chunk_id float64 0 165 | chunk_text stringlengths 1 11.1k | chunk_text_tokens float64 1 2k | serialized_text stringlengths 2 11.2k | serialized_text_tokens float64 1 2.03k |
|---|---|---|---|---|
0 | This is experimental HTML to improve accessibility. We invite you to report rendering errors. Use Alt+Y to toggle on accessible reporting links and Alt+Shift+Y to toggle off. Learn more about this project and help improve conversions. | 46 | This is experimental HTML to improve accessibility. We invite you to report rendering errors. Use Alt+Y to toggle on accessible reporting links and Alt+Shift+Y to toggle off. Learn more about this project and help improve conversions. | 46 |
1 | 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 ... | 480 | 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 ... | 484 |
2 | 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 classica... | 275 | 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 correla... | 291 |
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 detectio... | 1,455 | 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 ma... | 1,472 |
4 | 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 | 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... | 61 |
5 | 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... | 1,732 | 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_POSTS... | 1,763 |
6 | 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 := [ bol... | 1,180 | 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…... | 1,211 |
7 | 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_... | 1,219 | 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{... | 1,252 |
8 | 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 ... | 966 | 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_POSTSUPERSC... | 999 |
9 | 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 i... | 80 | 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. Ta... | 102 |
10 | 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_POSTSUBSCR... | 1,594 | 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,... | 1,627 |
11 | 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... | 711 | 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... | 742 |
12 | 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 correlate... | 746 | 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... | 779 |
13 | 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}i... | 995 | 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 w... | 1,029 |
14 | 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... | 1,685 | 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 o... | 1,707 |
15 | 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}... | 339 | 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_AR... | 361 |
16 | 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 ( i... | 1,550 | 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... | 1,572 |
17 | 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_POSTSUPE... | 713 | 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 sta... | 735 |
18 | 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, ... | 90 | 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 c... | 112 |
19 | 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. | 37 | 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 wh... | 71 |
20 | 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-sequence𝑑411supersc... | 724 | 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,\bol... | 770 |
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_... | 1,822 | 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... | 1,868 |
22 | \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_ital... | 1,594 | 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 ... | 1,640 |
23 | 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>0i... | 1,883 | 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 ro... | 1,930 |
24 | , 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)black... | 1,615 | 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-sequencesuperscript... | 1,662 |
25 | 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-ϵ... | 628 | 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,\bol... | 674 |
26 | 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 - itali... | 815 | 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 matchin... | 864 |
27 | 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 G... | 75 | 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... | 110 |
28 | 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)lognsuperscriptnorm𝝁2... | 1,523 | 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\text... | 1,571 |
29 | 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... | 665 | 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
Remark 3 (Comparison with Standard GMM Results).
When only X𝑋Xitalic_X is available, [14, 5] showed that exact community r... | 711 |
30 | 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\... | 440 | 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 4 (Impossibility for Exact Community Recovery).
Let (X,Y)∼CGMMs(n,𝛍,d,ρ)similar-to𝑋𝑌CGMMs𝑛𝛍𝑑𝜌(X,Y)\sim\text... | 488 |
31 | 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... | 217 | 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
Remark 4 (Information-Theoretic Gaps in Exact Community Recovery).
Theorem 3 assumes that either (7) or (8) is satisfied to... | 267 |
32 | 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... | 64 | Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
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 exa... | 88 |
33 | The following theorem provides sufficient conditions for exact matching in correlated CSBMs. | 15 | 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. | 44 |
34 | 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... | 1,231 | Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
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\textno... | 1,272 |
35 | 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_POSTSUB... | 522 | Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
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 ... | 564 |
36 | 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... | 1,522 | 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\textno... | 1,563 |
37 | 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 star... | 860 | 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 ro... | 890 |
38 | 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 ; itali... | 890 | 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_{... | 933 |
39 | 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 - squ... | 467 | 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... | 509 |
40 | 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 ; itali... | 604 | 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_{... | 647 |
41 | 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}\;... | 1,060 | 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-sequencesuperscript21𝜌𝑅221𝜌𝑅... | 1,100 |
42 | 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. | 93 | 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 permutatio... | 113 |
43 | 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}... | 1,745 | 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:=∥... | 1,775 |
44 | 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}}{... | 563 | 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(log... | 593 |
45 | 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... | 217 | 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 c... | 237 |
46 | 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 ro... | 611 | 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 ... | 645 |
47 | 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 ... | 362 | 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 ... | 397 |
48 | 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 fundamen... | 68 | 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 ... | 97 |
49 | 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 logn... | 158 | 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 Haje... | 185 |
50 | 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, exa... | 267 | 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... | 293 |
51 | 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 ... | 1,876 | 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 u... | 1,913 |
52 | 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... | 701 | 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... | 742 |
53 | 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 (... | 346 | 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>t0𝑡su... | 398 |
54 | 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_POSTSUBSC... | 511 | 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)^𝜋subscriptargmin𝜋subscript𝑆𝑛superscriptsubscript𝑖1𝑛subsc... | 550 |
55 | , 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... | 1,639 | 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)!absent... | 1,678 |
56 | 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_A... | 1,907 | 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 ... | 1,946 |
57 | 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-∥∥𝝁2𝑂... | 1,829 | 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}|... | 1,868 |
58 | 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... | 1,735 | 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 ine... | 1,774 |
59 | 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 it... | 1,357 | 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... | 1,396 |
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 ∥... | 1,331 | 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-\lambd... | 1,370 |
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}%
... | 1,796 | 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)ℙ𝜌𝜆2superscriptnormsubscript𝒛1subscript𝒛2𝝁1𝜆𝜌𝜆𝝁2�... | 1,835 |
62 | 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)superscript𝑑22𝜆𝜌𝜆s... | 949 | 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:
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,... | 988 |
63 | 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.... | 337 | Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
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 ... | 375 |
64 | 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 = d4... | 664 | Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
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}ital... | 715 |
65 | 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... | 685 | Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
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}i... | 736 |
66 | 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)2subscript𝒙𝑖subscri... | 512 | 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+πYs... | 550 |
67 | 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)ass... | 1,905 | 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{\boldsy... | 1,956 |
68 | , 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+ρ2oisubscript𝒙𝑖subscript𝒚subsc... | 1,231 | 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 ... | 1,282 |
69 | 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 ... | 363 | 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 matc... | 402 |
70 | 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 analyti... | 130 | 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-A The k𝑘kitalic_k-core matching and the proof of Theorem 5
The k𝑘kitalic_k-core matching has been extensively studied ... | 190 |
71 | 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}it... | 805 | 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-A The k𝑘kitalic_k-core matching and the proof of Theorem 5
Definition 1 (Matching).
