𝑖1superscript𝜌2subscript𝒘𝑖2similar-to𝝁subscript𝝈𝑖1𝜌2subscript𝑜𝑖\frac{{\boldsymbol{x}}_{i}+{\boldsymbol{y}}_{\pi_{*}}(i)}{2}=\boldsymbol{\mu}{%
\boldsymbol{\sigma}}_{i}+\frac{(1+\rho){\boldsymbol{z}}_{i}+\sqrt{1-\rho^{2}}{%
\boldsymbol{w}}_{i}}{2}\sim\boldsymbol{\mu}{\boldsymbol{\sigma}}_{i}+\sqrt{%
\frac{1+\rho}{2}}o_{i}divide start_ARG bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + bold_italic_y start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_i ) end_ARG start_ARG 2 end_ARG = bold_italic_μ bold_italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + divide start_ARG ( 1 + italic_ρ ) bold_italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + square-root start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG bold_italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG ∼ bold_italic_μ bold_italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + square-root start_ARG divide start_ARG 1 + italic_ρ end_ARG start_ARG 2 end_ARG end_ARG italic_o start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT, where 𝒛i,𝒘i,𝒐i∼𝒩(0,𝑰d)similar-tosubscript𝒛𝑖subscript𝒘𝑖subscript𝒐𝑖𝒩0subscript𝑰𝑑{\boldsymbol{z}}_{i},{\boldsymbol{w}}_{i},{\boldsymbol{o}}_{i}\sim\mathcal{N}(%
0,{\boldsymbol{I}}_{d})bold_italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_italic_o start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∼ caligraphic_N ( 0 , bold_italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ). Therefore, by Theorem 11, if ∥𝝁∥2≥(1+ϵ)1+ρ2(1+1+2dnlogn)lognsuperscriptdelimited-∥∥𝝁21italic-ϵ1𝜌2112𝑑𝑛𝑛𝑛\lVert\boldsymbol{\mu}\rVert^{2}\geq(1+\epsilon)\frac{1+\rho}{2}\left(1+\sqrt{%
1+\frac{2d}{n\log n}}\right)\log n∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ ( 1 + italic_ϵ ) divide start_ARG 1 + italic_ρ end_ARG start_ARG 2 end_ARG ( 1 + square-root start_ARG 1 + divide start_ARG 2 italic_d end_ARG start_ARG italic_n roman_log italic_n end_ARG end_ARG ) roman_log italic_n, then it holds that
, 1 = ℙ(𝐨𝐯(𝝈^(X+π∗Y),𝝈)≠1)=o(1).ℙ𝐨𝐯^𝝈subscriptsubscript𝜋𝑋𝑌𝝈1𝑜1\mathbb{P}\left(\mathbf{ov}(\hat{{\boldsymbol{\sigma}}}(X+_{\pi_{*}}Y),{%
\boldsymbol{\sigma}})\neq 1\right)=o(1).blackboard_P ( bold_ov ( over^ start_ARG bold_italic_σ end_ARG ( italic_X + start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_Y ) , bold_italic_σ ) ≠ 1 ) = italic_o ( 1 ) .. , 2 = . , 3 = (60)
By combining the results from (58), (59) and (60), the proof is complete.
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To prove Theorem 5, we consider a two-step procedure for exact matching. The first step utilizes the k𝑘kitalic_k-core matching based solely on edge information to recover the matching over n−n1−ns2(p+q)2logn+o(1)𝑛superscript𝑛1𝑛superscript𝑠2𝑝𝑞2𝑛𝑜1n-n^{1-\frac{ns^{2}(p+q)}{2\log n}+o(1)}italic_n - italic_n start_POSTSUPERSCRIPT 1 - divide start_ARG italic_n italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_p + italic_q ) end_ARG start_ARG 2 roman_log italic_n end_ARG + italic_o ( 1 ) end_POSTSUPERSCRIPT nodes. Then, the second step utilizes the node features to match the rest n1−ns2(p+q)2logn+o(1)superscript𝑛1𝑛superscript𝑠2𝑝𝑞2𝑛𝑜1n^{1-\frac{ns^{2}(p+q)}{2\log n}+o(1)}italic_n start_POSTSUPERSCRIPT 1 - divide start_ARG italic_n italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_p + italic_q ) end_ARG start_ARG 2 roman_log italic_n end_ARG + italic_o ( 1 ) end_POSTSUPERSCRIPT nodes, where we apply the estimator (31) used in proving Theorem 1.
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
IX Proof of Theorem 5:
Achievability of Exact Matching in Correlated Contextual Stochastic Block Models
To prove Theorem 5, we consider a two-step procedure for exact matching. The first step utilizes the k𝑘kitalic_k-core matching based solely on edge information to recover the matching over n−n1−ns2(p+q)2logn+o(1)𝑛superscript𝑛1𝑛superscript𝑠2𝑝𝑞2𝑛𝑜1n-n^{1-\frac{ns^{2}(p+q)}{2\log n}+o(1)}italic_n - italic_n start_POSTSUPERSCRIPT 1 - divide start_ARG italic_n italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_p + italic_q ) end_ARG start_ARG 2 roman_log italic_n end_ARG + italic_o ( 1 ) end_POSTSUPERSCRIPT nodes. Then, the second step utilizes the node features to match the rest n1−ns2(p+q)2logn+o(1)superscript𝑛1𝑛superscript𝑠2𝑝𝑞2𝑛𝑜1n^{1-\frac{ns^{2}(p+q)}{2\log n}+o(1)}italic_n start_POSTSUPERSCRIPT 1 - divide start_ARG italic_n italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_p + italic_q ) end_ARG start_ARG 2 roman_log italic_n end_ARG + italic_o ( 1 ) end_POSTSUPERSCRIPT nodes, where we apply the estimator (31) used in proving Theorem 1.
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The k𝑘kitalic_k-core matching has been extensively studied in recovering the latent vertex correspondence between the edge-correlated Erdős-Rényi graphs or more general inhomogeneous random graphs including the correlated Stochastic Block Models [11, 8, 35, 13]. In this subsection, we will demonstrate that the analytical techniques developed in the previous papers can be effectively applied to the general correlated SBMs we consider in this paper.
We restate the definitions of matching, k𝑘kitalic_k-core matching and k𝑘kitalic_k-core estimator, introduced in [11], for the completeness of the paper.
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IX Proof of Theorem 5:
Achievability of Exact Matching in Correlated Contextual Stochastic Block Models
IX-A The k𝑘kitalic_k-core matching and the proof of Theorem 5
The k𝑘kitalic_k-core matching has been extensively studied in recovering the latent vertex correspondence between the edge-correlated Erdős-Rényi graphs or more general inhomogeneous random graphs including the correlated Stochastic Block Models [11, 8, 35, 13]. In this subsection, we will demonstrate that the analytical techniques developed in the previous papers can be effectively applied to the general correlated SBMs we consider in this paper.
We restate the definitions of matching, k𝑘kitalic_k-core matching and k𝑘kitalic_k-core estimator, introduced in [11], for the completeness of the paper.
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Consider two graphs G1subscript𝐺1G_{1}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and G2subscript𝐺2G_{2}italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. (M,φ)𝑀𝜑(M,\varphi)( italic_M , italic_φ ) is a matching between G1subscript𝐺1G_{1}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and G2subscript𝐺2G_{2}italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT if M⊂[n]𝑀delimited-[]𝑛M\subset[n]italic_M ⊂ [ italic_n ] and φ𝜑\varphiitalic_φ : M→[n]→𝑀delimited-[]𝑛M\to[n]italic_M → [ italic_n ] is injective. For a matching (M,φ)𝑀𝜑(M,\varphi)( italic_M , italic_φ ), we define φ(M)𝜑𝑀\varphi(M)italic_φ ( italic_M ) as the image of M𝑀Mitalic_M under φ𝜑\varphiitalic_φ, and φ{M}:={(i,φ(i)):i∈[M]}assign𝜑𝑀conditional-set𝑖𝜑𝑖𝑖delimited-[]𝑀\varphi\{M\}:=\{(i,\varphi(i)):i\in[M]\}italic_φ { italic_M } := { ( italic_i , italic_φ ( italic_i ) ) : italic_i ∈ [ italic_M ] }.
Consider two graphs G1subscript𝐺1G_{1}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and G2subscript𝐺2G_{2}italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT with a matching (M,φ)𝑀𝜑(M,\varphi)( italic_M , italic_φ ). We define the intersection graph G1∧φG2subscript𝜑subscript𝐺1subscript𝐺2G_{1}\wedge_{\varphi}G_{2}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∧ start_POSTSUBSCRIPT italic_φ end_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT as follows:
•
For u,v∈M𝑢𝑣𝑀u,v\in Mitalic_u , italic_v ∈ italic_M, (u,v)𝑢𝑣(u,v)( italic_u , italic_v ) is an edge in G1∧φG2subscript𝜑subscript𝐺1subscript𝐺2G_{1}\wedge_{\varphi}G_{2}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∧ start_POSTSUBSCRIPT italic_φ end_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT if and only if (u,v)𝑢𝑣(u,v)( italic_u , italic_v ) is an edge in G1subscript𝐺1G_{1}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and (φ(u),φ(v))𝜑𝑢𝜑𝑣(\varphi(u),\varphi(v))( italic_φ ( italic_u ) , italic_φ ( italic_v ) ) is an edge in G2subscript𝐺2G_{2}italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT.Report issue for preceding element
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IX Proof of Theorem 5:
Achievability of Exact Matching in Correlated Contextual Stochastic Block Models
IX-A The k𝑘kitalic_k-core matching and the proof of Theorem 5
Definition 1 (Matching).
Consider two graphs G1subscript𝐺1G_{1}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and G2subscript𝐺2G_{2}italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. (M,φ)𝑀𝜑(M,\varphi)( italic_M , italic_φ ) is a matching between G1subscript𝐺1G_{1}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and G2subscript𝐺2G_{2}italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT if M⊂[n]𝑀delimited-[]𝑛M\subset[n]italic_M ⊂ [ italic_n ] and φ𝜑\varphiitalic_φ : M→[n]→𝑀delimited-[]𝑛M\to[n]italic_M → [ italic_n ] is injective. For a matching (M,φ)𝑀𝜑(M,\varphi)( italic_M , italic_φ ), we define φ(M)𝜑𝑀\varphi(M)italic_φ ( italic_M ) as the image of M𝑀Mitalic_M under φ𝜑\varphiitalic_φ, and φ{M}:={(i,φ(i)):i∈[M]}assign𝜑𝑀conditional-set𝑖𝜑𝑖𝑖delimited-[]𝑀\varphi\{M\}:=\{(i,\varphi(i)):i\in[M]\}italic_φ { italic_M } := { ( italic_i , italic_φ ( italic_i ) ) : italic_i ∈ [ italic_M ] }.
Consider two graphs G1subscript𝐺1G_{1}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and G2subscript𝐺2G_{2}italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT with a matching (M,φ)𝑀𝜑(M,\varphi)( italic_M , italic_φ ). We define the intersection graph G1∧φG2subscript𝜑subscript𝐺1subscript𝐺2G_{1}\wedge_{\varphi}G_{2}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∧ start_POSTSUBSCRIPT italic_φ end_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT as follows:
•
For u,v∈M𝑢𝑣𝑀u,v\in Mitalic_u , italic_v ∈ italic_M, (u,v)𝑢𝑣(u,v)( italic_u , italic_v ) is an edge in G1∧φG2subscript𝜑subscript𝐺1subscript𝐺2G_{1}\wedge_{\varphi}G_{2}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∧ start_POSTSUBSCRIPT italic_φ end_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT if and only if (u,v)𝑢𝑣(u,v)( italic_u , italic_v ) is an edge in G1subscript𝐺1G_{1}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and (φ(u),φ(v))𝜑𝑢𝜑𝑣(\varphi(u),\varphi(v))( italic_φ ( italic_u ) , italic_φ ( italic_v ) ) is an edge in G2subscript𝐺2G_{2}italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT.Report issue for preceding element
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Consider two graphs G1subscript𝐺1G_{1}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and G2subscript𝐺2G_{2}italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. A matching (M,φ)𝑀𝜑(M,\varphi)( italic_M , italic_φ ) is a k𝑘kitalic_k-core matching if dmin(G1∧φG2)≥ksubscript𝑑subscript𝜑subscript𝐺1subscript𝐺2𝑘d_{\min}(G_{1}\wedge_{\varphi}G_{2})\geq kitalic_d start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∧ start_POSTSUBSCRIPT italic_φ end_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ≥ italic_k. Furthermore, the k𝑘kitalic_k-core estimator (M^k,φ^k)subscript^𝑀𝑘subscript^𝜑𝑘(\widehat{M}_{k},\widehat{\varphi}_{k})( over^ start_ARG italic_M end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , over^ start_ARG italic_φ end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) is the k𝑘kitalic_k-core matching that includes the largest nodes among all the k𝑘kitalic_k-core matchings.
By using the k𝑘kitalic_k-core estimator, we can achieve a partial matching with no mismatched node pairs, as stated in the following theorem.
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IX Proof of Theorem 5:
Achievability of Exact Matching in Correlated Contextual Stochastic Block Models
IX-A The k𝑘kitalic_k-core matching and the proof of Theorem 5
Definition 2 (k𝑘kitalic_k-core matching and k𝑘kitalic_k-core estimator).
Consider two graphs G1subscript𝐺1G_{1}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and G2subscript𝐺2G_{2}italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. A matching (M,φ)𝑀𝜑(M,\varphi)( italic_M , italic_φ ) is a k𝑘kitalic_k-core matching if dmin(G1∧φG2)≥ksubscript𝑑subscript𝜑subscript𝐺1subscript𝐺2𝑘d_{\min}(G_{1}\wedge_{\varphi}G_{2})\geq kitalic_d start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∧ start_POSTSUBSCRIPT italic_φ end_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ≥ italic_k. Furthermore, the k𝑘kitalic_k-core estimator (M^k,φ^k)subscript^𝑀𝑘subscript^𝜑𝑘(\widehat{M}_{k},\widehat{\varphi}_{k})( over^ start_ARG italic_M end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , over^ start_ARG italic_φ end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) is the k𝑘kitalic_k-core matching that includes the largest nodes among all the k𝑘kitalic_k-core matchings.
By using the k𝑘kitalic_k-core estimator, we can achieve a partial matching with no mismatched node pairs, as stated in the following theorem.
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Consider the correlated Stochastic Block Models with two communities (G1,G2)∼CSBMs(n,p,q,s)similar-tosubscript𝐺1subscript𝐺2CSBMs𝑛𝑝𝑞𝑠(G_{1},G_{2})\sim\textnormal{CSBMs}(n,p,q,s)( italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ∼ CSBMs ( italic_n , italic_p , italic_q , italic_s ). Suppose that
, 1 = p≤O(1e(loglogn)3) and𝑝𝑂1superscript𝑒superscript𝑛3 andp\leq O\left(\frac{1}{e^{(\log\log n)^{3}}}\right)\text{ and}italic_p ≤ italic_O ( divide start_ARG 1 end_ARG start_ARG italic_e start_POSTSUPERSCRIPT ( roman_log roman_log italic_n ) start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT end_ARG ) and. , 2 = . , 3 = (61)
, 1 = k=nps2(lognps2)2∨logn(loglogn)2.𝑘𝑛𝑝superscript𝑠2superscript𝑛𝑝superscript𝑠22𝑛superscript𝑛2k=\frac{nps^{2}}{(\log nps^{2})^{2}}\vee\frac{\log n}{(\log\log n)^{2}}.italic_k = divide start_ARG italic_n italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG ( roman_log italic_n italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ∨ divide start_ARG roman_log italic_n end_ARG start_ARG ( roman_log roman_log italic_n ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG .. , 2 = . , 3 = (62)
Then, the k𝑘kitalic_k-core estimator (M^k,φ^k)subscript^𝑀𝑘subscript^𝜑𝑘(\widehat{M}_{k},\widehat{\varphi}_{k})( over^ start_ARG italic_M end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , over^ start_ARG italic_φ end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) satisfies that
, 1 = |M^k|≥n−n1−ns2(p+q)2logn+o(1) andsubscript^𝑀𝑘𝑛superscript𝑛1𝑛superscript𝑠2𝑝𝑞2𝑛𝑜1 and|\widehat{M}_{k}|\geq n-n^{1-\frac{ns^{2}(p+q)}{2\log n}+o(1)}\text{ and}| over^ start_ARG italic_M end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | ≥ italic_n - italic_n start_POSTSUPERSCRIPT 1 - divide start_ARG italic_n italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_p + italic_q ) end_ARG start_ARG 2 roman_log italic_n end_ARG + italic_o ( 1 ) end_POSTSUPERSCRIPT and. , 2 = . , 3 = (63)
, 1 = φ^k{M^k}=π∗{M^k}subscript^𝜑𝑘subscript^𝑀𝑘subscript𝜋subscript^𝑀𝑘\widehat{\varphi}_{k}\{\widehat{M}_{k}\}=\pi_{*}\{\widehat{M}_{k}\}over^ start_ARG italic_φ end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT { over^ start_ARG italic_M end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT } = italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT { over^ start_ARG italic_M end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT }. , 2 = . , 3 = (64)
with probability 1−o(1)1𝑜11-o(1)1 - italic_o ( 1 ).
Furthermore, the matched node set M^ksubscript^𝑀𝑘\widehat{M}_{k}over^ start_ARG italic_M end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT is equivalent to the k𝑘kitalic_k-core set of the graph G1∧π∗G2subscriptsubscript𝜋subscript𝐺1subscript𝐺2G_{1}\wedge_{\pi_{*}}G_{2}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∧ start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. A detailed explanation and the related lemmas for the k𝑘kitalic_k-core matching can be found in Section IX-B.
We also state the sufficient conditions for exact matching in the correlated Stochastic Block Models achievable by the k𝑘kitalic_k-core estimator.
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IX Proof of Theorem 5:
Achievability of Exact Matching in Correlated Contextual Stochastic Block Models
IX-A The k𝑘kitalic_k-core matching and the proof of Theorem 5
Theorem 12 (Partial matching achievable by the k𝑘kitalic_k-core estimator).
Consider the correlated Stochastic Block Models with two communities (G1,G2)∼CSBMs(n,p,q,s)similar-tosubscript𝐺1subscript𝐺2CSBMs𝑛𝑝𝑞𝑠(G_{1},G_{2})\sim\textnormal{CSBMs}(n,p,q,s)( italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ∼ CSBMs ( italic_n , italic_p , italic_q , italic_s ). Suppose that
, 1 = p≤O(1e(loglogn)3) and𝑝𝑂1superscript𝑒superscript𝑛3 andp\leq O\left(\frac{1}{e^{(\log\log n)^{3}}}\right)\text{ and}italic_p ≤ italic_O ( divide start_ARG 1 end_ARG start_ARG italic_e start_POSTSUPERSCRIPT ( roman_log roman_log italic_n ) start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT end_ARG ) and. , 2 = . , 3 = (61)
, 1 = k=nps2(lognps2)2∨logn(loglogn)2.𝑘𝑛𝑝superscript𝑠2superscript𝑛𝑝superscript𝑠22𝑛superscript𝑛2k=\frac{nps^{2}}{(\log nps^{2})^{2}}\vee\frac{\log n}{(\log\log n)^{2}}.italic_k = divide start_ARG italic_n italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG ( roman_log italic_n italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ∨ divide start_ARG roman_log italic_n end_ARG start_ARG ( roman_log roman_log italic_n ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG .. , 2 = . , 3 = (62)
Then, the k𝑘kitalic_k-core estimator (M^k,φ^k)subscript^𝑀𝑘subscript^𝜑𝑘(\widehat{M}_{k},\widehat{\varphi}_{k})( over^ start_ARG italic_M end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , over^ start_ARG italic_φ end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) satisfies that
, 1 = |M^k|≥n−n1−ns2(p+q)2logn+o(1) andsubscript^𝑀𝑘𝑛superscript𝑛1𝑛superscript𝑠2𝑝𝑞2𝑛𝑜1 and|\widehat{M}_{k}|\geq n-n^{1-\frac{ns^{2}(p+q)}{2\log n}+o(1)}\text{ and}| over^ start_ARG italic_M end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | ≥ italic_n - italic_n start_POSTSUPERSCRIPT 1 - divide start_ARG italic_n italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_p + italic_q ) end_ARG start_ARG 2 roman_log italic_n end_ARG + italic_o ( 1 ) end_POSTSUPERSCRIPT and. , 2 = . , 3 = (63)
, 1 = φ^k{M^k}=π∗{M^k}subscript^𝜑𝑘subscript^𝑀𝑘subscript𝜋subscript^𝑀𝑘\widehat{\varphi}_{k}\{\widehat{M}_{k}\}=\pi_{*}\{\widehat{M}_{k}\}over^ start_ARG italic_φ end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT { over^ start_ARG italic_M end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT } = italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT { over^ start_ARG italic_M end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT }. , 2 = . , 3 = (64)
with probability 1−o(1)1𝑜11-o(1)1 - italic_o ( 1 ).
Furthermore, the matched node set M^ksubscript^𝑀𝑘\widehat{M}_{k}over^ start_ARG italic_M end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT is equivalent to the k𝑘kitalic_k-core set of the graph G1∧π∗G2subscriptsubscript𝜋subscript𝐺1subscript𝐺2G_{1}\wedge_{\pi_{*}}G_{2}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∧ start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. A detailed explanation and the related lemmas for the k𝑘kitalic_k-core matching can be found in Section IX-B.
We also state the sufficient conditions for exact matching in the correlated Stochastic Block Models achievable by the k𝑘kitalic_k-core estimator.
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Consider the correlated Stochastic Block Models with two communities (G1,G2)∼CSBMs(n,p,q,s)similar-tosubscript𝐺1subscript𝐺2CSBMs𝑛𝑝𝑞𝑠(G_{1},G_{2})\sim\textnormal{CSBMs}(n,p,q,s)( italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ∼ CSBMs ( italic_n , italic_p , italic_q , italic_s ). Suppose that (61)italic-(61italic-)\eqref{eq:k-core p regmie}italic_( italic_) and (62) hold. Also, assume that
, 1 = ns2p+q2≥(1+ϵ)logn𝑛superscript𝑠2𝑝𝑞21italic-ϵ𝑛ns^{2}\frac{p+q}{2}\geq(1+\epsilon)\log nitalic_n italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divide start_ARG italic_p + italic_q end_ARG start_ARG 2 end_ARG ≥ ( 1 + italic_ϵ ) roman_log italic_n. , 2 = . , 3 = (65)
for an arbitrary small constant ϵ>0italic-ϵ0\epsilon>0italic_ϵ > 0. Then, φ^k=π∗subscript^𝜑𝑘subscript𝜋\widehat{\varphi}_{k}=\pi_{*}over^ start_ARG italic_φ end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT with probability 1−o(1)1𝑜11-o(1)1 - italic_o ( 1 ).
The proofs of Theorem 12 and 13 will be presented in Section IX-C.
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IX Proof of Theorem 5:
Achievability of Exact Matching in Correlated Contextual Stochastic Block Models
IX-A The k𝑘kitalic_k-core matching and the proof of Theorem 5
Theorem 13 (Exact matching achievable by the k𝑘kitalic_k-core estimator).
Consider the correlated Stochastic Block Models with two communities (G1,G2)∼CSBMs(n,p,q,s)similar-tosubscript𝐺1subscript𝐺2CSBMs𝑛𝑝𝑞𝑠(G_{1},G_{2})\sim\textnormal{CSBMs}(n,p,q,s)( italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ∼ CSBMs ( italic_n , italic_p , italic_q , italic_s ). Suppose that (61)italic-(61italic-)\eqref{eq:k-core p regmie}italic_( italic_) and (62) hold. Also, assume that
, 1 = ns2p+q2≥(1+ϵ)logn𝑛superscript𝑠2𝑝𝑞21italic-ϵ𝑛ns^{2}\frac{p+q}{2}\geq(1+\epsilon)\log nitalic_n italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divide start_ARG italic_p + italic_q end_ARG start_ARG 2 end_ARG ≥ ( 1 + italic_ϵ ) roman_log italic_n. , 2 = . , 3 = (65)
for an arbitrary small constant ϵ>0italic-ϵ0\epsilon>0italic_ϵ > 0. Then, φ^k=π∗subscript^𝜑𝑘subscript𝜋\widehat{\varphi}_{k}=\pi_{*}over^ start_ARG italic_φ end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT with probability 1−o(1)1𝑜11-o(1)1 - italic_o ( 1 ).
The proofs of Theorem 12 and 13 will be presented in Section IX-C.
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Let k=nps2(lognps2)2∨logn(loglogn)2𝑘𝑛𝑝superscript𝑠2superscript𝑛𝑝superscript𝑠22𝑛superscript𝑛2k=\frac{nps^{2}}{(\log nps^{2})^{2}}\vee\frac{\log n}{(\log\log n)^{2}}italic_k = divide start_ARG italic_n italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG ( roman_log italic_n italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ∨ divide start_ARG roman_log italic_n end_ARG start_ARG ( roman_log roman_log italic_n ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG. Let us first consider the case where nps2≥(1+ϵ)logn𝑛𝑝superscript𝑠21italic-ϵ𝑛nps^{2}\geq(1+\epsilon)\log nitalic_n italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ ( 1 + italic_ϵ ) roman_log italic_n for an arbitrary small constant ϵ>0italic-ϵ0\epsilon>0italic_ϵ > 0. By Theorem 13, we can confirm that the exact matching is achievable through the k𝑘kitalic_k-core matching using only edge information, without relying on node information.
Now, let us consider the case where nps2=O(logn)𝑛𝑝superscript𝑠2𝑂𝑛nps^{2}=O(\log n)italic_n italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = italic_O ( roman_log italic_n ). First, by Theorem 12, we can obtain a matching (M^k,φ^k)subscript^𝑀𝑘subscript^𝜑𝑘(\widehat{M}_{k},\widehat{\varphi}_{k})( over^ start_ARG italic_M end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , over^ start_ARG italic_φ end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) that satisfies (63) and (64)italic-(64italic-)\eqref{eq:k-core groundtruth}italic_( italic_) through the k𝑘kitalic_k-core matching. Let F𝐹Fitalic_F denote the set of nodes that remain unmatched after performing the k𝑘kitalic_k-core matching. That is, F:=[n]\M^kassign𝐹\delimited-[]𝑛subscript^𝑀𝑘F:=[n]\backslash\widehat{M}_{k}italic_F := [ italic_n ] \ over^ start_ARG italic_M end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT. From (63), we have
, 1 = |F|≤n1−ns2(p+q)2logn+o(1).𝐹superscript𝑛1𝑛superscript𝑠2𝑝𝑞2𝑛𝑜1|F|\leq n^{1-\frac{ns^{2}(p+q)}{2\log n}+o(1)}.| italic_F | ≤ italic_n start_POSTSUPERSCRIPT 1 - divide start_ARG italic_n italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_p + italic_q ) end_ARG start_ARG 2 roman_log italic_n end_ARG + italic_o ( 1 ) end_POSTSUPERSCRIPT .. , 2 = . , 3 = (66)
From (66) and (23), we can also have
, 1 = d4log11−ρ2≥(1+ϵ)logn−ns2p+q2≥(1+ϵ/2)log|F|𝑑411superscript𝜌21italic-ϵ𝑛𝑛superscript𝑠2𝑝𝑞21italic-ϵ2𝐹\frac{d}{4}\log\frac{1}{1-\rho^{2}}\geq(1+\epsilon)\log n-ns^{2}\frac{p+q}{2}%
\geq\left(1+\epsilon/2\right)\log|F|divide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≥ ( 1 + italic_ϵ ) roman_log italic_n - italic_n italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divide start_ARG italic_p + italic_q end_ARG start_ARG 2 end_ARG ≥ ( 1 + italic_ϵ / 2 ) roman_log | italic_F |. , 2 = . , 3 = (67)
for a sufficiently large n𝑛nitalic_n. Therefore, if ∥𝝁∥2≥2logn+ω(1)superscriptdelimited-∥∥𝝁22𝑛𝜔1\lVert\boldsymbol{\mu}\rVert^{2}\geq 2\log n+\omega(1)∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ 2 roman_log italic_n + italic_ω ( 1 ) or d=ω(logn)𝑑𝜔𝑛d=\omega(\log n)italic_d = italic_ω ( roman_log italic_n ), then the exact matching is possible between the unmatched nodes belonging to F𝐹Fitalic_F solely using the node features by Theorem 1. Thus, the proof is complete.
∎
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IX Proof of Theorem 5:
Achievability of Exact Matching in Correlated Contextual Stochastic Block Models
IX-A The k𝑘kitalic_k-core matching and the proof of Theorem 5
Proof:
Let k=nps2(lognps2)2∨logn(loglogn)2𝑘𝑛𝑝superscript𝑠2superscript𝑛𝑝superscript𝑠22𝑛superscript𝑛2k=\frac{nps^{2}}{(\log nps^{2})^{2}}\vee\frac{\log n}{(\log\log n)^{2}}italic_k = divide start_ARG italic_n italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG ( roman_log italic_n italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ∨ divide start_ARG roman_log italic_n end_ARG start_ARG ( roman_log roman_log italic_n ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG. Let us first consider the case where nps2≥(1+ϵ)logn𝑛𝑝superscript𝑠21italic-ϵ𝑛nps^{2}\geq(1+\epsilon)\log nitalic_n italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ ( 1 + italic_ϵ ) roman_log italic_n for an arbitrary small constant ϵ>0italic-ϵ0\epsilon>0italic_ϵ > 0. By Theorem 13, we can confirm that the exact matching is achievable through the k𝑘kitalic_k-core matching using only edge information, without relying on node information.
Now, let us consider the case where nps2=O(logn)𝑛𝑝superscript𝑠2𝑂𝑛nps^{2}=O(\log n)italic_n italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = italic_O ( roman_log italic_n ). First, by Theorem 12, we can obtain a matching (M^k,φ^k)subscript^𝑀𝑘subscript^𝜑𝑘(\widehat{M}_{k},\widehat{\varphi}_{k})( over^ start_ARG italic_M end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , over^ start_ARG italic_φ end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) that satisfies (63) and (64)italic-(64italic-)\eqref{eq:k-core groundtruth}italic_( italic_) through the k𝑘kitalic_k-core matching. Let F𝐹Fitalic_F denote the set of nodes that remain unmatched after performing the k𝑘kitalic_k-core matching. That is, F:=[n]\M^kassign𝐹\delimited-[]𝑛subscript^𝑀𝑘F:=[n]\backslash\widehat{M}_{k}italic_F := [ italic_n ] \ over^ start_ARG italic_M end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT. From (63), we have
, 1 = |F|≤n1−ns2(p+q)2logn+o(1).𝐹superscript𝑛1𝑛superscript𝑠2𝑝𝑞2𝑛𝑜1|F|\leq n^{1-\frac{ns^{2}(p+q)}{2\log n}+o(1)}.| italic_F | ≤ italic_n start_POSTSUPERSCRIPT 1 - divide start_ARG italic_n italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_p + italic_q ) end_ARG start_ARG 2 roman_log italic_n end_ARG + italic_o ( 1 ) end_POSTSUPERSCRIPT .. , 2 = . , 3 = (66)
From (66) and (23), we can also have
, 1 = d4log11−ρ2≥(1+ϵ)logn−ns2p+q2≥(1+ϵ/2)log|F|𝑑411superscript𝜌21italic-ϵ𝑛𝑛superscript𝑠2𝑝𝑞21italic-ϵ2𝐹\frac{d}{4}\log\frac{1}{1-\rho^{2}}\geq(1+\epsilon)\log n-ns^{2}\frac{p+q}{2}%
\geq\left(1+\epsilon/2\right)\log|F|divide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≥ ( 1 + italic_ϵ ) roman_log italic_n - italic_n italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divide start_ARG italic_p + italic_q end_ARG start_ARG 2 end_ARG ≥ ( 1 + italic_ϵ / 2 ) roman_log | italic_F |. , 2 = . , 3 = (67)
for a sufficiently large n𝑛nitalic_n. Therefore, if ∥𝝁∥2≥2logn+ω(1)superscriptdelimited-∥∥𝝁22𝑛𝜔1\lVert\boldsymbol{\mu}\rVert^{2}\geq 2\log n+\omega(1)∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ 2 roman_log italic_n + italic_ω ( 1 ) or d=ω(logn)𝑑𝜔𝑛d=\omega(\log n)italic_d = italic_ω ( roman_log italic_n ), then the exact matching is possible between the unmatched nodes belonging to F𝐹Fitalic_F solely using the node features by Theorem 1. Thus, the proof is complete.
∎
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For a matching (M,φ)𝑀𝜑(M,\varphi)( italic_M , italic_φ ), define
, 1 = f(M,φ)=Σi∈M:φ(i)≠π∗(i)degG1∧φG2(i).𝑓𝑀𝜑subscriptΣ:𝑖𝑀𝜑𝑖subscript𝜋𝑖subscriptdegreesubscript𝜑subscript𝐺1subscript𝐺2𝑖f(M,\varphi)=\Sigma_{i\in M:\varphi(i)\neq\pi_{*}(i)}\deg_{G_{1}\wedge_{%
\varphi}G_{2}}(i).italic_f ( italic_M , italic_φ ) = roman_Σ start_POSTSUBSCRIPT italic_i ∈ italic_M : italic_φ ( italic_i ) ≠ italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_i ) end_POSTSUBSCRIPT roman_deg start_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∧ start_POSTSUBSCRIPT italic_φ end_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_i ) .. , 2 = . , 3 = (68)
The weak k𝑘kitalic_k-core matching, π∗subscript𝜋\pi_{*}italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT-maximal matching and k𝑘kitalic_k-core set were defined in [11, 8, 35, 13]. For the completeness, we present the corresponding definitions below.
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IX Proof of Theorem 5:
Achievability of Exact Matching in Correlated Contextual Stochastic Block Models
IX-B Lemmas for the analysis of k𝑘kitalic_k-core matching
For a matching (M,φ)𝑀𝜑(M,\varphi)( italic_M , italic_φ ), define
, 1 = f(M,φ)=Σi∈M:φ(i)≠π∗(i)degG1∧φG2(i).𝑓𝑀𝜑subscriptΣ:𝑖𝑀𝜑𝑖subscript𝜋𝑖subscriptdegreesubscript𝜑subscript𝐺1subscript𝐺2𝑖f(M,\varphi)=\Sigma_{i\in M:\varphi(i)\neq\pi_{*}(i)}\deg_{G_{1}\wedge_{%
\varphi}G_{2}}(i).italic_f ( italic_M , italic_φ ) = roman_Σ start_POSTSUBSCRIPT italic_i ∈ italic_M : italic_φ ( italic_i ) ≠ italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_i ) end_POSTSUBSCRIPT roman_deg start_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∧ start_POSTSUBSCRIPT italic_φ end_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_i ) .. , 2 = . , 3 = (68)
The weak k𝑘kitalic_k-core matching, π∗subscript𝜋\pi_{*}italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT-maximal matching and k𝑘kitalic_k-core set were defined in [11, 8, 35, 13]. For the completeness, we present the corresponding definitions below.
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We say that a matching (M,φ)𝑀𝜑(M,\varphi)( italic_M , italic_φ ) is a weak k𝑘kitalic_k-core matching if
, 1 = f(M,φ)≥k|{i∈M:φ(i)≠π∗(i)}|.𝑓𝑀𝜑𝑘conditional-set𝑖𝑀𝜑𝑖subscript𝜋𝑖f(M,\varphi)\geq k|\{i\in M:\varphi(i)\neq\pi_{*}(i)\}|.italic_f ( italic_M , italic_φ ) ≥ italic_k | { italic_i ∈ italic_M : italic_φ ( italic_i ) ≠ italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_i ) } | .. , 2 = . , 3 = (69)
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IX Proof of Theorem 5:
Achievability of Exact Matching in Correlated Contextual Stochastic Block Models
IX-B Lemmas for the analysis of k𝑘kitalic_k-core matching
Definition 3 (Weak k𝑘kitalic_k-core matching).
We say that a matching (M,φ)𝑀𝜑(M,\varphi)( italic_M , italic_φ ) is a weak k𝑘kitalic_k-core matching if
, 1 = f(M,φ)≥k|{i∈M:φ(i)≠π∗(i)}|.𝑓𝑀𝜑𝑘conditional-set𝑖𝑀𝜑𝑖subscript𝜋𝑖f(M,\varphi)\geq k|\{i\in M:\varphi(i)\neq\pi_{*}(i)\}|.italic_f ( italic_M , italic_φ ) ≥ italic_k | { italic_i ∈ italic_M : italic_φ ( italic_i ) ≠ italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_i ) } | .. , 2 = . , 3 = (69)
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We say that a matching (M,φ)𝑀𝜑(M,\varphi)( italic_M , italic_φ ) is π∗subscript𝜋\pi_{*}italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT-maximal if for every i∈[n]𝑖delimited-[]𝑛i\in[n]italic_i ∈ [ italic_n ], either i∈M𝑖𝑀i\in Mitalic_i ∈ italic_M or π∗(i)∈φ(M)subscript𝜋𝑖𝜑𝑀\pi_{*}(i)\in\varphi(M)italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_i ) ∈ italic_φ ( italic_M ), where φ(M)𝜑𝑀\varphi(M)italic_φ ( italic_M ) is the image of M𝑀Mitalic_M under φ𝜑\varphiitalic_φ. Additionally, let us define ℳ(t)ℳ𝑡\mathcal{M}(t)caligraphic_M ( italic_t ) as the set of π∗subscript𝜋\pi_{*}italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT-maximal matchings that have t𝑡titalic_t errors. It means that
, 1 = ℳ(t):={(M,φ):(M,φ) is π∗-maximal and |{i∈M:φ(i)≠π∗(i)}|=t}.assignℳ𝑡conditional-set𝑀𝜑𝑀𝜑 is subscript𝜋-maximal and conditional-set𝑖𝑀𝜑𝑖subscript𝜋𝑖𝑡\displaystyle\mathcal{M}(t):=\{(M,\varphi):(M,\varphi)\text{ is }\pi_{*}\text{%
-maximal and }|\{i\in M:\varphi(i)\neq\pi_{*}(i)\}|=t\}.caligraphic_M ( italic_t ) := { ( italic_M , italic_φ ) : ( italic_M , italic_φ ) is italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT -maximal and | { italic_i ∈ italic_M : italic_φ ( italic_i ) ≠ italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_i ) } | = italic_t } .. , 2 = . , 3 = (70)
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IX Proof of Theorem 5:
Achievability of Exact Matching in Correlated Contextual Stochastic Block Models
IX-B Lemmas for the analysis of k𝑘kitalic_k-core matching
Definition 4 (π∗subscript𝜋\pi_{*}italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT-maximal matching).
We say that a matching (M,φ)𝑀𝜑(M,\varphi)( italic_M , italic_φ ) is π∗subscript𝜋\pi_{*}italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT-maximal if for every i∈[n]𝑖delimited-[]𝑛i\in[n]italic_i ∈ [ italic_n ], either i∈M𝑖𝑀i\in Mitalic_i ∈ italic_M or π∗(i)∈φ(M)subscript𝜋𝑖𝜑𝑀\pi_{*}(i)\in\varphi(M)italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_i ) ∈ italic_φ ( italic_M ), where φ(M)𝜑𝑀\varphi(M)italic_φ ( italic_M ) is the image of M𝑀Mitalic_M under φ𝜑\varphiitalic_φ. Additionally, let us define ℳ(t)ℳ𝑡\mathcal{M}(t)caligraphic_M ( italic_t ) as the set of π∗subscript𝜋\pi_{*}italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT-maximal matchings that have t𝑡titalic_t errors. It means that
, 1 = ℳ(t):={(M,φ):(M,φ) is π∗-maximal and |{i∈M:φ(i)≠π∗(i)}|=t}.assignℳ𝑡conditional-set𝑀𝜑𝑀𝜑 is subscript𝜋-maximal and conditional-set𝑖𝑀𝜑𝑖subscript𝜋𝑖𝑡\displaystyle\mathcal{M}(t):=\{(M,\varphi):(M,\varphi)\text{ is }\pi_{*}\text{%
-maximal and }|\{i\in M:\varphi(i)\neq\pi_{*}(i)\}|=t\}.caligraphic_M ( italic_t ) := { ( italic_M , italic_φ ) : ( italic_M , italic_φ ) is italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT -maximal and | { italic_i ∈ italic_M : italic_φ ( italic_i ) ≠ italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_i ) } | = italic_t } .. , 2 = . , 3 = (70)
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For a graph G𝐺Gitalic_G, a vertex set M𝑀Mitalic_M is referred to as the k𝑘kitalic_k-core of G𝐺Gitalic_G if it is the largest set such that dmin(G{M})≥ksubscript𝑑𝐺𝑀𝑘d_{\min}(G\{M\})\geq kitalic_d start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( italic_G { italic_M } ) ≥ italic_k.