Consider two graphs G1subscript𝐺1G... | 871 |
72 | 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_{\mi... | 365 | 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-A The k𝑘kitalic_k-core matching and the proof of Theorem 5
Definition 2 (k𝑘kitalic_k-core matching and k𝑘kitalic_k-co... | 449 |
73 | 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 , i... | 1,219 | 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-A The k𝑘kitalic_k-core matching and the proof of Theorem 5
Theorem 12 (Partial matching achievable by the k𝑘kitalic_k-... | 1,299 |
74 | 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 , i... | 437 | 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-A The k𝑘kitalic_k-core matching and the proof of Theorem 5
Theorem 13 (Exact matching achievable by the k𝑘kitalic_k-co... | 517 |
75 | 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 it... | 1,357 | 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-A The k𝑘kitalic_k-core matching and the proof of Theorem 5
Proof:
Let k=nps2(lognps2)2∨logn(loglogn)2𝑘𝑛𝑝supe... | 1,419 |
76 | 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 , ita... | 373 | 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
For a matching (M,φ)𝑀𝜑(M,\varphi)( italic_M , italic_φ ), defi... | 430 |
77 | 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... | 207 | 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
Definition 3 (Weak k𝑘kitalic_k-core matching).
We say that a ma... | 279 |
78 | 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 sta... | 587 | 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
Definition 4 (π∗subscript𝜋\pi_{*}italic_π start_POSTSUBSCRIPT ∗... | 678 |
79 | 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}itali... | 418 | 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
Definition 5 (k𝑘kitalic_k-core set).
For a graph G𝐺Gitalic_G, ... | 489 |
80 | 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 k𝑘kitali... | 1,001 | 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,... | 1,073 |
81 | 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)𝑀𝜑ℳ�... | 460 | 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 4 (Lemma A.4 in [35]).
Consider the (G1,G2)∼CSBMs(n,p,q,s... | 529 |
82 | , 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... | 295 | 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
Definition 6 (Balanced communities).
, 1 = ℬ:={n2−n2/3≤|V+|,|V−|... | 360 |
83 | It holds that ℙ(ℬ)=1−o(1).ℙℬ1𝑜1\mathbb{P}(\mathcal{B})=1-o(1).blackboard_P ( caligraphic_B ) = 1 - italic_o ( 1 ) . | 65 | 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 5.
It holds that ℙ(ℬ)=1−o(1).ℙℬ1𝑜1\mathbb{P}(\mathcal{B... | 126 |
84 | 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_POSTSUBSC... | 529 | 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 6.
Let G∼SBM(n,p,q)similar-to𝐺SBM𝑛𝑝𝑞G\sim\textnormal{... | 590 |
85 | 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 de... | 469 | 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 7 (Lemma IV.6 in [35]).
Let G∼SBM(n,p,q)similar-to𝐺SBM𝑛... | 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>0ital... | 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}\... | 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 =... | 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).superscript𝑒2𝜃𝑝superscript𝑠2su... | 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_POSTSUP... | 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)subscr... | 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 st... | 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 t... | 1,945 |
91 | , 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_POSTSU... | 1,827 | 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-sequence𝑛2superscript𝑛23... | 1,890 |
92 | 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 npita... | 186 | 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, th... | 249 |
93 | 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 introduc... | 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 d... | 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 subscript𝜎𝑖subscript𝜎𝑗𝑝𝑠1... | 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... | 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),ℙsubscript𝜋conditional𝜋𝐴𝐵𝝈𝑐superscriptsubscript𝑝00subscript𝑝11subscript𝑝01subscript𝑝10sub... | 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_POSTSUB... | 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 ita... | 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 a... | 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\del... | 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_POSTSU... | 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_... | 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 ∈ caligraphi... | 571 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.