Let Mksubscript𝑀𝑘M_{k}italic_M start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT denote the k𝑘kitalic_k-core of G1∧π∗G2subscriptsubscript𝜋subscript𝐺1subscript𝐺2G_{1}\wedge_{\pi_{*}}G_{2}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∧ start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. The following lemma provides a lower bound on the probability to obtain the k𝑘kitalic_k-core of G1∧π∗G2subscriptsubscript𝜋subscript𝐺1subscript𝐺2G_{1}\wedge_{\pi_{*}}G_{2}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∧ start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT and the correct matching over the k𝑘kitalic_k-core as the result of k𝑘kitalic_k-core matching. Since Gaudio et al. [8] presented this result for general pairs of random graphs in Corollary 20, it can also be applied to the correlated SBMs.
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IX Proof of Theorem 5:
Achievability of Exact Matching in Correlated Contextual Stochastic Block Models
IX-B Lemmas for the analysis of k𝑘kitalic_k-core matching
Definition 5 (k𝑘kitalic_k-core set).
For a graph G𝐺Gitalic_G, a vertex set M𝑀Mitalic_M is referred to as the k𝑘kitalic_k-core of G𝐺Gitalic_G if it is the largest set such that dmin(G{M})≥ksubscript𝑑𝐺𝑀𝑘d_{\min}(G\{M\})\geq kitalic_d start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( italic_G { italic_M } ) ≥ italic_k.
Let Mksubscript𝑀𝑘M_{k}italic_M start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT denote the k𝑘kitalic_k-core of G1∧π∗G2subscriptsubscript𝜋subscript𝐺1subscript𝐺2G_{1}\wedge_{\pi_{*}}G_{2}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∧ start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. The following lemma provides a lower bound on the probability to obtain the k𝑘kitalic_k-core of G1∧π∗G2subscriptsubscript𝜋subscript𝐺1subscript𝐺2G_{1}\wedge_{\pi_{*}}G_{2}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∧ start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT and the correct matching over the k𝑘kitalic_k-core as the result of k𝑘kitalic_k-core matching. Since Gaudio et al. [8] presented this result for general pairs of random graphs in Corollary 20, it can also be applied to the correlated SBMs.
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Consider the (G1,G2)∼CSBMs(n,p,q,s)similar-tosubscript𝐺1subscript𝐺2CSBMs𝑛𝑝𝑞𝑠(G_{1},G_{2})\sim\textnormal{CSBMs}(n,p,q,s)( italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ∼ CSBMs ( italic_n , italic_p , italic_q , italic_s ). For any positive integer k𝑘kitalic_k, define the quantity
, 1 = ξ:=max1≤t≤nmax(M,φ)∈ℳ(t)ℙ(f(M,φ)≥kt)1/t.assign𝜉subscript1𝑡𝑛subscript𝑀𝜑ℳ𝑡ℙsuperscript𝑓𝑀𝜑𝑘𝑡1𝑡\xi:=\max_{1\leq t\leq n}\max_{(M,\varphi)\in\mathcal{M}(t)}\mathbb{P}(f(M,%
\varphi)\geq kt)^{1/t}.italic_ξ := roman_max start_POSTSUBSCRIPT 1 ≤ italic_t ≤ italic_n end_POSTSUBSCRIPT roman_max start_POSTSUBSCRIPT ( italic_M , italic_φ ) ∈ caligraphic_M ( italic_t ) end_POSTSUBSCRIPT blackboard_P ( italic_f ( italic_M , italic_φ ) ≥ italic_k italic_t ) start_POSTSUPERSCRIPT 1 / italic_t end_POSTSUPERSCRIPT .. , 2 = . , 3 = (71)
Then, the k𝑘kitalic_k-core estimator (M^k,φ^k)subscript^𝑀𝑘subscript^𝜑𝑘(\widehat{M}_{k},\widehat{\varphi}_{k})( over^ start_ARG italic_M end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , over^ start_ARG italic_φ end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) satisfies that
, 1 = ℙ(M^k=Mk and φ^k{M^k}\displaystyle\mathbb{P}\left(\widehat{M}_{k}=M_{k}\text{ and }\widehat{\varphi%
}_{k}\{\widehat{M}_{k}\}\right.blackboard_P ( over^ start_ARG italic_M end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_M start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT and over^ start_ARG italic_φ end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT { over^ start_ARG italic_M end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT }. , 2 = =π∗{M^k})≥2−exp(n2ξ).\displaystyle\left.=\pi_{*}\{\widehat{M}_{k}\}\right)\geq 2-\exp\left(n^{2}\xi%
\right).= italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT { over^ start_ARG italic_M end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT } ) ≥ 2 - roman_exp ( italic_n start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_ξ ) .. , 3 = . , 4 = (72)
From the above lemma, we can see that if ξ=o(n−2)𝜉𝑜superscript𝑛2\xi=o(n^{-2})italic_ξ = italic_o ( italic_n start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT ), the matching obtained through the k𝑘kitalic_k-core estimator is the correct matching over the k𝑘kitalic_k-core of G1∧π∗G2subscriptsubscript𝜋subscript𝐺1subscript𝐺2G_{1}\wedge_{\pi_{*}}G_{2}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∧ start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT with high probability.
The next lemma represents a upper bound on ξ𝜉\xiitalic_ξ. This result has been proven in [8, 13, 35], and in particular [35] provides a proof for general random graph pairs in Lemma A.4. We obtain the similar results in the correlated Stochastic Block Models.
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
IX Proof of Theorem 5:
Achievability of Exact Matching in Correlated Contextual Stochastic Block Models
IX-B Lemmas for the analysis of k𝑘kitalic_k-core matching
Lemma 3 (Corollary 4.7 in [8]).
Consider the (G1,G2)∼CSBMs(n,p,q,s)similar-tosubscript𝐺1subscript𝐺2CSBMs𝑛𝑝𝑞𝑠(G_{1},G_{2})\sim\textnormal{CSBMs}(n,p,q,s)( italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ∼ CSBMs ( italic_n , italic_p , italic_q , italic_s ). For any positive integer k𝑘kitalic_k, define the quantity
, 1 = ξ:=max1≤t≤nmax(M,φ)∈ℳ(t)ℙ(f(M,φ)≥kt)1/t.assign𝜉subscript1𝑡𝑛subscript𝑀𝜑ℳ𝑡ℙsuperscript𝑓𝑀𝜑𝑘𝑡1𝑡\xi:=\max_{1\leq t\leq n}\max_{(M,\varphi)\in\mathcal{M}(t)}\mathbb{P}(f(M,%
\varphi)\geq kt)^{1/t}.italic_ξ := roman_max start_POSTSUBSCRIPT 1 ≤ italic_t ≤ italic_n end_POSTSUBSCRIPT roman_max start_POSTSUBSCRIPT ( italic_M , italic_φ ) ∈ caligraphic_M ( italic_t ) end_POSTSUBSCRIPT blackboard_P ( italic_f ( italic_M , italic_φ ) ≥ italic_k italic_t ) start_POSTSUPERSCRIPT 1 / italic_t end_POSTSUPERSCRIPT .. , 2 = . , 3 = (71)
Then, the k𝑘kitalic_k-core estimator (M^k,φ^k)subscript^𝑀𝑘subscript^𝜑𝑘(\widehat{M}_{k},\widehat{\varphi}_{k})( over^ start_ARG italic_M end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , over^ start_ARG italic_φ end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) satisfies that
, 1 = ℙ(M^k=Mk and φ^k{M^k}\displaystyle\mathbb{P}\left(\widehat{M}_{k}=M_{k}\text{ and }\widehat{\varphi%
}_{k}\{\widehat{M}_{k}\}\right.blackboard_P ( over^ start_ARG italic_M end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_M start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT and over^ start_ARG italic_φ end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT { over^ start_ARG italic_M end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT }. , 2 = =π∗{M^k})≥2−exp(n2ξ).\displaystyle\left.=\pi_{*}\{\widehat{M}_{k}\}\right)\geq 2-\exp\left(n^{2}\xi%
\right).= italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT { over^ start_ARG italic_M end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT } ) ≥ 2 - roman_exp ( italic_n start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_ξ ) .. , 3 = . , 4 = (72)
From the above lemma, we can see that if ξ=o(n−2)𝜉𝑜superscript𝑛2\xi=o(n^{-2})italic_ξ = italic_o ( italic_n start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT ), the matching obtained through the k𝑘kitalic_k-core estimator is the correct matching over the k𝑘kitalic_k-core of G1∧π∗G2subscriptsubscript𝜋subscript𝐺1subscript𝐺2G_{1}\wedge_{\pi_{*}}G_{2}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∧ start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT with high probability.
The next lemma represents a upper bound on ξ𝜉\xiitalic_ξ. This result has been proven in [8, 13, 35], and in particular [35] provides a proof for general random graph pairs in Lemma A.4. We obtain the similar results in the correlated Stochastic Block Models.
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Consider the (G1,G2)∼CSBMs(n,p,q,s)similar-tosubscript𝐺1subscript𝐺2CSBMs𝑛𝑝𝑞𝑠(G_{1},G_{2})\sim\textnormal{CSBMs}(n,p,q,s)( italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ∼ CSBMs ( italic_n , italic_p , italic_q , italic_s ).
For any matching (M,φ)∈ℳ(t)𝑀𝜑ℳ𝑡(M,\varphi)\in\mathcal{M}(t)( italic_M , italic_φ ) ∈ caligraphic_M ( italic_t ) and any θ>0𝜃0\theta>0italic_θ > 0, we have that
, 1 = ℙ(f(M,φ)\displaystyle\mathbb{P}(f(M,\varphi)blackboard_P ( italic_f ( italic_M , italic_φ ). , 2 = ≥kt)≤3exp(−t(θk−e2θps2−ne6θp2s2)).\displaystyle\geq kt)\leq 3\exp\left(-t\left(\theta k-e^{2\theta}ps^{2}-ne^{6%
\theta}p^{2}s^{2}\right)\right).≥ italic_k italic_t ) ≤ 3 roman_exp ( - italic_t ( italic_θ italic_k - italic_e start_POSTSUPERSCRIPT 2 italic_θ end_POSTSUPERSCRIPT italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_n italic_e start_POSTSUPERSCRIPT 6 italic_θ end_POSTSUPERSCRIPT italic_p start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ) .. , 3 = . , 4 = (73)
Now, let us analyze the size of the matched node set.
Let us introduce a good event that will be useful in our analysis.
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IX Proof of Theorem 5:
Achievability of Exact Matching in Correlated Contextual Stochastic Block Models
IX-B Lemmas for the analysis of k𝑘kitalic_k-core matching
Lemma 4 (Lemma A.4 in [35]).
Consider the (G1,G2)∼CSBMs(n,p,q,s)similar-tosubscript𝐺1subscript𝐺2CSBMs𝑛𝑝𝑞𝑠(G_{1},G_{2})\sim\textnormal{CSBMs}(n,p,q,s)( italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ∼ CSBMs ( italic_n , italic_p , italic_q , italic_s ).
For any matching (M,φ)∈ℳ(t)𝑀𝜑ℳ𝑡(M,\varphi)\in\mathcal{M}(t)( italic_M , italic_φ ) ∈ caligraphic_M ( italic_t ) and any θ>0𝜃0\theta>0italic_θ > 0, we have that
, 1 = ℙ(f(M,φ)\displaystyle\mathbb{P}(f(M,\varphi)blackboard_P ( italic_f ( italic_M , italic_φ ). , 2 = ≥kt)≤3exp(−t(θk−e2θps2−ne6θp2s2)).\displaystyle\geq kt)\leq 3\exp\left(-t\left(\theta k-e^{2\theta}ps^{2}-ne^{6%
\theta}p^{2}s^{2}\right)\right).≥ italic_k italic_t ) ≤ 3 roman_exp ( - italic_t ( italic_θ italic_k - italic_e start_POSTSUPERSCRIPT 2 italic_θ end_POSTSUPERSCRIPT italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_n italic_e start_POSTSUPERSCRIPT 6 italic_θ end_POSTSUPERSCRIPT italic_p start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ) .. , 3 = . , 4 = (73)
Now, let us analyze the size of the matched node set.
Let us introduce a good event that will be useful in our analysis.
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, 1 = ℬ:={n2−n2/3≤|V+|,|V−|≤n2+n2/3}.assignℬformulae-sequence𝑛2superscript𝑛23superscript𝑉superscript𝑉𝑛2superscript𝑛23\mathcal{B}:=\left\{\frac{n}{2}-n^{2/3}\leq|V^{+}|,|V^{-}|\leq\frac{n}{2}+n^{2%
/3}\right\}.caligraphic_B := { divide start_ARG italic_n end_ARG start_ARG 2 end_ARG - italic_n start_POSTSUPERSCRIPT 2 / 3 end_POSTSUPERSCRIPT ≤ | italic_V start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT | , | italic_V start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT | ≤ divide start_ARG italic_n end_ARG start_ARG 2 end_ARG + italic_n start_POSTSUPERSCRIPT 2 / 3 end_POSTSUPERSCRIPT } .. , 2 = . , 3 = (74)
When +++ or −-- label is assigned to each node with equal probabilities, the probability of the event ℬℬ\mathcal{B}caligraphic_B is 1−o(1)1𝑜11-o(1)1 - italic_o ( 1 ).
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IX Proof of Theorem 5:
Achievability of Exact Matching in Correlated Contextual Stochastic Block Models
IX-B Lemmas for the analysis of k𝑘kitalic_k-core matching
Definition 6 (Balanced communities).
, 1 = ℬ:={n2−n2/3≤|V+|,|V−|≤n2+n2/3}.assignℬformulae-sequence𝑛2superscript𝑛23superscript𝑉superscript𝑉𝑛2superscript𝑛23\mathcal{B}:=\left\{\frac{n}{2}-n^{2/3}\leq|V^{+}|,|V^{-}|\leq\frac{n}{2}+n^{2%
/3}\right\}.caligraphic_B := { divide start_ARG italic_n end_ARG start_ARG 2 end_ARG - italic_n start_POSTSUPERSCRIPT 2 / 3 end_POSTSUPERSCRIPT ≤ | italic_V start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT | , | italic_V start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT | ≤ divide start_ARG italic_n end_ARG start_ARG 2 end_ARG + italic_n start_POSTSUPERSCRIPT 2 / 3 end_POSTSUPERSCRIPT } .. , 2 = . , 3 = (74)
When +++ or −-- label is assigned to each node with equal probabilities, the probability of the event ℬℬ\mathcal{B}caligraphic_B is 1−o(1)1𝑜11-o(1)1 - italic_o ( 1 ).
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It holds that ℙ(ℬ)=1−o(1).ℙℬ1𝑜1\mathbb{P}(\mathcal{B})=1-o(1).blackboard_P ( caligraphic_B ) = 1 - italic_o ( 1 ) .
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IX Proof of Theorem 5:
Achievability of Exact Matching in Correlated Contextual Stochastic Block Models
IX-B Lemmas for the analysis of k𝑘kitalic_k-core matching
Lemma 5.
It holds that ℙ(ℬ)=1−o(1).ℙℬ1𝑜1\mathbb{P}(\mathcal{B})=1-o(1).blackboard_P ( caligraphic_B ) = 1 - italic_o ( 1 ) .
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Let G∼SBM(n,p,q)similar-to𝐺SBM𝑛𝑝𝑞G\sim\textnormal{SBM}(n,p,q)italic_G ∼ SBM ( italic_n , italic_p , italic_q ). Define the set
, 1 = Lk:={i∈[n]:degG(i)≤k}.assignsubscript𝐿𝑘conditional-set𝑖delimited-[]𝑛subscriptdegree𝐺𝑖𝑘L_{k}:=\{i\in[n]:\deg_{G}(i)\leq k\}.italic_L start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT := { italic_i ∈ [ italic_n ] : roman_deg start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_i ) ≤ italic_k } .. , 2 = . , 3 = (75)
Then, on the event ℬℬ\mathcal{B}caligraphic_B, it holds that
, 1 = 𝔼[|Lk|]≤nexp(−np+q2+o(np+q2)+klognp+1).𝔼delimited-[]subscript𝐿𝑘𝑛𝑛𝑝𝑞2𝑜𝑛𝑝𝑞2𝑘𝑛𝑝1\mathbb{E}[|L_{k}|]\leq n\exp\left(-n\frac{p+q}{2}+o\left(n\frac{p+q}{2}\right%
)+k\log np+1\right).blackboard_E [ | italic_L start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | ] ≤ italic_n roman_exp ( - italic_n divide start_ARG italic_p + italic_q end_ARG start_ARG 2 end_ARG + italic_o ( italic_n divide start_ARG italic_p + italic_q end_ARG start_ARG 2 end_ARG ) + italic_k roman_log italic_n italic_p + 1 ) .. , 2 = . , 3 = (76)
Using the Luczak expansion [38], the size of the k𝑘kitalic_k-core of the graph G𝐺Gitalic_G can be obtained as follows. In [35], this result was proven for a general case.
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IX Proof of Theorem 5:
Achievability of Exact Matching in Correlated Contextual Stochastic Block Models
IX-B Lemmas for the analysis of k𝑘kitalic_k-core matching
Lemma 6.
Let G∼SBM(n,p,q)similar-to𝐺SBM𝑛𝑝𝑞G\sim\textnormal{SBM}(n,p,q)italic_G ∼ SBM ( italic_n , italic_p , italic_q ). Define the set
, 1 = Lk:={i∈[n]:degG(i)≤k}.assignsubscript𝐿𝑘conditional-set𝑖delimited-[]𝑛subscriptdegree𝐺𝑖𝑘L_{k}:=\{i\in[n]:\deg_{G}(i)\leq k\}.italic_L start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT := { italic_i ∈ [ italic_n ] : roman_deg start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_i ) ≤ italic_k } .. , 2 = . , 3 = (75)
Then, on the event ℬℬ\mathcal{B}caligraphic_B, it holds that
, 1 = 𝔼[|Lk|]≤nexp(−np+q2+o(np+q2)+klognp+1).𝔼delimited-[]subscript𝐿𝑘𝑛𝑛𝑝𝑞2𝑜𝑛𝑝𝑞2𝑘𝑛𝑝1\mathbb{E}[|L_{k}|]\leq n\exp\left(-n\frac{p+q}{2}+o\left(n\frac{p+q}{2}\right%
)+k\log np+1\right).blackboard_E [ | italic_L start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | ] ≤ italic_n roman_exp ( - italic_n divide start_ARG italic_p + italic_q end_ARG start_ARG 2 end_ARG + italic_o ( italic_n divide start_ARG italic_p + italic_q end_ARG start_ARG 2 end_ARG ) + italic_k roman_log italic_n italic_p + 1 ) .. , 2 = . , 3 = (76)
Using the Luczak expansion [38], the size of the k𝑘kitalic_k-core of the graph G𝐺Gitalic_G can be obtained as follows. In [35], this result was proven for a general case.
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Let G∼SBM(n,p,q)similar-to𝐺SBM𝑛𝑝𝑞G\sim\textnormal{SBM}(n,p,q)italic_G ∼ SBM ( italic_n , italic_p , italic_q ) with np,nq=O(logn)𝑛𝑝𝑛𝑞𝑂𝑛np,nq=O(\log n)italic_n italic_p , italic_n italic_q = italic_O ( roman_log italic_n ). Let Jksubscript𝐽𝑘J_{k}italic_J start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT denote the k𝑘kitalic_k-core of G𝐺Gitalic_G, and let J¯k:=[n]\Jkassignsubscript¯𝐽𝑘\delimited-[]𝑛subscript𝐽𝑘\bar{J}_{k}:=[n]\backslash J_{k}over¯ start_ARG italic_J end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT := [ italic_n ] \ italic_J start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT. Recall Lksubscript𝐿𝑘L_{k}italic_L start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT defined in (75). If |Lk|≤ncsubscript𝐿𝑘superscript𝑛𝑐|L_{k}|\leq n^{c}| italic_L start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | ≤ italic_n start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT for c∈(0,1)𝑐01c\in(0,1)italic_c ∈ ( 0 , 1 ), then |J¯k|≤3|Lk|subscript¯𝐽𝑘3subscript𝐿𝑘|\bar{J}_{k}|\leq 3|L_{k}|| over¯ start_ARG italic_J end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | ≤ 3 | italic_L start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT |.
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IX Proof of Theorem 5:
Achievability of Exact Matching in Correlated Contextual Stochastic Block Models
IX-B Lemmas for the analysis of k𝑘kitalic_k-core matching
Lemma 7 (Lemma IV.6 in [35]).
Let G∼SBM(n,p,q)similar-to𝐺SBM𝑛𝑝𝑞G\sim\textnormal{SBM}(n,p,q)italic_G ∼ SBM ( italic_n , italic_p , italic_q ) with np,nq=O(logn)𝑛𝑝𝑛𝑞𝑂𝑛np,nq=O(\log n)italic_n italic_p , italic_n italic_q = italic_O ( roman_log italic_n ). Let Jksubscript𝐽𝑘J_{k}italic_J start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT denote the k𝑘kitalic_k-core of G𝐺Gitalic_G, and let J¯k:=[n]\Jkassignsubscript¯𝐽𝑘\delimited-[]𝑛subscript𝐽𝑘\bar{J}_{k}:=[n]\backslash J_{k}over¯ start_ARG italic_J end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT := [ italic_n ] \ italic_J start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT. Recall Lksubscript𝐿𝑘L_{k}italic_L start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT defined in (75). If |Lk|≤ncsubscript𝐿𝑘superscript𝑛𝑐|L_{k}|\leq n^{c}| italic_L start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | ≤ italic_n start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT for c∈(0,1)𝑐01c\in(0,1)italic_c ∈ ( 0 , 1 ), then |J¯k|≤3|Lk|subscript¯𝐽𝑘3subscript𝐿𝑘|\bar{J}_{k}|\leq 3|L_{k}|| over¯ start_ARG italic_J end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | ≤ 3 | italic_L start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT |.
| 538
|
86
|
Based on the lemmas above, we can now prove Theorem 12 as follows.
| 18
|
Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
IX Proof of Theorem 5:
Achievability of Exact Matching in Correlated Contextual Stochastic Block Models
IX-C Proofs of Theorems 12 and 13 and Lemmas 5 and 6
Based on the lemmas above, we can now prove Theorem 12 as follows.
| 79
|
87
|
If ns2p+q2≥(1+ϵ)logn𝑛superscript𝑠2𝑝𝑞21italic-ϵ𝑛ns^{2}\frac{p+q}{2}\geq(1+\epsilon)\log nitalic_n italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divide start_ARG italic_p + italic_q end_ARG start_ARG 2 end_ARG ≥ ( 1 + italic_ϵ ) roman_log italic_n for an arbitrary small constant ϵ>0italic-ϵ0\epsilon>0italic_ϵ > 0, then we can obtain φ^k=π∗subscript^𝜑𝑘subscript𝜋\widehat{\varphi}_{k}=\pi_{*}over^ start_ARG italic_φ end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT by Theorem 13.
Now, let us consider the case where nps2,nqs2=O(logn)𝑛𝑝superscript𝑠2𝑛𝑞superscript𝑠2𝑂𝑛nps^{2},nqs^{2}=O(\log n)italic_n italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , italic_n italic_q italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = italic_O ( roman_log italic_n ).
By Lemma 4 and the definition of ξ𝜉\xiitalic_ξ in (71), we have that
, 1 = ξ𝜉\displaystyle\xiitalic_ξ. , 2 = ≤31/texp(−θk+e2θps2+ne6θp2s2)≤3exp(−θk+e2θps2+ne6θp2s2)absentsuperscript31𝑡𝜃𝑘superscript𝑒2𝜃𝑝superscript𝑠2𝑛superscript𝑒6𝜃superscript𝑝2superscript𝑠23𝜃𝑘superscript𝑒2𝜃𝑝superscript𝑠2𝑛superscript𝑒6𝜃superscript𝑝2superscript𝑠2\displaystyle\leq 3^{1/t}\exp\left(-\theta k+e^{2\theta}ps^{2}+ne^{6\theta}p^{%
2}s^{2}\right)\leq 3\exp\left(-\theta k+e^{2\theta}ps^{2}+ne^{6\theta}p^{2}s^{%
2}\right)≤ 3 start_POSTSUPERSCRIPT 1 / italic_t end_POSTSUPERSCRIPT roman_exp ( - italic_θ italic_k + italic_e start_POSTSUPERSCRIPT 2 italic_θ end_POSTSUPERSCRIPT italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_n italic_e start_POSTSUPERSCRIPT 6 italic_θ end_POSTSUPERSCRIPT italic_p start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ≤ 3 roman_exp ( - italic_θ italic_k + italic_e start_POSTSUPERSCRIPT 2 italic_θ end_POSTSUPERSCRIPT italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_n italic_e start_POSTSUPERSCRIPT 6 italic_θ end_POSTSUPERSCRIPT italic_p start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ). , 3 = . , 4 = (77)
for any θ>0𝜃0\theta>0italic_θ > 0. Through Lemma 3 and (77), we can show that if there exists an θ>0𝜃0\theta>0italic_θ > 0 such that θk−e2θps2−ne6θp2s2≥2logn+ω(1)𝜃𝑘superscript𝑒2𝜃𝑝superscript𝑠2𝑛superscript𝑒6𝜃superscript𝑝2superscript𝑠22𝑛𝜔1\theta k-e^{2\theta}ps^{2}-ne^{6\theta}p^{2}s^{2}\geq 2\log n+\omega(1)italic_θ italic_k - italic_e start_POSTSUPERSCRIPT 2 italic_θ end_POSTSUPERSCRIPT italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_n italic_e start_POSTSUPERSCRIPT 6 italic_θ end_POSTSUPERSCRIPT italic_p start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ 2 roman_log italic_n + italic_ω ( 1 ), then M^k=Mk and φ^k{M^k}=π∗{M^k}subscript^𝑀𝑘subscript𝑀𝑘 and subscript^𝜑𝑘subscript^𝑀𝑘subscript𝜋subscript^𝑀𝑘\widehat{M}_{k}=M_{k}\text{ and }\widehat{\varphi}_{k}\{\widehat{M}_{k}\}=\pi_%
{*}\{\widehat{M}_{k}\}over^ start_ARG italic_M end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_M start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT and over^ start_ARG italic_φ end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT { over^ start_ARG italic_M end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT } = italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT { over^ start_ARG italic_M end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT } hold with probability 1−o(1)1𝑜11-o(1)1 - italic_o ( 1 ). Let θ=(loglogn)2.5𝜃superscript𝑛2.5\theta=(\log\log n)^{2.5}italic_θ = ( roman_log roman_log italic_n ) start_POSTSUPERSCRIPT 2.5 end_POSTSUPERSCRIPT and recall that p≤O(e−(loglogn)3)𝑝𝑂superscript𝑒superscript𝑛3p\leq O\left(e^{-(\log\log n)^{3}}\right)italic_p ≤ italic_O ( italic_e start_POSTSUPERSCRIPT - ( roman_log roman_log italic_n ) start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT ) and k=nps2(lognps2)2∨logn(loglogn)2𝑘𝑛𝑝superscript𝑠2superscript𝑛𝑝superscript𝑠22𝑛superscript𝑛2k=\frac{nps^{2}}{(\log nps^{2})^{2}}\vee\frac{\log n}{(\log\log n)^{2}}italic_k = divide start_ARG italic_n italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG ( roman_log italic_n italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ∨ divide start_ARG roman_log italic_n end_ARG start_ARG ( roman_log roman_log italic_n ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG. Then, we can have that
, 1 = θk≥logn⋅(loglogn)0.5𝜃𝑘⋅𝑛superscript𝑛0.5\theta k\geq\log n\cdot(\log\log n)^{0.5}italic_θ italic_k ≥ roman_log italic_n ⋅ ( roman_log roman_log italic_n ) start_POSTSUPERSCRIPT 0.5 end_POSTSUPERSCRIPT. , 2 = . , 3 = (78)
and
| 1,926
|
Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
IX Proof of Theorem 5:
Achievability of Exact Matching in Correlated Contextual Stochastic Block Models
IX-C Proofs of Theorems 12 and 13 and Lemmas 5 and 6
Proof:
If ns2p+q2≥(1+ϵ)logn𝑛superscript𝑠2𝑝𝑞21italic-ϵ𝑛ns^{2}\frac{p+q}{2}\geq(1+\epsilon)\log nitalic_n italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divide start_ARG italic_p + italic_q end_ARG start_ARG 2 end_ARG ≥ ( 1 + italic_ϵ ) roman_log italic_n for an arbitrary small constant ϵ>0italic-ϵ0\epsilon>0italic_ϵ > 0, then we can obtain φ^k=π∗subscript^𝜑𝑘subscript𝜋\widehat{\varphi}_{k}=\pi_{*}over^ start_ARG italic_φ end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT by Theorem 13.
Now, let us consider the case where nps2,nqs2=O(logn)𝑛𝑝superscript𝑠2𝑛𝑞superscript𝑠2𝑂𝑛nps^{2},nqs^{2}=O(\log n)italic_n italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , italic_n italic_q italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = italic_O ( roman_log italic_n ).
By Lemma 4 and the definition of ξ𝜉\xiitalic_ξ in (71), we have that
, 1 = ξ𝜉\displaystyle\xiitalic_ξ. , 2 = ≤31/texp(−θk+e2θps2+ne6θp2s2)≤3exp(−θk+e2θps2+ne6θp2s2)absentsuperscript31𝑡𝜃𝑘superscript𝑒2𝜃𝑝superscript𝑠2𝑛superscript𝑒6𝜃superscript𝑝2superscript𝑠23𝜃𝑘superscript𝑒2𝜃𝑝superscript𝑠2𝑛superscript𝑒6𝜃superscript𝑝2superscript𝑠2\displaystyle\leq 3^{1/t}\exp\left(-\theta k+e^{2\theta}ps^{2}+ne^{6\theta}p^{%
2}s^{2}\right)\leq 3\exp\left(-\theta k+e^{2\theta}ps^{2}+ne^{6\theta}p^{2}s^{%
2}\right)≤ 3 start_POSTSUPERSCRIPT 1 / italic_t end_POSTSUPERSCRIPT roman_exp ( - italic_θ italic_k + italic_e start_POSTSUPERSCRIPT 2 italic_θ end_POSTSUPERSCRIPT italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_n italic_e start_POSTSUPERSCRIPT 6 italic_θ end_POSTSUPERSCRIPT italic_p start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ≤ 3 roman_exp ( - italic_θ italic_k + italic_e start_POSTSUPERSCRIPT 2 italic_θ end_POSTSUPERSCRIPT italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_n italic_e start_POSTSUPERSCRIPT 6 italic_θ end_POSTSUPERSCRIPT italic_p start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ). , 3 = . , 4 = (77)
for any θ>0𝜃0\theta>0italic_θ > 0. Through Lemma 3 and (77), we can show that if there exists an θ>0𝜃0\theta>0italic_θ > 0 such that θk−e2θps2−ne6θp2s2≥2logn+ω(1)𝜃𝑘superscript𝑒2𝜃𝑝superscript𝑠2𝑛superscript𝑒6𝜃superscript𝑝2superscript𝑠22𝑛𝜔1\theta k-e^{2\theta}ps^{2}-ne^{6\theta}p^{2}s^{2}\geq 2\log n+\omega(1)italic_θ italic_k - italic_e start_POSTSUPERSCRIPT 2 italic_θ end_POSTSUPERSCRIPT italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_n italic_e start_POSTSUPERSCRIPT 6 italic_θ end_POSTSUPERSCRIPT italic_p start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ 2 roman_log italic_n + italic_ω ( 1 ), then M^k=Mk and φ^k{M^k}=π∗{M^k}subscript^𝑀𝑘subscript𝑀𝑘 and subscript^𝜑𝑘subscript^𝑀𝑘subscript𝜋subscript^𝑀𝑘\widehat{M}_{k}=M_{k}\text{ and }\widehat{\varphi}_{k}\{\widehat{M}_{k}\}=\pi_%
{*}\{\widehat{M}_{k}\}over^ start_ARG italic_M end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_M start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT and over^ start_ARG italic_φ end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT { over^ start_ARG italic_M end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT } = italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT { over^ start_ARG italic_M end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT } hold with probability 1−o(1)1𝑜11-o(1)1 - italic_o ( 1 ). Let θ=(loglogn)2.5𝜃superscript𝑛2.5\theta=(\log\log n)^{2.5}italic_θ = ( roman_log roman_log italic_n ) start_POSTSUPERSCRIPT 2.5 end_POSTSUPERSCRIPT and recall that p≤O(e−(loglogn)3)𝑝𝑂superscript𝑒superscript𝑛3p\leq O\left(e^{-(\log\log n)^{3}}\right)italic_p ≤ italic_O ( italic_e start_POSTSUPERSCRIPT - ( roman_log roman_log italic_n ) start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT ) and k=nps2(lognps2)2∨logn(loglogn)2𝑘𝑛𝑝superscript𝑠2superscript𝑛𝑝superscript𝑠22𝑛superscript𝑛2k=\frac{nps^{2}}{(\log nps^{2})^{2}}\vee\frac{\log n}{(\log\log n)^{2}}italic_k = divide start_ARG italic_n italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG ( roman_log italic_n italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ∨ divide start_ARG roman_log italic_n end_ARG start_ARG ( roman_log roman_log italic_n ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG. Then, we can have that
, 1 = θk≥logn⋅(loglogn)0.5𝜃𝑘⋅𝑛superscript𝑛0.5\theta k\geq\log n\cdot(\log\log n)^{0.5}italic_θ italic_k ≥ roman_log italic_n ⋅ ( roman_log roman_log italic_n ) start_POSTSUPERSCRIPT 0.5 end_POSTSUPERSCRIPT. , 2 = . , 3 = (78)
and
| 1,989
|
88
|
, 1 = e2θps2≤e2θp=o(1).superscript𝑒2𝜃𝑝superscript𝑠2superscript𝑒2𝜃𝑝𝑜1e^{2\theta}ps^{2}\leq e^{2\theta}p=o(1).italic_e start_POSTSUPERSCRIPT 2 italic_θ end_POSTSUPERSCRIPT italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≤ italic_e start_POSTSUPERSCRIPT 2 italic_θ end_POSTSUPERSCRIPT italic_p = italic_o ( 1 ) .. , 2 = . , 3 = (79)
Moreover, we can obtain
, 1 = θk≥θnps2(lognps2)2≥(loglogn)2.5nps2(logn)2≥2ne6θp2s2.𝜃𝑘𝜃𝑛𝑝superscript𝑠2superscript𝑛𝑝superscript𝑠22superscript𝑛2.5𝑛𝑝superscript𝑠2superscript𝑛22𝑛superscript𝑒6𝜃superscript𝑝2superscript𝑠2\theta k\geq\theta\frac{nps^{2}}{(\log nps^{2})^{2}}\geq(\log\log n)^{2.5}%
\frac{nps^{2}}{(\log n)^{2}}\geq 2ne^{6\theta}p^{2}s^{2}.italic_θ italic_k ≥ italic_θ divide start_ARG italic_n italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG ( roman_log italic_n italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≥ ( roman_log roman_log italic_n ) start_POSTSUPERSCRIPT 2.5 end_POSTSUPERSCRIPT divide start_ARG italic_n italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG ( roman_log italic_n ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≥ 2 italic_n italic_e start_POSTSUPERSCRIPT 6 italic_θ end_POSTSUPERSCRIPT italic_p start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT .. , 2 = . , 3 = (80)
The last inequality holds by p≤O(e−(loglogn)3)𝑝𝑂superscript𝑒superscript𝑛3p\leq O\left(e^{-(\log\log n)^{3}}\right)italic_p ≤ italic_O ( italic_e start_POSTSUPERSCRIPT - ( roman_log roman_log italic_n ) start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT ). Therefore, we have
, 1 = θk−e2θps2−ne6θp2s2≥13θk=ω(logn).𝜃𝑘superscript𝑒2𝜃𝑝superscript𝑠2𝑛superscript𝑒6𝜃superscript𝑝2superscript𝑠213𝜃𝑘𝜔𝑛\theta k-e^{2\theta}ps^{2}-ne^{6\theta}p^{2}s^{2}\geq\frac{1}{3}\theta k=%
\omega(\log n).italic_θ italic_k - italic_e start_POSTSUPERSCRIPT 2 italic_θ end_POSTSUPERSCRIPT italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_n italic_e start_POSTSUPERSCRIPT 6 italic_θ end_POSTSUPERSCRIPT italic_p start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ divide start_ARG 1 end_ARG start_ARG 3 end_ARG italic_θ italic_k = italic_ω ( roman_log italic_n ) .. , 2 = . , 3 = (81)
| 988
|
Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
IX Proof of Theorem 5:
Achievability of Exact Matching in Correlated Contextual Stochastic Block Models
IX-C Proofs of Theorems 12 and 13 and Lemmas 5 and 6
Proof:
, 1 = e2θps2≤e2θp=o(1).superscript𝑒2𝜃𝑝superscript𝑠2superscript𝑒2𝜃𝑝𝑜1e^{2\theta}ps^{2}\leq e^{2\theta}p=o(1).italic_e start_POSTSUPERSCRIPT 2 italic_θ end_POSTSUPERSCRIPT italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≤ italic_e start_POSTSUPERSCRIPT 2 italic_θ end_POSTSUPERSCRIPT italic_p = italic_o ( 1 ) .. , 2 = . , 3 = (79)
Moreover, we can obtain
, 1 = θk≥θnps2(lognps2)2≥(loglogn)2.5nps2(logn)2≥2ne6θp2s2.𝜃𝑘𝜃𝑛𝑝superscript𝑠2superscript𝑛𝑝superscript𝑠22superscript𝑛2.5𝑛𝑝superscript𝑠2superscript𝑛22𝑛superscript𝑒6𝜃superscript𝑝2superscript𝑠2\theta k\geq\theta\frac{nps^{2}}{(\log nps^{2})^{2}}\geq(\log\log n)^{2.5}%
\frac{nps^{2}}{(\log n)^{2}}\geq 2ne^{6\theta}p^{2}s^{2}.italic_θ italic_k ≥ italic_θ divide start_ARG italic_n italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG ( roman_log italic_n italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≥ ( roman_log roman_log italic_n ) start_POSTSUPERSCRIPT 2.5 end_POSTSUPERSCRIPT divide start_ARG italic_n italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG ( roman_log italic_n ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≥ 2 italic_n italic_e start_POSTSUPERSCRIPT 6 italic_θ end_POSTSUPERSCRIPT italic_p start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT .. , 2 = . , 3 = (80)
The last inequality holds by p≤O(e−(loglogn)3)𝑝𝑂superscript𝑒superscript𝑛3p\leq O\left(e^{-(\log\log n)^{3}}\right)italic_p ≤ italic_O ( italic_e start_POSTSUPERSCRIPT - ( roman_log roman_log italic_n ) start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT ). Therefore, we have
, 1 = θk−e2θps2−ne6θp2s2≥13θk=ω(logn).𝜃𝑘superscript𝑒2𝜃𝑝superscript𝑠2𝑛superscript𝑒6𝜃superscript𝑝2superscript𝑠213𝜃𝑘𝜔𝑛\theta k-e^{2\theta}ps^{2}-ne^{6\theta}p^{2}s^{2}\geq\frac{1}{3}\theta k=%
\omega(\log n).italic_θ italic_k - italic_e start_POSTSUPERSCRIPT 2 italic_θ end_POSTSUPERSCRIPT italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_n italic_e start_POSTSUPERSCRIPT 6 italic_θ end_POSTSUPERSCRIPT italic_p start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ divide start_ARG 1 end_ARG start_ARG 3 end_ARG italic_θ italic_k = italic_ω ( roman_log italic_n ) .. , 2 = . , 3 = (81)
| 1,051
|
89
|
Now, we will prove that |Mk|≥n−n1−ns2(p+q)2logn+o(1)subscript𝑀𝑘𝑛superscript𝑛1𝑛superscript𝑠2𝑝𝑞2𝑛𝑜1|M_{k}|\geq n-n^{1-\frac{ns^{2}(p+q)}{2\log n}+o(1)}| italic_M start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | ≥ italic_n - italic_n start_POSTSUPERSCRIPT 1 - divide start_ARG italic_n italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_p + italic_q ) end_ARG start_ARG 2 roman_log italic_n end_ARG + italic_o ( 1 ) end_POSTSUPERSCRIPT. If ns2(p+q)=o(logn)𝑛superscript𝑠2𝑝𝑞𝑜𝑛ns^{2}(p+q)=o(\log n)italic_n italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_p + italic_q ) = italic_o ( roman_log italic_n ), the right-hand side converges to 0, making the result trivial. So, let us consider the case where ns2(p+q)=Θ(logn)𝑛superscript𝑠2𝑝𝑞Θ𝑛ns^{2}(p+q)=\Theta(\log n)italic_n italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_p + italic_q ) = roman_Θ ( roman_log italic_n ).
Recall that Mksubscript𝑀𝑘M_{k}italic_M start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT is the k𝑘kitalic_k-core of G1∧π∗G2subscriptsubscript𝜋subscript𝐺1subscript𝐺2G_{1}\wedge_{\pi_{*}}G_{2}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∧ start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. Let L¯k:={i∈[n]:degG1∧π∗G2(i)≤k}assignsubscript¯𝐿𝑘conditional-set𝑖delimited-[]𝑛subscriptdegreesubscriptsubscript𝜋subscript𝐺1subscript𝐺2𝑖𝑘\overline{L}_{k}:=\{i\in[n]:\deg_{G_{1}\wedge_{\pi_{*}}G_{2}}(i)\leq k\}over¯ start_ARG italic_L end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT := { italic_i ∈ [ italic_n ] : roman_deg start_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∧ start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_i ) ≤ italic_k } and M¯k:=[n]\Mkassignsubscript¯𝑀𝑘\delimited-[]𝑛subscript𝑀𝑘\overline{M}_{k}:=[n]\backslash M_{k}over¯ start_ARG italic_M end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT := [ italic_n ] \ italic_M start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT.
Since G1∧π∗G2∼SBM(n,ps2,qs2)similar-tosubscriptsubscript𝜋subscript𝐺1subscript𝐺2SBM𝑛𝑝superscript𝑠2𝑞superscript𝑠2G_{1}\wedge_{\pi_{*}}G_{2}\sim\textnormal{SBM}(n,ps^{2},qs^{2})italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∧ start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∼ SBM ( italic_n , italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , italic_q italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ), combining Lemma 5 and Lemma 6 allows us to conclude that
, 1 = 𝔼[|L¯k|]≤nexp(−ns2p+q2+o(ns2p+q2)+klognps2+1)𝔼delimited-[]subscript¯𝐿𝑘𝑛𝑛superscript𝑠2𝑝𝑞2𝑜𝑛superscript𝑠2𝑝𝑞2𝑘𝑛𝑝superscript𝑠21\mathbb{E}[|\overline{L}_{k}|]\leq n\exp\left(-ns^{2}\frac{p+q}{2}+o\left(ns^{%
2}\frac{p+q}{2}\right)+k\log nps^{2}+1\right)blackboard_E [ | over¯ start_ARG italic_L end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | ] ≤ italic_n roman_exp ( - italic_n italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divide start_ARG italic_p + italic_q end_ARG start_ARG 2 end_ARG + italic_o ( italic_n italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divide start_ARG italic_p + italic_q end_ARG start_ARG 2 end_ARG ) + italic_k roman_log italic_n italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 1 ). , 2 = . , 3 = (82)
with probability 1−o(1)1𝑜11-o(1)1 - italic_o ( 1 ). By Markov’s inequality, we can also obtain that
, 1 = |L¯k|≤(logn)𝔼[|L¯k|]subscript¯𝐿𝑘𝑛𝔼delimited-[]subscript¯𝐿𝑘|\overline{L}_{k}|\leq(\log n)\mathbb{E}[|\overline{L}_{k}|]| over¯ start_ARG italic_L end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | ≤ ( roman_log italic_n ) blackboard_E [ | over¯ start_ARG italic_L end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | ]. , 2 = . , 3 = (83)
with probability at least 1−1logn11𝑛1-\frac{1}{\log n}1 - divide start_ARG 1 end_ARG start_ARG roman_log italic_n end_ARG. By applying Lemma 7 with (62), (82) and (83), we can obtain
, 1 = M¯k≤3|L¯k|≤n1−ns2(p+q)2logn+o(1).subscript¯𝑀𝑘3subscript¯𝐿𝑘superscript𝑛1𝑛superscript𝑠2𝑝𝑞2𝑛𝑜1\overline{M}_{k}\leq 3|\overline{L}_{k}|\leq n^{1-\frac{ns^{2}(p+q)}{2\log n}+%
o(1)}.over¯ start_ARG italic_M end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ≤ 3 | over¯ start_ARG italic_L end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | ≤ italic_n start_POSTSUPERSCRIPT 1 - divide start_ARG italic_n italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_p + italic_q ) end_ARG start_ARG 2 roman_log italic_n end_ARG + italic_o ( 1 ) end_POSTSUPERSCRIPT .. , 2 = . , 3 = (84)
Hence, the proof is complete.
∎
| 1,886
|
Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
IX Proof of Theorem 5:
Achievability of Exact Matching in Correlated Contextual Stochastic Block Models
IX-C Proofs of Theorems 12 and 13 and Lemmas 5 and 6
Proof:
Now, we will prove that |Mk|≥n−n1−ns2(p+q)2logn+o(1)subscript𝑀𝑘𝑛superscript𝑛1𝑛superscript𝑠2𝑝𝑞2𝑛𝑜1|M_{k}|\geq n-n^{1-\frac{ns^{2}(p+q)}{2\log n}+o(1)}| italic_M start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | ≥ italic_n - italic_n start_POSTSUPERSCRIPT 1 - divide start_ARG italic_n italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_p + italic_q ) end_ARG start_ARG 2 roman_log italic_n end_ARG + italic_o ( 1 ) end_POSTSUPERSCRIPT. If ns2(p+q)=o(logn)𝑛superscript𝑠2𝑝𝑞𝑜𝑛ns^{2}(p+q)=o(\log n)italic_n italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_p + italic_q ) = italic_o ( roman_log italic_n ), the right-hand side converges to 0, making the result trivial. So, let us consider the case where ns2(p+q)=Θ(logn)𝑛superscript𝑠2𝑝𝑞Θ𝑛ns^{2}(p+q)=\Theta(\log n)italic_n italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_p + italic_q ) = roman_Θ ( roman_log italic_n ).
Recall that Mksubscript𝑀𝑘M_{k}italic_M start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT is the k𝑘kitalic_k-core of G1∧π∗G2subscriptsubscript𝜋subscript𝐺1subscript𝐺2G_{1}\wedge_{\pi_{*}}G_{2}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∧ start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. Let L¯k:={i∈[n]:degG1∧π∗G2(i)≤k}assignsubscript¯𝐿𝑘conditional-set𝑖delimited-[]𝑛subscriptdegreesubscriptsubscript𝜋subscript𝐺1subscript𝐺2𝑖𝑘\overline{L}_{k}:=\{i\in[n]:\deg_{G_{1}\wedge_{\pi_{*}}G_{2}}(i)\leq k\}over¯ start_ARG italic_L end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT := { italic_i ∈ [ italic_n ] : roman_deg start_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∧ start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_i ) ≤ italic_k } and M¯k:=[n]\Mkassignsubscript¯𝑀𝑘\delimited-[]𝑛subscript𝑀𝑘\overline{M}_{k}:=[n]\backslash M_{k}over¯ start_ARG italic_M end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT := [ italic_n ] \ italic_M start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT.
Since G1∧π∗G2∼SBM(n,ps2,qs2)similar-tosubscriptsubscript𝜋subscript𝐺1subscript𝐺2SBM𝑛𝑝superscript𝑠2𝑞superscript𝑠2G_{1}\wedge_{\pi_{*}}G_{2}\sim\textnormal{SBM}(n,ps^{2},qs^{2})italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∧ start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∼ SBM ( italic_n , italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , italic_q italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ), combining Lemma 5 and Lemma 6 allows us to conclude that
, 1 = 𝔼[|L¯k|]≤nexp(−ns2p+q2+o(ns2p+q2)+klognps2+1)𝔼delimited-[]subscript¯𝐿𝑘𝑛𝑛superscript𝑠2𝑝𝑞2𝑜𝑛superscript𝑠2𝑝𝑞2𝑘𝑛𝑝superscript𝑠21\mathbb{E}[|\overline{L}_{k}|]\leq n\exp\left(-ns^{2}\frac{p+q}{2}+o\left(ns^{%
2}\frac{p+q}{2}\right)+k\log nps^{2}+1\right)blackboard_E [ | over¯ start_ARG italic_L end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | ] ≤ italic_n roman_exp ( - italic_n italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divide start_ARG italic_p + italic_q end_ARG start_ARG 2 end_ARG + italic_o ( italic_n italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divide start_ARG italic_p + italic_q end_ARG start_ARG 2 end_ARG ) + italic_k roman_log italic_n italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 1 ). , 2 = . , 3 = (82)
with probability 1−o(1)1𝑜11-o(1)1 - italic_o ( 1 ). By Markov’s inequality, we can also obtain that
, 1 = |L¯k|≤(logn)𝔼[|L¯k|]subscript¯𝐿𝑘𝑛𝔼delimited-[]subscript¯𝐿𝑘|\overline{L}_{k}|\leq(\log n)\mathbb{E}[|\overline{L}_{k}|]| over¯ start_ARG italic_L end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | ≤ ( roman_log italic_n ) blackboard_E [ | over¯ start_ARG italic_L end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | ]. , 2 = . , 3 = (83)
with probability at least 1−1logn11𝑛1-\frac{1}{\log n}1 - divide start_ARG 1 end_ARG start_ARG roman_log italic_n end_ARG. By applying Lemma 7 with (62), (82) and (83), we can obtain
, 1 = M¯k≤3|L¯k|≤n1−ns2(p+q)2logn+o(1).subscript¯𝑀𝑘3subscript¯𝐿𝑘superscript𝑛1𝑛superscript𝑠2𝑝𝑞2𝑛𝑜1\overline{M}_{k}\leq 3|\overline{L}_{k}|\leq n^{1-\frac{ns^{2}(p+q)}{2\log n}+%
o(1)}.over¯ start_ARG italic_M end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ≤ 3 | over¯ start_ARG italic_L end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | ≤ italic_n start_POSTSUPERSCRIPT 1 - divide start_ARG italic_n italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_p + italic_q ) end_ARG start_ARG 2 roman_log italic_n end_ARG + italic_o ( 1 ) end_POSTSUPERSCRIPT .. , 2 = . , 3 = (84)
Hence, the proof is complete.
∎
| 1,949
|
90
|
For the same reasons as in the proof of Theorem 12, we obtain that M^k=Mk and φ^k{M^k}=π∗{M^k}subscript^𝑀𝑘subscript𝑀𝑘 and subscript^𝜑𝑘subscript^𝑀𝑘subscript𝜋subscript^𝑀𝑘\widehat{M}_{k}=M_{k}\text{ and }\widehat{\varphi}_{k}\{\widehat{M}_{k}\}=\pi_%
{*}\{\widehat{M}_{k}\}over^ start_ARG italic_M end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_M start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT and over^ start_ARG italic_φ end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT { over^ start_ARG italic_M end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT } = italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT { over^ start_ARG italic_M end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT } with probability 1−o(1)1𝑜11-o(1)1 - italic_o ( 1 ). Thus, if we can show that |Mk|=nsubscript𝑀𝑘𝑛|M_{k}|=n| italic_M start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | = italic_n with probability 1−o(1)1𝑜11-o(1)1 - italic_o ( 1 ), the proof will be complete.
On the event ℬℬ\mathcal{B}caligraphic_B, we can obtain
, 1 = ℙ(|Mk|≠n)ℙsubscript𝑀𝑘𝑛\displaystyle\mathbb{P}(|M_{k}|\neq n)blackboard_P ( | italic_M start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | ≠ italic_n ). , 2 = =ℙ(dmin(G1∧π∗G2)<k)absentℙsubscript𝑑subscriptsubscript𝜋subscript𝐺1subscript𝐺2𝑘\displaystyle=\mathbb{P}(d_{\min}(G_{1}\wedge_{\pi_{*}}G_{2})<k)= blackboard_P ( italic_d start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∧ start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) < italic_k ). , 3 = . , 4 = (85). , 1 = . , 2 = ≤(a)nℙ(degG1∧π∗G2(i)<k)superscript𝑎absent𝑛ℙsubscriptdegreesubscriptsubscript𝜋subscript𝐺1subscript𝐺2𝑖𝑘\displaystyle\stackrel{{\scriptstyle(a)}}{{\leq}}n\mathbb{P}(\deg_{G_{1}\wedge%
_{\pi_{*}}G_{2}}(i)<k)start_RELOP SUPERSCRIPTOP start_ARG ≤ end_ARG start_ARG ( italic_a ) end_ARG end_RELOP italic_n blackboard_P ( roman_deg start_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∧ start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_i ) < italic_k ). , 3 = . , 4 = (85). , 1 = . , 2 = ≤(b)nexp(−ns2p+q2+o(ns2p+q2)+klognps2+1)superscript𝑏absent𝑛𝑛superscript𝑠2𝑝𝑞2𝑜𝑛superscript𝑠2𝑝𝑞2𝑘𝑛𝑝superscript𝑠21\displaystyle\stackrel{{\scriptstyle(b)}}{{\leq}}n\exp\left(-ns^{2}\frac{p+q}{%
2}+o\left(ns^{2}\frac{p+q}{2}\right)+k\log nps^{2}+1\right)start_RELOP SUPERSCRIPTOP start_ARG ≤ end_ARG start_ARG ( italic_b ) end_ARG end_RELOP italic_n roman_exp ( - italic_n italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divide start_ARG italic_p + italic_q end_ARG start_ARG 2 end_ARG + italic_o ( italic_n italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divide start_ARG italic_p + italic_q end_ARG start_ARG 2 end_ARG ) + italic_k roman_log italic_n italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 1 ). , 3 = . , 4 = (85). , 1 = . , 2 = =(c)o(1).superscript𝑐absent𝑜1\displaystyle\stackrel{{\scriptstyle(c)}}{{=}}o(1).start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG ( italic_c ) end_ARG end_RELOP italic_o ( 1 ) .. , 3 = . , 4 = (85)
The inequality (a)𝑎(a)( italic_a ) holds by taking union bound over i∈[n]𝑖delimited-[]𝑛i\in[n]italic_i ∈ [ italic_n ], the inequality (b)𝑏(b)( italic_b ) holds by Lemma 6, and the equality (c)𝑐(c)( italic_c ) holds by (62) and (65). Therefore, combining (85) and Lemma 5, the proof is complete.
∎
We know that |V+|∼Bin(n,1/2)similar-tosuperscript𝑉Bin𝑛12|V^{+}|\sim\operatorname{Bin}(n,1/2)| italic_V start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT | ∼ roman_Bin ( italic_n , 1 / 2 ). By applying Hoeffding’s inequality (stated in Lemma 20), we can obtain that
, 1 = ℙ(||V+|−n/2|≥n2/3)≤2exp(−2n1/3).ℙsuperscript𝑉𝑛2superscript𝑛2322superscript𝑛13\mathbb{P}\left(\left||V^{+}|-n/2\right|\geq n^{2/3}\right)\leq 2\exp(-2n^{1/3%
}).blackboard_P ( | | italic_V start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT | - italic_n / 2 | ≥ italic_n start_POSTSUPERSCRIPT 2 / 3 end_POSTSUPERSCRIPT ) ≤ 2 roman_exp ( - 2 italic_n start_POSTSUPERSCRIPT 1 / 3 end_POSTSUPERSCRIPT ) .. , 2 = . , 3 = (86)
Since |V−|superscript𝑉|V^{-}|| italic_V start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT | also follows a Bin(n,1/2)Bin𝑛12\operatorname{Bin}(n,1/2)roman_Bin ( italic_n , 1 / 2 ), the proof is complete.
∎
Let n1:=|V+|assignsubscript𝑛1superscript𝑉n_{1}:=|V^{+}|italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT := | italic_V start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT | and n2:=|V−|assignsubscript𝑛2superscript𝑉n_{2}:=|V^{-}|italic_n start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT := | italic_V start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT |. On the event ℬℬ\mathcal{B}caligraphic_B, we can obtain that
| 1,882
|
Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
IX Proof of Theorem 5:
Achievability of Exact Matching in Correlated Contextual Stochastic Block Models
IX-C Proofs of Theorems 12 and 13 and Lemmas 5 and 6
Proof:
For the same reasons as in the proof of Theorem 12, we obtain that M^k=Mk and φ^k{M^k}=π∗{M^k}subscript^𝑀𝑘subscript𝑀𝑘 and subscript^𝜑𝑘subscript^𝑀𝑘subscript𝜋subscript^𝑀𝑘\widehat{M}_{k}=M_{k}\text{ and }\widehat{\varphi}_{k}\{\widehat{M}_{k}\}=\pi_%
{*}\{\widehat{M}_{k}\}over^ start_ARG italic_M end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_M start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT and over^ start_ARG italic_φ end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT { over^ start_ARG italic_M end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT } = italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT { over^ start_ARG italic_M end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT } with probability 1−o(1)1𝑜11-o(1)1 - italic_o ( 1 ). Thus, if we can show that |Mk|=nsubscript𝑀𝑘𝑛|M_{k}|=n| italic_M start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | = italic_n with probability 1−o(1)1𝑜11-o(1)1 - italic_o ( 1 ), the proof will be complete.
On the event ℬℬ\mathcal{B}caligraphic_B, we can obtain
, 1 = ℙ(|Mk|≠n)ℙsubscript𝑀𝑘𝑛\displaystyle\mathbb{P}(|M_{k}|\neq n)blackboard_P ( | italic_M start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | ≠ italic_n ). , 2 = =ℙ(dmin(G1∧π∗G2)<k)absentℙsubscript𝑑subscriptsubscript𝜋subscript𝐺1subscript𝐺2𝑘\displaystyle=\mathbb{P}(d_{\min}(G_{1}\wedge_{\pi_{*}}G_{2})<k)= blackboard_P ( italic_d start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∧ start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) < italic_k ). , 3 = . , 4 = (85). , 1 = . , 2 = ≤(a)nℙ(degG1∧π∗G2(i)<k)superscript𝑎absent𝑛ℙsubscriptdegreesubscriptsubscript𝜋subscript𝐺1subscript𝐺2𝑖𝑘\displaystyle\stackrel{{\scriptstyle(a)}}{{\leq}}n\mathbb{P}(\deg_{G_{1}\wedge%
_{\pi_{*}}G_{2}}(i)<k)start_RELOP SUPERSCRIPTOP start_ARG ≤ end_ARG start_ARG ( italic_a ) end_ARG end_RELOP italic_n blackboard_P ( roman_deg start_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∧ start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_i ) < italic_k ). , 3 = . , 4 = (85). , 1 = . , 2 = ≤(b)nexp(−ns2p+q2+o(ns2p+q2)+klognps2+1)superscript𝑏absent𝑛𝑛superscript𝑠2𝑝𝑞2𝑜𝑛superscript𝑠2𝑝𝑞2𝑘𝑛𝑝superscript𝑠21\displaystyle\stackrel{{\scriptstyle(b)}}{{\leq}}n\exp\left(-ns^{2}\frac{p+q}{%
2}+o\left(ns^{2}\frac{p+q}{2}\right)+k\log nps^{2}+1\right)start_RELOP SUPERSCRIPTOP start_ARG ≤ end_ARG start_ARG ( italic_b ) end_ARG end_RELOP italic_n roman_exp ( - italic_n italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divide start_ARG italic_p + italic_q end_ARG start_ARG 2 end_ARG + italic_o ( italic_n italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divide start_ARG italic_p + italic_q end_ARG start_ARG 2 end_ARG ) + italic_k roman_log italic_n italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 1 ). , 3 = . , 4 = (85). , 1 = . , 2 = =(c)o(1).superscript𝑐absent𝑜1\displaystyle\stackrel{{\scriptstyle(c)}}{{=}}o(1).start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG ( italic_c ) end_ARG end_RELOP italic_o ( 1 ) .. , 3 = . , 4 = (85)
The inequality (a)𝑎(a)( italic_a ) holds by taking union bound over i∈[n]𝑖delimited-[]𝑛i\in[n]italic_i ∈ [ italic_n ], the inequality (b)𝑏(b)( italic_b ) holds by Lemma 6, and the equality (c)𝑐(c)( italic_c ) holds by (62) and (65). Therefore, combining (85) and Lemma 5, the proof is complete.
∎
We know that |V+|∼Bin(n,1/2)similar-tosuperscript𝑉Bin𝑛12|V^{+}|\sim\operatorname{Bin}(n,1/2)| italic_V start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT | ∼ roman_Bin ( italic_n , 1 / 2 ). By applying Hoeffding’s inequality (stated in Lemma 20), we can obtain that
, 1 = ℙ(||V+|−n/2|≥n2/3)≤2exp(−2n1/3).ℙsuperscript𝑉𝑛2superscript𝑛2322superscript𝑛13\mathbb{P}\left(\left||V^{+}|-n/2\right|\geq n^{2/3}\right)\leq 2\exp(-2n^{1/3%
}).blackboard_P ( | | italic_V start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT | - italic_n / 2 | ≥ italic_n start_POSTSUPERSCRIPT 2 / 3 end_POSTSUPERSCRIPT ) ≤ 2 roman_exp ( - 2 italic_n start_POSTSUPERSCRIPT 1 / 3 end_POSTSUPERSCRIPT ) .. , 2 = . , 3 = (86)
Since |V−|superscript𝑉|V^{-}|| italic_V start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT | also follows a Bin(n,1/2)Bin𝑛12\operatorname{Bin}(n,1/2)roman_Bin ( italic_n , 1 / 2 ), the proof is complete.
∎
Let n1:=|V+|assignsubscript𝑛1superscript𝑉n_{1}:=|V^{+}|italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT := | italic_V start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT | and n2:=|V−|assignsubscript𝑛2superscript𝑉n_{2}:=|V^{-}|italic_n start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT := | italic_V start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT |. On the event ℬℬ\mathcal{B}caligraphic_B, we can obtain that
| 1,945
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91
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, 1 = n2−n2/3≤n1,n2≤n2+n2/3.formulae-sequence𝑛2superscript𝑛23subscript𝑛1subscript𝑛2𝑛2superscript𝑛23\frac{n}{2}-n^{2/3}\leq n_{1},n_{2}\leq\frac{n}{2}+n^{2/3}.divide start_ARG italic_n end_ARG start_ARG 2 end_ARG - italic_n start_POSTSUPERSCRIPT 2 / 3 end_POSTSUPERSCRIPT ≤ italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_n start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ≤ divide start_ARG italic_n end_ARG start_ARG 2 end_ARG + italic_n start_POSTSUPERSCRIPT 2 / 3 end_POSTSUPERSCRIPT .. , 2 = . , 3 = (87)
For i∈V+𝑖superscript𝑉i\in V^{+}italic_i ∈ italic_V start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT, it holds that degG(i)=∑j=1n1−1Xj+∑j=1n2Yjsubscriptdegree𝐺𝑖subscriptsuperscriptsubscript𝑛11𝑗1subscript𝑋𝑗subscriptsuperscriptsubscript𝑛2𝑗1subscript𝑌𝑗\deg_{G}(i)=\sum^{n_{1}-1}_{j=1}X_{j}+\sum^{n_{2}}_{j=1}Y_{j}roman_deg start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_i ) = ∑ start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT italic_X start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT + ∑ start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT italic_Y start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT, where Xj∼Bern(p)similar-tosubscript𝑋𝑗Bern𝑝X_{j}\sim\operatorname{Bern}(p)italic_X start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∼ roman_Bern ( italic_p ) and Yj∼Bern(q)similar-tosubscript𝑌𝑗Bern𝑞Y_{j}\sim\operatorname{Bern}(q)italic_Y start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∼ roman_Bern ( italic_q ). Therefore, we can obtain that
, 1 = ℙ(degG(i)≤k)ℙsubscriptdegree𝐺𝑖𝑘\displaystyle\mathbb{P}(\deg_{G}(i)\leq k)blackboard_P ( roman_deg start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_i ) ≤ italic_k ). , 2 = =ℙ(∑j=1n1−1Xj+∑j=1n2Yj≤k)absentℙsubscriptsuperscriptsubscript𝑛11𝑗1subscript𝑋𝑗subscriptsuperscriptsubscript𝑛2𝑗1subscript𝑌𝑗𝑘\displaystyle=\mathbb{P}\left(\sum^{n_{1}-1}_{j=1}X_{j}+\sum^{n_{2}}_{j=1}Y_{j%
}\leq k\right)= blackboard_P ( ∑ start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT italic_X start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT + ∑ start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT italic_Y start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ≤ italic_k ). , 3 = . , 4 = (88). , 1 = . , 2 = ≤(a)inft>0(1−p+pe−t)n1−1(1−q+qe−t)n2ektsuperscript𝑎absentsubscriptinfimum𝑡0superscript1𝑝𝑝superscript𝑒𝑡subscript𝑛11superscript1𝑞𝑞superscript𝑒𝑡subscript𝑛2superscript𝑒𝑘𝑡\displaystyle\stackrel{{\scriptstyle(a)}}{{\leq}}\inf_{t>0}(1-p+pe^{-t})^{n_{1%
}-1}(1-q+qe^{-t})^{n_{2}}e^{kt}start_RELOP SUPERSCRIPTOP start_ARG ≤ end_ARG start_ARG ( italic_a ) end_ARG end_RELOP roman_inf start_POSTSUBSCRIPT italic_t > 0 end_POSTSUBSCRIPT ( 1 - italic_p + italic_p italic_e start_POSTSUPERSCRIPT - italic_t end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - 1 end_POSTSUPERSCRIPT ( 1 - italic_q + italic_q italic_e start_POSTSUPERSCRIPT - italic_t end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT italic_k italic_t end_POSTSUPERSCRIPT. , 3 = . , 4 = (88). , 1 = . , 2 = ≤(b)inft>0exp(−(n1−1)p(1−e−t)−n2q(1−e−t)+kt)superscript𝑏absentsubscriptinfimum𝑡0subscript𝑛11𝑝1superscript𝑒𝑡subscript𝑛2𝑞1superscript𝑒𝑡𝑘𝑡\displaystyle\stackrel{{\scriptstyle(b)}}{{\leq}}\inf_{t>0}\exp\left(-(n_{1}-1%
)p(1-e^{-t})-n_{2}q(1-e^{-t})+kt\right)start_RELOP SUPERSCRIPTOP start_ARG ≤ end_ARG start_ARG ( italic_b ) end_ARG end_RELOP roman_inf start_POSTSUBSCRIPT italic_t > 0 end_POSTSUBSCRIPT roman_exp ( - ( italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - 1 ) italic_p ( 1 - italic_e start_POSTSUPERSCRIPT - italic_t end_POSTSUPERSCRIPT ) - italic_n start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_q ( 1 - italic_e start_POSTSUPERSCRIPT - italic_t end_POSTSUPERSCRIPT ) + italic_k italic_t ). , 3 = . , 4 = (88). , 1 = . , 2 = ≤(c)exp(−np+q2+o(np+q2)+klognp+1).superscript𝑐absent𝑛𝑝𝑞2𝑜𝑛𝑝𝑞2𝑘𝑛𝑝1\displaystyle\stackrel{{\scriptstyle(c)}}{{\leq}}\exp\left(-n\frac{p+q}{2}+o%
\left(n\frac{p+q}{2}\right)+k\log np+1\right).start_RELOP SUPERSCRIPTOP start_ARG ≤ end_ARG start_ARG ( italic_c ) end_ARG end_RELOP roman_exp ( - italic_n divide start_ARG italic_p + italic_q end_ARG start_ARG 2 end_ARG + italic_o ( italic_n divide start_ARG italic_p + italic_q end_ARG start_ARG 2 end_ARG ) + italic_k roman_log italic_n italic_p + 1 ) .. , 3 = . , 4 = (88)
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
IX Proof of Theorem 5:
Achievability of Exact Matching in Correlated Contextual Stochastic Block Models
IX-C Proofs of Theorems 12 and 13 and Lemmas 5 and 6
Proof:
, 1 = n2−n2/3≤n1,n2≤n2+n2/3.formulae-sequence𝑛2superscript𝑛23subscript𝑛1subscript𝑛2𝑛2superscript𝑛23\frac{n}{2}-n^{2/3}\leq n_{1},n_{2}\leq\frac{n}{2}+n^{2/3}.divide start_ARG italic_n end_ARG start_ARG 2 end_ARG - italic_n start_POSTSUPERSCRIPT 2 / 3 end_POSTSUPERSCRIPT ≤ italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_n start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ≤ divide start_ARG italic_n end_ARG start_ARG 2 end_ARG + italic_n start_POSTSUPERSCRIPT 2 / 3 end_POSTSUPERSCRIPT .. , 2 = . , 3 = (87)
For i∈V+𝑖superscript𝑉i\in V^{+}italic_i ∈ italic_V start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT, it holds that degG(i)=∑j=1n1−1Xj+∑j=1n2Yjsubscriptdegree𝐺𝑖subscriptsuperscriptsubscript𝑛11𝑗1subscript𝑋𝑗subscriptsuperscriptsubscript𝑛2𝑗1subscript𝑌𝑗\deg_{G}(i)=\sum^{n_{1}-1}_{j=1}X_{j}+\sum^{n_{2}}_{j=1}Y_{j}roman_deg start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_i ) = ∑ start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT italic_X start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT + ∑ start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT italic_Y start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT, where Xj∼Bern(p)similar-tosubscript𝑋𝑗Bern𝑝X_{j}\sim\operatorname{Bern}(p)italic_X start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∼ roman_Bern ( italic_p ) and Yj∼Bern(q)similar-tosubscript𝑌𝑗Bern𝑞Y_{j}\sim\operatorname{Bern}(q)italic_Y start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∼ roman_Bern ( italic_q ). Therefore, we can obtain that
, 1 = ℙ(degG(i)≤k)ℙsubscriptdegree𝐺𝑖𝑘\displaystyle\mathbb{P}(\deg_{G}(i)\leq k)blackboard_P ( roman_deg start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_i ) ≤ italic_k ). , 2 = =ℙ(∑j=1n1−1Xj+∑j=1n2Yj≤k)absentℙsubscriptsuperscriptsubscript𝑛11𝑗1subscript𝑋𝑗subscriptsuperscriptsubscript𝑛2𝑗1subscript𝑌𝑗𝑘\displaystyle=\mathbb{P}\left(\sum^{n_{1}-1}_{j=1}X_{j}+\sum^{n_{2}}_{j=1}Y_{j%
}\leq k\right)= blackboard_P ( ∑ start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT italic_X start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT + ∑ start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT italic_Y start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ≤ italic_k ). , 3 = . , 4 = (88). , 1 = . , 2 = ≤(a)inft>0(1−p+pe−t)n1−1(1−q+qe−t)n2ektsuperscript𝑎absentsubscriptinfimum𝑡0superscript1𝑝𝑝superscript𝑒𝑡subscript𝑛11superscript1𝑞𝑞superscript𝑒𝑡subscript𝑛2superscript𝑒𝑘𝑡\displaystyle\stackrel{{\scriptstyle(a)}}{{\leq}}\inf_{t>0}(1-p+pe^{-t})^{n_{1%
}-1}(1-q+qe^{-t})^{n_{2}}e^{kt}start_RELOP SUPERSCRIPTOP start_ARG ≤ end_ARG start_ARG ( italic_a ) end_ARG end_RELOP roman_inf start_POSTSUBSCRIPT italic_t > 0 end_POSTSUBSCRIPT ( 1 - italic_p + italic_p italic_e start_POSTSUPERSCRIPT - italic_t end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - 1 end_POSTSUPERSCRIPT ( 1 - italic_q + italic_q italic_e start_POSTSUPERSCRIPT - italic_t end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT italic_k italic_t end_POSTSUPERSCRIPT. , 3 = . , 4 = (88). , 1 = . , 2 = ≤(b)inft>0exp(−(n1−1)p(1−e−t)−n2q(1−e−t)+kt)superscript𝑏absentsubscriptinfimum𝑡0subscript𝑛11𝑝1superscript𝑒𝑡subscript𝑛2𝑞1superscript𝑒𝑡𝑘𝑡\displaystyle\stackrel{{\scriptstyle(b)}}{{\leq}}\inf_{t>0}\exp\left(-(n_{1}-1%
)p(1-e^{-t})-n_{2}q(1-e^{-t})+kt\right)start_RELOP SUPERSCRIPTOP start_ARG ≤ end_ARG start_ARG ( italic_b ) end_ARG end_RELOP roman_inf start_POSTSUBSCRIPT italic_t > 0 end_POSTSUBSCRIPT roman_exp ( - ( italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - 1 ) italic_p ( 1 - italic_e start_POSTSUPERSCRIPT - italic_t end_POSTSUPERSCRIPT ) - italic_n start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_q ( 1 - italic_e start_POSTSUPERSCRIPT - italic_t end_POSTSUPERSCRIPT ) + italic_k italic_t ). , 3 = . , 4 = (88). , 1 = . , 2 = ≤(c)exp(−np+q2+o(np+q2)+klognp+1).superscript𝑐absent𝑛𝑝𝑞2𝑜𝑛𝑝𝑞2𝑘𝑛𝑝1\displaystyle\stackrel{{\scriptstyle(c)}}{{\leq}}\exp\left(-n\frac{p+q}{2}+o%
\left(n\frac{p+q}{2}\right)+k\log np+1\right).start_RELOP SUPERSCRIPTOP start_ARG ≤ end_ARG start_ARG ( italic_c ) end_ARG end_RELOP roman_exp ( - italic_n divide start_ARG italic_p + italic_q end_ARG start_ARG 2 end_ARG + italic_o ( italic_n divide start_ARG italic_p + italic_q end_ARG start_ARG 2 end_ARG ) + italic_k roman_log italic_n italic_p + 1 ) .. , 3 = . , 4 = (88)
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The inequality (a)𝑎(a)( italic_a ) holds by Chernoff bound, the inequality (b)𝑏(b)( italic_b ) holds since 1−x≥e−x1𝑥superscript𝑒𝑥1-x\geq e^{-x}1 - italic_x ≥ italic_e start_POSTSUPERSCRIPT - italic_x end_POSTSUPERSCRIPT, and the inequality (c)𝑐(c)( italic_c ) holds by (87) and choosing t=lognp𝑡𝑛𝑝t=\log npitalic_t = roman_log italic_n italic_p.
Since the same result can be obtained for i∈V−𝑖superscript𝑉i\in V^{-}italic_i ∈ italic_V start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT as well, the proof is complete.
∎
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
IX Proof of Theorem 5:
Achievability of Exact Matching in Correlated Contextual Stochastic Block Models
IX-C Proofs of Theorems 12 and 13 and Lemmas 5 and 6
Proof:
The inequality (a)𝑎(a)( italic_a ) holds by Chernoff bound, the inequality (b)𝑏(b)( italic_b ) holds since 1−x≥e−x1𝑥superscript𝑒𝑥1-x\geq e^{-x}1 - italic_x ≥ italic_e start_POSTSUPERSCRIPT - italic_x end_POSTSUPERSCRIPT, and the inequality (c)𝑐(c)( italic_c ) holds by (87) and choosing t=lognp𝑡𝑛𝑝t=\log npitalic_t = roman_log italic_n italic_p.
Since the same result can be obtained for i∈V−𝑖superscript𝑉i\in V^{-}italic_i ∈ italic_V start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT as well, the proof is complete.
∎
| 249
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93
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To prove Theorem 6, we will analyze the MAP estimator and find the conditions where the MAP estimator fails. The posterior distribution of the permutation π∈Sn𝜋subscript𝑆𝑛\pi\in S_{n}italic_π ∈ italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT in the correlated SBMs was analyzed in [7, 6, 39]. We will introduce this result and the related lemmas and explain how these previous results can be utilized to prove Theorem 6.
Given community labels 𝝈𝝈{\boldsymbol{\sigma}}bold_italic_σ, for a,b∈{0,1}𝑎𝑏01a,b\in\{0,1\}italic_a , italic_b ∈ { 0 , 1 }, define
, 1 = ψab+(π):=∑(i,j)∈ℰ+(𝝈)𝟙{(Ai,j,Bπ(i),π(j))=(a,b)},assignsubscriptsuperscript𝜓𝑎𝑏𝜋subscript𝑖𝑗superscriptℰ𝝈1subscript𝐴𝑖𝑗subscript𝐵𝜋𝑖𝜋𝑗𝑎𝑏\displaystyle\psi^{+}_{ab}(\pi):=\sum_{(i,j)\in\mathcal{E}^{+}({\boldsymbol{%
\sigma}})}\mathds{1}\{(A_{i,j},B_{\pi(i),\pi(j)})=(a,b)\},italic_ψ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ( italic_π ) := ∑ start_POSTSUBSCRIPT ( italic_i , italic_j ) ∈ caligraphic_E start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ( bold_italic_σ ) end_POSTSUBSCRIPT blackboard_1 { ( italic_A start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT , italic_B start_POSTSUBSCRIPT italic_π ( italic_i ) , italic_π ( italic_j ) end_POSTSUBSCRIPT ) = ( italic_a , italic_b ) } ,. , 2 = . , 3 = (89)
, 1 = ψab−(π):=∑(i,j)∈ℰ−(𝝈)𝟙{(Ai,j,Bπ(i),π(j))=(a,b)},assignsubscriptsuperscript𝜓𝑎𝑏𝜋subscript𝑖𝑗superscriptℰ𝝈1subscript𝐴𝑖𝑗subscript𝐵𝜋𝑖𝜋𝑗𝑎𝑏\displaystyle\psi^{-}_{ab}(\pi):=\sum_{(i,j)\in\mathcal{E}^{-}({\boldsymbol{%
\sigma}})}\mathds{1}\{(A_{i,j},B_{\pi(i),\pi(j)})=(a,b)\},italic_ψ start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ( italic_π ) := ∑ start_POSTSUBSCRIPT ( italic_i , italic_j ) ∈ caligraphic_E start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ( bold_italic_σ ) end_POSTSUBSCRIPT blackboard_1 { ( italic_A start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT , italic_B start_POSTSUBSCRIPT italic_π ( italic_i ) , italic_π ( italic_j ) end_POSTSUBSCRIPT ) = ( italic_a , italic_b ) } ,. , 2 = . , 3 = (90)
, 1 = χ+(π):=∑(i,j)∈ℰ+(𝝈)Bπ(i),π(j),assignsuperscript𝜒𝜋subscript𝑖𝑗superscriptℰ𝝈subscript𝐵𝜋𝑖𝜋𝑗\displaystyle\chi^{+}(\pi):=\sum_{(i,j)\in\mathcal{E}^{+}({\boldsymbol{\sigma}%
})}B_{\pi(i),\pi(j)},italic_χ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ( italic_π ) := ∑ start_POSTSUBSCRIPT ( italic_i , italic_j ) ∈ caligraphic_E start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ( bold_italic_σ ) end_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT italic_π ( italic_i ) , italic_π ( italic_j ) end_POSTSUBSCRIPT ,. , 2 = . , 3 = (91)
, 1 = χ−(π):=∑(i,j)∈ℰ+(𝝈)Bπ(i),π(j),assignsuperscript𝜒𝜋subscript𝑖𝑗superscriptℰ𝝈subscript𝐵𝜋𝑖𝜋𝑗\displaystyle\chi^{-}(\pi):=\sum_{(i,j)\in\mathcal{E}^{+}({\boldsymbol{\sigma}%
})}B_{\pi(i),\pi(j)},italic_χ start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ( italic_π ) := ∑ start_POSTSUBSCRIPT ( italic_i , italic_j ) ∈ caligraphic_E start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ( bold_italic_σ ) end_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT italic_π ( italic_i ) , italic_π ( italic_j ) end_POSTSUBSCRIPT ,. , 2 = . , 3 = (92)
where ℰ+(𝝈)superscriptℰ𝝈\mathcal{E}^{+}({\boldsymbol{\sigma}})caligraphic_E start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ( bold_italic_σ ) represents the set of node pairs belonging to the same community, while ℰ−(𝝈)superscriptℰ𝝈\mathcal{E}^{-}({\boldsymbol{\sigma}})caligraphic_E start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ( bold_italic_σ ) represents the set of node pairs belonging to different communities.
We will also use the notation for the edge probabilities as follows.
| 1,441
|
Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
X Proof of Theorem 6:
Impossibility of Exact Matching in Correlated Contextual Stochastic Block Models
To prove Theorem 6, we will analyze the MAP estimator and find the conditions where the MAP estimator fails. The posterior distribution of the permutation π∈Sn𝜋subscript𝑆𝑛\pi\in S_{n}italic_π ∈ italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT in the correlated SBMs was analyzed in [7, 6, 39]. We will introduce this result and the related lemmas and explain how these previous results can be utilized to prove Theorem 6.
Given community labels 𝝈𝝈{\boldsymbol{\sigma}}bold_italic_σ, for a,b∈{0,1}𝑎𝑏01a,b\in\{0,1\}italic_a , italic_b ∈ { 0 , 1 }, define
, 1 = ψab+(π):=∑(i,j)∈ℰ+(𝝈)𝟙{(Ai,j,Bπ(i),π(j))=(a,b)},assignsubscriptsuperscript𝜓𝑎𝑏𝜋subscript𝑖𝑗superscriptℰ𝝈1subscript𝐴𝑖𝑗subscript𝐵𝜋𝑖𝜋𝑗𝑎𝑏\displaystyle\psi^{+}_{ab}(\pi):=\sum_{(i,j)\in\mathcal{E}^{+}({\boldsymbol{%
\sigma}})}\mathds{1}\{(A_{i,j},B_{\pi(i),\pi(j)})=(a,b)\},italic_ψ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ( italic_π ) := ∑ start_POSTSUBSCRIPT ( italic_i , italic_j ) ∈ caligraphic_E start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ( bold_italic_σ ) end_POSTSUBSCRIPT blackboard_1 { ( italic_A start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT , italic_B start_POSTSUBSCRIPT italic_π ( italic_i ) , italic_π ( italic_j ) end_POSTSUBSCRIPT ) = ( italic_a , italic_b ) } ,. , 2 = . , 3 = (89)
, 1 = ψab−(π):=∑(i,j)∈ℰ−(𝝈)𝟙{(Ai,j,Bπ(i),π(j))=(a,b)},assignsubscriptsuperscript𝜓𝑎𝑏𝜋subscript𝑖𝑗superscriptℰ𝝈1subscript𝐴𝑖𝑗subscript𝐵𝜋𝑖𝜋𝑗𝑎𝑏\displaystyle\psi^{-}_{ab}(\pi):=\sum_{(i,j)\in\mathcal{E}^{-}({\boldsymbol{%
\sigma}})}\mathds{1}\{(A_{i,j},B_{\pi(i),\pi(j)})=(a,b)\},italic_ψ start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ( italic_π ) := ∑ start_POSTSUBSCRIPT ( italic_i , italic_j ) ∈ caligraphic_E start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ( bold_italic_σ ) end_POSTSUBSCRIPT blackboard_1 { ( italic_A start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT , italic_B start_POSTSUBSCRIPT italic_π ( italic_i ) , italic_π ( italic_j ) end_POSTSUBSCRIPT ) = ( italic_a , italic_b ) } ,. , 2 = . , 3 = (90)
, 1 = χ+(π):=∑(i,j)∈ℰ+(𝝈)Bπ(i),π(j),assignsuperscript𝜒𝜋subscript𝑖𝑗superscriptℰ𝝈subscript𝐵𝜋𝑖𝜋𝑗\displaystyle\chi^{+}(\pi):=\sum_{(i,j)\in\mathcal{E}^{+}({\boldsymbol{\sigma}%
})}B_{\pi(i),\pi(j)},italic_χ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ( italic_π ) := ∑ start_POSTSUBSCRIPT ( italic_i , italic_j ) ∈ caligraphic_E start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ( bold_italic_σ ) end_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT italic_π ( italic_i ) , italic_π ( italic_j ) end_POSTSUBSCRIPT ,. , 2 = . , 3 = (91)
, 1 = χ−(π):=∑(i,j)∈ℰ+(𝝈)Bπ(i),π(j),assignsuperscript𝜒𝜋subscript𝑖𝑗superscriptℰ𝝈subscript𝐵𝜋𝑖𝜋𝑗\displaystyle\chi^{-}(\pi):=\sum_{(i,j)\in\mathcal{E}^{+}({\boldsymbol{\sigma}%
})}B_{\pi(i),\pi(j)},italic_χ start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ( italic_π ) := ∑ start_POSTSUBSCRIPT ( italic_i , italic_j ) ∈ caligraphic_E start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ( bold_italic_σ ) end_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT italic_π ( italic_i ) , italic_π ( italic_j ) end_POSTSUBSCRIPT ,. , 2 = . , 3 = (92)
where ℰ+(𝝈)superscriptℰ𝝈\mathcal{E}^{+}({\boldsymbol{\sigma}})caligraphic_E start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ( bold_italic_σ ) represents the set of node pairs belonging to the same community, while ℰ−(𝝈)superscriptℰ𝝈\mathcal{E}^{-}({\boldsymbol{\sigma}})caligraphic_E start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ( bold_italic_σ ) represents the set of node pairs belonging to different communities.
We will also use the notation for the edge probabilities as follows.
| 1,480
|
94
|
, 1 = pab:=ℙ((Ai,j,Bπ∗(i),π∗(j))=(a,b)|𝝈)={ps2if (a,b)=(1,1) and σi=σj;ps(1−s)if (a,b)=(1,0),(0,1) and σi=σj;1−2ps+ps2if (a,b)=(0,0) and σi=σj.assignsubscript𝑝𝑎𝑏ℙsubscript𝐴𝑖𝑗subscript𝐵subscript𝜋𝑖subscript𝜋𝑗conditional𝑎𝑏𝝈cases𝑝superscript𝑠2if 𝑎𝑏11 and subscript𝜎𝑖subscript𝜎𝑗𝑝𝑠1𝑠formulae-sequenceif 𝑎𝑏1001 and subscript𝜎𝑖subscript𝜎𝑗12𝑝𝑠𝑝superscript𝑠2if 𝑎𝑏00 and subscript𝜎𝑖subscript𝜎𝑗p_{ab}:=\mathbb{P}\left((A_{i,j},B_{\pi_{*}(i),\pi_{*}(j)})=(a,b)|{\boldsymbol%
{\sigma}}\right)=\begin{cases}ps^{2}&\text{if }(a,b)=(1,1)\text{ and }\sigma_{%
i}=\sigma_{j};\\
ps(1-s)&\text{if }(a,b)=(1,0),(0,1)\text{ and }\sigma_{i}=\sigma_{j};\\
1-2ps+ps^{2}&\text{if }(a,b)=(0,0)\text{ and }\sigma_{i}=\sigma_{j}.\end{cases}italic_p start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT := blackboard_P ( ( italic_A start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT , italic_B start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_i ) , italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_j ) end_POSTSUBSCRIPT ) = ( italic_a , italic_b ) | bold_italic_σ ) = { start_ROW start_CELL italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL start_CELL if ( italic_a , italic_b ) = ( 1 , 1 ) and italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ; end_CELL end_ROW start_ROW start_CELL italic_p italic_s ( 1 - italic_s ) end_CELL start_CELL if ( italic_a , italic_b ) = ( 1 , 0 ) , ( 0 , 1 ) and italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ; end_CELL end_ROW start_ROW start_CELL 1 - 2 italic_p italic_s + italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL start_CELL if ( italic_a , italic_b ) = ( 0 , 0 ) and italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT . end_CELL end_ROW. , 2 = . , 3 = (93)
, 1 = qab:=ℙ((Ai,j,Bπ∗(i),π∗(j))=(a,b)|𝝈)={qs2if (a,b)=(1,1) and σi≠σj;qs(1−s)if (a,b)=(1,0),(0,1) and σi≠σj;1−2qs+qs2if (a,b)=(0,0) and σi≠σj.assignsubscript𝑞𝑎𝑏ℙsubscript𝐴𝑖𝑗subscript𝐵subscript𝜋𝑖subscript𝜋𝑗conditional𝑎𝑏𝝈cases𝑞superscript𝑠2if 𝑎𝑏11 and subscript𝜎𝑖subscript𝜎𝑗𝑞𝑠1𝑠formulae-sequenceif 𝑎𝑏1001 and subscript𝜎𝑖subscript𝜎𝑗12𝑞𝑠𝑞superscript𝑠2if 𝑎𝑏00 and subscript𝜎𝑖subscript𝜎𝑗q_{ab}:=\mathbb{P}\left((A_{i,j},B_{\pi_{*}(i),\pi_{*}(j)})=(a,b)|{\boldsymbol%
{\sigma}}\right)=\begin{cases}qs^{2}&\text{if }(a,b)=(1,1)\text{ and }\sigma_{%
i}\neq\sigma_{j};\\
qs(1-s)&\text{if }(a,b)=(1,0),(0,1)\text{ and }\sigma_{i}\neq\sigma_{j};\\
1-2qs+qs^{2}&\text{if }(a,b)=(0,0)\text{ and }\sigma_{i}\neq\sigma_{j}.\end{cases}italic_q start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT := blackboard_P ( ( italic_A start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT , italic_B start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_i ) , italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_j ) end_POSTSUBSCRIPT ) = ( italic_a , italic_b ) | bold_italic_σ ) = { start_ROW start_CELL italic_q italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL start_CELL if ( italic_a , italic_b ) = ( 1 , 1 ) and italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≠ italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ; end_CELL end_ROW start_ROW start_CELL italic_q italic_s ( 1 - italic_s ) end_CELL start_CELL if ( italic_a , italic_b ) = ( 1 , 0 ) , ( 0 , 1 ) and italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≠ italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ; end_CELL end_ROW start_ROW start_CELL 1 - 2 italic_q italic_s + italic_q italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL start_CELL if ( italic_a , italic_b ) = ( 0 , 0 ) and italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≠ italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT . end_CELL end_ROW. , 2 = . , 3 = (94)
That is, pabsubscript𝑝𝑎𝑏p_{ab}italic_p start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT represents the edge probability between nodes within the same community, while qabsubscript𝑞𝑎𝑏q_{ab}italic_q start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT represents the edge probability between nodes in different communities.
Given A,B𝐴𝐵A,Bitalic_A , italic_B and 𝝈𝝈{\boldsymbol{\sigma}}bold_italic_σ, the posterior distribution for π𝜋\piitalic_π can be written as follows:
| 1,814
|
Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
X Proof of Theorem 6:
Impossibility of Exact Matching in Correlated Contextual Stochastic Block Models
, 1 = pab:=ℙ((Ai,j,Bπ∗(i),π∗(j))=(a,b)|𝝈)={ps2if (a,b)=(1,1) and σi=σj;ps(1−s)if (a,b)=(1,0),(0,1) and σi=σj;1−2ps+ps2if (a,b)=(0,0) and σi=σj.assignsubscript𝑝𝑎𝑏ℙsubscript𝐴𝑖𝑗subscript𝐵subscript𝜋𝑖subscript𝜋𝑗conditional𝑎𝑏𝝈cases𝑝superscript𝑠2if 𝑎𝑏11 and subscript𝜎𝑖subscript𝜎𝑗𝑝𝑠1𝑠formulae-sequenceif 𝑎𝑏1001 and subscript𝜎𝑖subscript𝜎𝑗12𝑝𝑠𝑝superscript𝑠2if 𝑎𝑏00 and subscript𝜎𝑖subscript𝜎𝑗p_{ab}:=\mathbb{P}\left((A_{i,j},B_{\pi_{*}(i),\pi_{*}(j)})=(a,b)|{\boldsymbol%
{\sigma}}\right)=\begin{cases}ps^{2}&\text{if }(a,b)=(1,1)\text{ and }\sigma_{%
i}=\sigma_{j};\\
ps(1-s)&\text{if }(a,b)=(1,0),(0,1)\text{ and }\sigma_{i}=\sigma_{j};\\
1-2ps+ps^{2}&\text{if }(a,b)=(0,0)\text{ and }\sigma_{i}=\sigma_{j}.\end{cases}italic_p start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT := blackboard_P ( ( italic_A start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT , italic_B start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_i ) , italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_j ) end_POSTSUBSCRIPT ) = ( italic_a , italic_b ) | bold_italic_σ ) = { start_ROW start_CELL italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL start_CELL if ( italic_a , italic_b ) = ( 1 , 1 ) and italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ; end_CELL end_ROW start_ROW start_CELL italic_p italic_s ( 1 - italic_s ) end_CELL start_CELL if ( italic_a , italic_b ) = ( 1 , 0 ) , ( 0 , 1 ) and italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ; end_CELL end_ROW start_ROW start_CELL 1 - 2 italic_p italic_s + italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL start_CELL if ( italic_a , italic_b ) = ( 0 , 0 ) and italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT . end_CELL end_ROW. , 2 = . , 3 = (93)
, 1 = qab:=ℙ((Ai,j,Bπ∗(i),π∗(j))=(a,b)|𝝈)={qs2if (a,b)=(1,1) and σi≠σj;qs(1−s)if (a,b)=(1,0),(0,1) and σi≠σj;1−2qs+qs2if (a,b)=(0,0) and σi≠σj.assignsubscript𝑞𝑎𝑏ℙsubscript𝐴𝑖𝑗subscript𝐵subscript𝜋𝑖subscript𝜋𝑗conditional𝑎𝑏𝝈cases𝑞superscript𝑠2if 𝑎𝑏11 and subscript𝜎𝑖subscript𝜎𝑗𝑞𝑠1𝑠formulae-sequenceif 𝑎𝑏1001 and subscript𝜎𝑖subscript𝜎𝑗12𝑞𝑠𝑞superscript𝑠2if 𝑎𝑏00 and subscript𝜎𝑖subscript𝜎𝑗q_{ab}:=\mathbb{P}\left((A_{i,j},B_{\pi_{*}(i),\pi_{*}(j)})=(a,b)|{\boldsymbol%
{\sigma}}\right)=\begin{cases}qs^{2}&\text{if }(a,b)=(1,1)\text{ and }\sigma_{%
i}\neq\sigma_{j};\\
qs(1-s)&\text{if }(a,b)=(1,0),(0,1)\text{ and }\sigma_{i}\neq\sigma_{j};\\
1-2qs+qs^{2}&\text{if }(a,b)=(0,0)\text{ and }\sigma_{i}\neq\sigma_{j}.\end{cases}italic_q start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT := blackboard_P ( ( italic_A start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT , italic_B start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_i ) , italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_j ) end_POSTSUBSCRIPT ) = ( italic_a , italic_b ) | bold_italic_σ ) = { start_ROW start_CELL italic_q italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL start_CELL if ( italic_a , italic_b ) = ( 1 , 1 ) and italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≠ italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ; end_CELL end_ROW start_ROW start_CELL italic_q italic_s ( 1 - italic_s ) end_CELL start_CELL if ( italic_a , italic_b ) = ( 1 , 0 ) , ( 0 , 1 ) and italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≠ italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ; end_CELL end_ROW start_ROW start_CELL 1 - 2 italic_q italic_s + italic_q italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL start_CELL if ( italic_a , italic_b ) = ( 0 , 0 ) and italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≠ italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT . end_CELL end_ROW. , 2 = . , 3 = (94)
That is, pabsubscript𝑝𝑎𝑏p_{ab}italic_p start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT represents the edge probability between nodes within the same community, while qabsubscript𝑞𝑎𝑏q_{ab}italic_q start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT represents the edge probability between nodes in different communities.
Given A,B𝐴𝐵A,Bitalic_A , italic_B and 𝝈𝝈{\boldsymbol{\sigma}}bold_italic_σ, the posterior distribution for π𝜋\piitalic_π can be written as follows:
| 1,853
|
95
|
Let π∈Sn𝜋subscript𝑆𝑛\pi\in S_{n}italic_π ∈ italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT. Then, we have
, 1 = ℙ(π∗=π∣A,B,𝝈)=c(p00p11p01p10)ψ11+(π)(q00q11q01q10)ψ11−(π)(p01p00)χ+(π)(q01q00)χ−(π),ℙsubscript𝜋conditional𝜋𝐴𝐵𝝈𝑐superscriptsubscript𝑝00subscript𝑝11subscript𝑝01subscript𝑝10subscriptsuperscript𝜓11𝜋superscriptsubscript𝑞00subscript𝑞11subscript𝑞01subscript𝑞10subscriptsuperscript𝜓11𝜋superscriptsubscript𝑝01subscript𝑝00superscript𝜒𝜋superscriptsubscript𝑞01subscript𝑞00superscript𝜒𝜋\mathbb{P}\left(\pi_{*}=\pi\mid A,B,{\boldsymbol{\sigma}}\right)=c\left(\frac{%
p_{00}p_{11}}{p_{01}p_{10}}\right)^{\psi^{+}_{11}(\pi)}\left(\frac{q_{00}q_{11%
}}{q_{01}q_{10}}\right)^{\psi^{-}_{11}(\pi)}\left(\frac{p_{01}}{p_{00}}\right)%
^{\chi^{+}(\pi)}\left(\frac{q_{01}}{q_{00}}\right)^{\chi^{-}(\pi)},blackboard_P ( italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT = italic_π ∣ italic_A , italic_B , bold_italic_σ ) = italic_c ( divide start_ARG italic_p start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT end_ARG start_ARG italic_p start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT end_ARG ) start_POSTSUPERSCRIPT italic_ψ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT ( italic_π ) end_POSTSUPERSCRIPT ( divide start_ARG italic_q start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT italic_q start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT end_ARG start_ARG italic_q start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT italic_q start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT end_ARG ) start_POSTSUPERSCRIPT italic_ψ start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT ( italic_π ) end_POSTSUPERSCRIPT ( divide start_ARG italic_p start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT end_ARG start_ARG italic_p start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT end_ARG ) start_POSTSUPERSCRIPT italic_χ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ( italic_π ) end_POSTSUPERSCRIPT ( divide start_ARG italic_q start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT end_ARG start_ARG italic_q start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT end_ARG ) start_POSTSUPERSCRIPT italic_χ start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ( italic_π ) end_POSTSUPERSCRIPT ,. , 2 = . , 3 = (95)
where c𝑐citalic_c is a constant depending on A,B𝐴𝐵A,Bitalic_A , italic_B and 𝛔𝛔{\boldsymbol{\sigma}}bold_italic_σ.
For a permutation π∈Sn𝜋subscript𝑆𝑛\pi\in S_{n}italic_π ∈ italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, let us define
, 1 = ℋ(π):={i∈[n]:∀j∈[n],Ai,jBπ(i),π(j)=0},assignℋ𝜋conditional-set𝑖delimited-[]𝑛formulae-sequencefor-all𝑗delimited-[]𝑛subscript𝐴𝑖𝑗subscript𝐵𝜋𝑖𝜋𝑗0\mathcal{H}(\pi):=\left\{i\in[n]:\forall j\in[n],A_{i,j}B_{\pi(i),\pi(j)}=0%
\right\},caligraphic_H ( italic_π ) := { italic_i ∈ [ italic_n ] : ∀ italic_j ∈ [ italic_n ] , italic_A start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT italic_π ( italic_i ) , italic_π ( italic_j ) end_POSTSUBSCRIPT = 0 } ,. , 2 = . , 3 = (96)
where A𝐴Aitalic_A and B𝐵Bitalic_B are the adjacency matrices of the graphs G1subscript𝐺1G_{1}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and G2subscript𝐺2G_{2}italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT.
Additionally, let ℋ(π)+:=ℋ(π)∩V+assignℋsuperscript𝜋ℋ𝜋superscript𝑉\mathcal{H}(\pi)^{+}:=\mathcal{H}(\pi)\cap V^{+}caligraphic_H ( italic_π ) start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT := caligraphic_H ( italic_π ) ∩ italic_V start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT and ℋ(π)−:=ℋπ∩V−assignℋsuperscript𝜋superscriptℋ𝜋superscript𝑉\mathcal{H}(\pi)^{-}:=\mathcal{H}^{\pi}\cap V^{-}caligraphic_H ( italic_π ) start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT := caligraphic_H start_POSTSUPERSCRIPT italic_π end_POSTSUPERSCRIPT ∩ italic_V start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT. To simplify the notation, let the node sets corresponding to π∗subscript𝜋\pi_{*}italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT be denoted by ℋ∗,ℋ∗+subscriptℋsuperscriptsubscriptℋ\mathcal{H}_{*},\mathcal{H}_{*}^{+}caligraphic_H start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT , caligraphic_H start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT and ℋ∗−superscriptsubscriptℋ\mathcal{H}_{*}^{-}caligraphic_H start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT.
Now, we will derive a lower bound for |ℋ∗+|superscriptsubscriptℋ|\mathcal{H}_{*}^{+}|| caligraphic_H start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT | and |ℋ∗−|superscriptsubscriptℋ|\mathcal{H}_{*}^{-}|| caligraphic_H start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT |. Recall that
, 1 = ℬ:={n2−n2/3≤|V+|,|V−|≤n2+n2/3}assignℬformulae-sequence𝑛2superscript𝑛23superscript𝑉superscript𝑉𝑛2superscript𝑛23\mathcal{B}:=\left\{\frac{n}{2}-n^{2/3}\leq|V^{+}|,|V^{-}|\leq\frac{n}{2}+n^{2%
/3}\right\}caligraphic_B := { divide start_ARG italic_n end_ARG start_ARG 2 end_ARG - italic_n start_POSTSUPERSCRIPT 2 / 3 end_POSTSUPERSCRIPT ≤ | italic_V start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT | , | italic_V start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT | ≤ divide start_ARG italic_n end_ARG start_ARG 2 end_ARG + italic_n start_POSTSUPERSCRIPT 2 / 3 end_POSTSUPERSCRIPT }. , 2 = . , 3 = (97)
| 1,949
|
Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
X Proof of Theorem 6:
Impossibility of Exact Matching in Correlated Contextual Stochastic Block Models
Lemma 8 (Lemma C.1 in [39]).
Let π∈Sn𝜋subscript𝑆𝑛\pi\in S_{n}italic_π ∈ italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT. Then, we have
, 1 = ℙ(π∗=π∣A,B,𝝈)=c(p00p11p01p10)ψ11+(π)(q00q11q01q10)ψ11−(π)(p01p00)χ+(π)(q01q00)χ−(π),ℙsubscript𝜋conditional𝜋𝐴𝐵𝝈𝑐superscriptsubscript𝑝00subscript𝑝11subscript𝑝01subscript𝑝10subscriptsuperscript𝜓11𝜋superscriptsubscript𝑞00subscript𝑞11subscript𝑞01subscript𝑞10subscriptsuperscript𝜓11𝜋superscriptsubscript𝑝01subscript𝑝00superscript𝜒𝜋superscriptsubscript𝑞01subscript𝑞00superscript𝜒𝜋\mathbb{P}\left(\pi_{*}=\pi\mid A,B,{\boldsymbol{\sigma}}\right)=c\left(\frac{%
p_{00}p_{11}}{p_{01}p_{10}}\right)^{\psi^{+}_{11}(\pi)}\left(\frac{q_{00}q_{11%
}}{q_{01}q_{10}}\right)^{\psi^{-}_{11}(\pi)}\left(\frac{p_{01}}{p_{00}}\right)%
^{\chi^{+}(\pi)}\left(\frac{q_{01}}{q_{00}}\right)^{\chi^{-}(\pi)},blackboard_P ( italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT = italic_π ∣ italic_A , italic_B , bold_italic_σ ) = italic_c ( divide start_ARG italic_p start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT end_ARG start_ARG italic_p start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT end_ARG ) start_POSTSUPERSCRIPT italic_ψ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT ( italic_π ) end_POSTSUPERSCRIPT ( divide start_ARG italic_q start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT italic_q start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT end_ARG start_ARG italic_q start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT italic_q start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT end_ARG ) start_POSTSUPERSCRIPT italic_ψ start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT ( italic_π ) end_POSTSUPERSCRIPT ( divide start_ARG italic_p start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT end_ARG start_ARG italic_p start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT end_ARG ) start_POSTSUPERSCRIPT italic_χ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ( italic_π ) end_POSTSUPERSCRIPT ( divide start_ARG italic_q start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT end_ARG start_ARG italic_q start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT end_ARG ) start_POSTSUPERSCRIPT italic_χ start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ( italic_π ) end_POSTSUPERSCRIPT ,. , 2 = . , 3 = (95)
where c𝑐citalic_c is a constant depending on A,B𝐴𝐵A,Bitalic_A , italic_B and 𝛔𝛔{\boldsymbol{\sigma}}bold_italic_σ.
For a permutation π∈Sn𝜋subscript𝑆𝑛\pi\in S_{n}italic_π ∈ italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, let us define
, 1 = ℋ(π):={i∈[n]:∀j∈[n],Ai,jBπ(i),π(j)=0},assignℋ𝜋conditional-set𝑖delimited-[]𝑛formulae-sequencefor-all𝑗delimited-[]𝑛subscript𝐴𝑖𝑗subscript𝐵𝜋𝑖𝜋𝑗0\mathcal{H}(\pi):=\left\{i\in[n]:\forall j\in[n],A_{i,j}B_{\pi(i),\pi(j)}=0%
\right\},caligraphic_H ( italic_π ) := { italic_i ∈ [ italic_n ] : ∀ italic_j ∈ [ italic_n ] , italic_A start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT italic_π ( italic_i ) , italic_π ( italic_j ) end_POSTSUBSCRIPT = 0 } ,. , 2 = . , 3 = (96)
where A𝐴Aitalic_A and B𝐵Bitalic_B are the adjacency matrices of the graphs G1subscript𝐺1G_{1}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and G2subscript𝐺2G_{2}italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT.
Additionally, let ℋ(π)+:=ℋ(π)∩V+assignℋsuperscript𝜋ℋ𝜋superscript𝑉\mathcal{H}(\pi)^{+}:=\mathcal{H}(\pi)\cap V^{+}caligraphic_H ( italic_π ) start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT := caligraphic_H ( italic_π ) ∩ italic_V start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT and ℋ(π)−:=ℋπ∩V−assignℋsuperscript𝜋superscriptℋ𝜋superscript𝑉\mathcal{H}(\pi)^{-}:=\mathcal{H}^{\pi}\cap V^{-}caligraphic_H ( italic_π ) start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT := caligraphic_H start_POSTSUPERSCRIPT italic_π end_POSTSUPERSCRIPT ∩ italic_V start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT. To simplify the notation, let the node sets corresponding to π∗subscript𝜋\pi_{*}italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT be denoted by ℋ∗,ℋ∗+subscriptℋsuperscriptsubscriptℋ\mathcal{H}_{*},\mathcal{H}_{*}^{+}caligraphic_H start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT , caligraphic_H start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT and ℋ∗−superscriptsubscriptℋ\mathcal{H}_{*}^{-}caligraphic_H start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT.
Now, we will derive a lower bound for |ℋ∗+|superscriptsubscriptℋ|\mathcal{H}_{*}^{+}|| caligraphic_H start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT | and |ℋ∗−|superscriptsubscriptℋ|\mathcal{H}_{*}^{-}|| caligraphic_H start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT |. Recall that
, 1 = ℬ:={n2−n2/3≤|V+|,|V−|≤n2+n2/3}assignℬformulae-sequence𝑛2superscript𝑛23superscript𝑉superscript𝑉𝑛2superscript𝑛23\mathcal{B}:=\left\{\frac{n}{2}-n^{2/3}\leq|V^{+}|,|V^{-}|\leq\frac{n}{2}+n^{2%
/3}\right\}caligraphic_B := { divide start_ARG italic_n end_ARG start_ARG 2 end_ARG - italic_n start_POSTSUPERSCRIPT 2 / 3 end_POSTSUPERSCRIPT ≤ | italic_V start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT | , | italic_V start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT | ≤ divide start_ARG italic_n end_ARG start_ARG 2 end_ARG + italic_n start_POSTSUPERSCRIPT 2 / 3 end_POSTSUPERSCRIPT }. , 2 = . , 3 = (97)
| 2,000
|
96
|
as defined in Definition 6.
| 7
|
Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
X Proof of Theorem 6:
Impossibility of Exact Matching in Correlated Contextual Stochastic Block Models
Lemma 8 (Lemma C.1 in [39]).
as defined in Definition 6.
| 58
|
97
|
Suppose that the community label σ∗subscript𝜎\sigma_{*}italic_σ start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT is given and the event ℬℬ\mathcal{B}caligraphic_B holds. If ns2p+q2=O(logn)𝑛superscript𝑠2𝑝𝑞2𝑂𝑛ns^{2}\frac{p+q}{2}=O(\log n)italic_n italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divide start_ARG italic_p + italic_q end_ARG start_ARG 2 end_ARG = italic_O ( roman_log italic_n ), then it holds that
, 1 = |ℋ∗+|,|ℋ∗−|≥18n1−ns2(p+q)2lognsuperscriptsubscriptℋsuperscriptsubscriptℋ18superscript𝑛1𝑛superscript𝑠2𝑝𝑞2𝑛|\mathcal{H}_{*}^{+}|,|\mathcal{H}_{*}^{-}|\geq\frac{1}{8}n^{1-\frac{ns^{2}(p+%
q)}{2\log n}}| caligraphic_H start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT | , | caligraphic_H start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT | ≥ divide start_ARG 1 end_ARG start_ARG 8 end_ARG italic_n start_POSTSUPERSCRIPT 1 - divide start_ARG italic_n italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_p + italic_q ) end_ARG start_ARG 2 roman_log italic_n end_ARG end_POSTSUPERSCRIPT. , 2 = . , 3 = (98)
with probability 1−o(1)1𝑜11-o(1)1 - italic_o ( 1 ).
We define the set of permutations 𝒯∗subscript𝒯\mathcal{T}_{*}caligraphic_T start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT as follows. The permutations belonging to 𝒯∗subscript𝒯\mathcal{T}_{*}caligraphic_T start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT will have a posterior probability greater than or equal to that of π∗subscript𝜋\pi_{*}italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT
| 566
|
Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
X Proof of Theorem 6:
Impossibility of Exact Matching in Correlated Contextual Stochastic Block Models
Lemma 9.
Suppose that the community label σ∗subscript𝜎\sigma_{*}italic_σ start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT is given and the event ℬℬ\mathcal{B}caligraphic_B holds. If ns2p+q2=O(logn)𝑛superscript𝑠2𝑝𝑞2𝑂𝑛ns^{2}\frac{p+q}{2}=O(\log n)italic_n italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divide start_ARG italic_p + italic_q end_ARG start_ARG 2 end_ARG = italic_O ( roman_log italic_n ), then it holds that
, 1 = |ℋ∗+|,|ℋ∗−|≥18n1−ns2(p+q)2lognsuperscriptsubscriptℋsuperscriptsubscriptℋ18superscript𝑛1𝑛superscript𝑠2𝑝𝑞2𝑛|\mathcal{H}_{*}^{+}|,|\mathcal{H}_{*}^{-}|\geq\frac{1}{8}n^{1-\frac{ns^{2}(p+%
q)}{2\log n}}| caligraphic_H start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT | , | caligraphic_H start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT | ≥ divide start_ARG 1 end_ARG start_ARG 8 end_ARG italic_n start_POSTSUPERSCRIPT 1 - divide start_ARG italic_n italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_p + italic_q ) end_ARG start_ARG 2 roman_log italic_n end_ARG end_POSTSUPERSCRIPT. , 2 = . , 3 = (98)
with probability 1−o(1)1𝑜11-o(1)1 - italic_o ( 1 ).
We define the set of permutations 𝒯∗subscript𝒯\mathcal{T}_{*}caligraphic_T start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT as follows. The permutations belonging to 𝒯∗subscript𝒯\mathcal{T}_{*}caligraphic_T start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT will have a posterior probability greater than or equal to that of π∗subscript𝜋\pi_{*}italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT
| 609
|
98
|
A permutation π∈𝒯∗𝜋subscript𝒯\pi\in\mathcal{T}_{*}italic_π ∈ caligraphic_T start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT if and only if the following conditions hold:
•
π(i)=π∗(i)𝜋𝑖subscript𝜋𝑖\pi(i)=\pi_{*}(i)italic_π ( italic_i ) = italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_i ) if i∈[n]\ℋ∗𝑖\delimited-[]𝑛subscriptℋi\in[n]\backslash\mathcal{H}_{*}italic_i ∈ [ italic_n ] \ caligraphic_H start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPTReport issue for preceding element
•
π(i)=π∗(ρ+(i))𝜋𝑖subscript𝜋superscript𝜌𝑖\pi(i)=\pi_{*}(\rho^{+}(i))italic_π ( italic_i ) = italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ( italic_i ) ) if i∈ℋ∗+𝑖superscriptsubscriptℋi\in\mathcal{H}_{*}^{+}italic_i ∈ caligraphic_H start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT,Report issue for preceding element
•
π(i)=π∗(ρ−(i))𝜋𝑖subscript𝜋superscript𝜌𝑖\pi(i)=\pi_{*}(\rho^{-}(i))italic_π ( italic_i ) = italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ( italic_i ) ) if i∈ℋ∗−𝑖superscriptsubscriptℋi\in\mathcal{H}_{*}^{-}italic_i ∈ caligraphic_H start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT,Report issue for preceding element
where ρ+superscript𝜌\rho^{+}italic_ρ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT and ρ−superscript𝜌\rho^{-}italic_ρ start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT are any permutations over ℋ∗+subscriptsuperscriptℋ\mathcal{H}^{+}_{*}caligraphic_H start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT and ℋ∗−subscriptsuperscriptℋ\mathcal{H}^{-}_{*}caligraphic_H start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT, respectively.
The permutations in set 𝒯∗subscript𝒯\mathcal{T}_{*}caligraphic_T start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT satisfy the following properties:
| 707
|
Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
X Proof of Theorem 6:
Impossibility of Exact Matching in Correlated Contextual Stochastic Block Models
Definition 7.
A permutation π∈𝒯∗𝜋subscript𝒯\pi\in\mathcal{T}_{*}italic_π ∈ caligraphic_T start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT if and only if the following conditions hold:
•
π(i)=π∗(i)𝜋𝑖subscript𝜋𝑖\pi(i)=\pi_{*}(i)italic_π ( italic_i ) = italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_i ) if i∈[n]\ℋ∗𝑖\delimited-[]𝑛subscriptℋi\in[n]\backslash\mathcal{H}_{*}italic_i ∈ [ italic_n ] \ caligraphic_H start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPTReport issue for preceding element
•
π(i)=π∗(ρ+(i))𝜋𝑖subscript𝜋superscript𝜌𝑖\pi(i)=\pi_{*}(\rho^{+}(i))italic_π ( italic_i ) = italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ( italic_i ) ) if i∈ℋ∗+𝑖superscriptsubscriptℋi\in\mathcal{H}_{*}^{+}italic_i ∈ caligraphic_H start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT,Report issue for preceding element
•
π(i)=π∗(ρ−(i))𝜋𝑖subscript𝜋superscript𝜌𝑖\pi(i)=\pi_{*}(\rho^{-}(i))italic_π ( italic_i ) = italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ( italic_i ) ) if i∈ℋ∗−𝑖superscriptsubscriptℋi\in\mathcal{H}_{*}^{-}italic_i ∈ caligraphic_H start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT,Report issue for preceding element
where ρ+superscript𝜌\rho^{+}italic_ρ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT and ρ−superscript𝜌\rho^{-}italic_ρ start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT are any permutations over ℋ∗+subscriptsuperscriptℋ\mathcal{H}^{+}_{*}caligraphic_H start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT and ℋ∗−subscriptsuperscriptℋ\mathcal{H}^{-}_{*}caligraphic_H start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT, respectively.
The permutations in set 𝒯∗subscript𝒯\mathcal{T}_{*}caligraphic_T start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT satisfy the following properties:
| 750
|
99
|
For any permutation π∈𝒯∗𝜋subscript𝒯\pi\in\mathcal{T}_{*}italic_π ∈ caligraphic_T start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT, we have that ψ11+(π)≥ψ11+(π∗)subscriptsuperscript𝜓11𝜋subscriptsuperscript𝜓11subscript𝜋\psi^{+}_{11}(\pi)\geq\psi^{+}_{11}(\pi_{*})italic_ψ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT ( italic_π ) ≥ italic_ψ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT ( italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ), ψ11−(π)≥ψ11−(π∗)subscriptsuperscript𝜓11𝜋subscriptsuperscript𝜓11subscript𝜋\psi^{-}_{11}(\pi)\geq\psi^{-}_{11}(\pi_{*})italic_ψ start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT ( italic_π ) ≥ italic_ψ start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT ( italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ), χ+(π)=χ+(π∗)superscript𝜒𝜋superscript𝜒subscript𝜋\chi^{+}(\pi)=\chi^{+}(\pi_{*})italic_χ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ( italic_π ) = italic_χ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ( italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ) and χ−(π)=χ−(π∗)superscript𝜒𝜋superscript𝜒subscript𝜋\chi^{-}(\pi)=\chi^{-}(\pi_{*})italic_χ start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ( italic_π ) = italic_χ start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ( italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ).
| 515
|
Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
X Proof of Theorem 6:
Impossibility of Exact Matching in Correlated Contextual Stochastic Block Models
Lemma 10 (Proposition C.2 and C.3 in [39]).
For any permutation π∈𝒯∗𝜋subscript𝒯\pi\in\mathcal{T}_{*}italic_π ∈ caligraphic_T start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT, we have that ψ11+(π)≥ψ11+(π∗)subscriptsuperscript𝜓11𝜋subscriptsuperscript𝜓11subscript𝜋\psi^{+}_{11}(\pi)\geq\psi^{+}_{11}(\pi_{*})italic_ψ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT ( italic_π ) ≥ italic_ψ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT ( italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ), ψ11−(π)≥ψ11−(π∗)subscriptsuperscript𝜓11𝜋subscriptsuperscript𝜓11subscript𝜋\psi^{-}_{11}(\pi)\geq\psi^{-}_{11}(\pi_{*})italic_ψ start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT ( italic_π ) ≥ italic_ψ start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT ( italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ), χ+(π)=χ+(π∗)superscript𝜒𝜋superscript𝜒subscript𝜋\chi^{+}(\pi)=\chi^{+}(\pi_{*})italic_χ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ( italic_π ) = italic_χ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ( italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ) and χ−(π)=χ−(π∗)superscript𝜒𝜋superscript𝜒subscript𝜋\chi^{-}(\pi)=\chi^{-}(\pi_{*})italic_χ start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ( italic_π ) = italic_χ start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ( italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ).
| 571
|
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