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[EQUATION] where [MATH] is a function on the graph nodes (so it is an [MATH] -element vector) and [MATH] is again a positive number. Expression ( 11 [MATH] -converges to graph TV |
In the multiphase case, we represent the community assignments [MATH] in terms of an [MATH] matrix whose [MATH] entry is [MATH] , where [MATH] is the Kroneker delta. Instead of a double-well potential, we use a multi-well potential on [MATH] whose value is minimized by arguments with exactly one nonzero entry in each r... |
proposed the following potential: [EQUATION] where [MATH] is the [MATH] th row of the [MATH] matrix [MATH] and [MATH] is an [MATH] -element vector that is equal to [MATH] except for a [MATH] in the [MATH] th entry. |
For the particular case of surface-tension dynamics, we proceed as follows. Additionally, we assume in this subsection and the next that we have already eliminated the diagonal of [MATH] (see Appendix |
). Given community assignments (and hence a partition of a network), if [MATH] is the corresponding [MATH] matrix, one can show that [MATH] , where [MATH] is the entry-wise product. |
Therefore, an appropriate GL functional for our problem is [EQUATION] Because [MATH] gives the vector of volumes, one can rewrite Eq. |
12 as [EQUATION] where [MATH] is the entry-wise exponential. As in a continuum setting, one can prove [MATH] -convergence. Theorem 4.1 |
Let [MATH] . The functionals in Eq. 13 [MATH] -converge (as functions on [MATH] ) to Eq. as [MATH] See Appendix for a proof. As far as we are aware, this is the first [MATH] -convergence result for a multiphase graph energy on arbitrary graphs. However, see |
for [MATH] -convergence applied to consistency of multiphase geometric graph energies. The resulting AC equation is [EQUATION] See Appendix |
for further details on the numerical solution of Eq. 14 4.3 MBO Iteration Input the initial domain. Initialize [MATH] as the characteristic function of the initial domain. |
for [MATH] do [MATH] is the solution at time [MATH] of [MATH] with initial condition [MATH] [MATH] , where [MATH] is the floor function. |
end for Output the set of points for which [MATH] Algorithm 2 A two-phase, continuum MBO scheme. In , Merriman, Bence, and Osher showed that continuum MCF is well-approximated by the simple iteration in Algorithm |
. In a rectangular domain, the iteration is extremely efficient, as one can use a fast Fourier transform when solving the heat equation. Esedoglu and Otto |
developed a generalized version of the MBO scheme (see Algorithm ) for computing the evolution of multiphase systems that are modeled by Eq. |
Input the initial state of the domain. Initialize [MATH] as the characteristic functions of the initial domains. for [MATH] do for |
[MATH] do [MATH] is the solution at time [MATH] to [MATH] with initial condition [MATH] end for for each point [MATH] do [MATH] // [Choose uniformly at random in case of a tie] |
[MATH] and [MATH] if [MATH] end for end for Output [MATH] Algorithm 3 A multiphase, continuum MBO scheme. One can apply the MBO idea to community detection in networks by replacing the continuum Laplacian with the (negative) combinatorial graph Laplacian, replacing [MATH] with [MATH] , changing [MATH] to [MATH] , and a... |
for additional implementation details. 4.4 Learning [MATH] The MCF, AC, and MBO algorithms are able to produce a good partition of a network, given [MATH] , but they do not include a way to find [MATH] . A simple way to address this issue is to use an expectation-maximization (EM) algorithm, in which one alternates bet... |
with respect to any component of [MATH] and setting the result to [MATH] One must be careful, however, because the optimal [MATH] is infinite when [MATH] . This is problematic, because once one of the entries in [MATH] is infinite, it prevents [MATH] in subsequent iterations from taking any nonzero value of [MATH] ; th... |
for a discussion of these examples.) We address this issue by modifying the EM algorithm to reset all infinite values of [MATH] to [MATH] , where [MATH] is the largest non-infinite element of [MATH] and [MATH] is a (hand-tuned) parameter that allows moderate growth in [MATH] |
We also need to address another practical issue for an EM approach to work. Specifically, the algorithm that we have described thus far in this section often finds bad local minima in which communities are merged erroneously or a single community is split inappropriately. |
To overcome this issue, we implement a wrapper function (see Algorithm ) that checks each community that is returned by MCF, AC, or MBO for further possible splitting or merging with other communities. Whenever we call MCF, AC, or MBO on a subgraph, we use the values of [MATH] and [MATH] for the whole graph rather than... |
for their recursive partitioning procedures. There is also a danger of overfitting by setting [MATH] , which gives a likelihood of [MATH] in Eq. |
. The proper selection of [MATH] is a complicated problem, both algorithmically and theoretically . For our tests, we were very successful by using a simple heuristic approach. (Our framework is also compatible with more sophisticated methods for selecting [MATH] .) For each data set, we supply an expected value of [MA... |
Input [MATH] [MATH] Place all nodes in the same community and add this community to a queue. while the queue is not empty do Save [MATH] and [MATH] |
Save the current objective-function value as [MATH] Partition the next community (as an induced subgraph, as we include all associated edges ) in the queue into [MATH] communities using MCF, AC, or MBO with [MATH] |
while it is possible to improve the objective-function value by merging two partition elements do Perform the merge that most improves the objective function. |
end while if the objective-function value is larger than [MATH] then Add any newly created communities to the queue. else Set [MATH] and [MATH] |
Remove the current community from the queue. end if end while Output [MATH] [MATH] Algorithm 4 Our splitting–merging wrapper for escaping from bad local minima. In this algorithm, [MATH] is the objective function from Eq. |
multiplied by [MATH] , where we chose the value [MATH] based on hand-tuning. Empirical Results We now discuss our results from several numerical experiments to (1) confirm that our algorithms can successfully recover [MATH] and [MATH] from networks that we generate using SBMs and (2) explore their applicability to real... |
), and an example related to hyperspectral video segmentation. Because of the random initialization in our approach, we perform three trials on each of the networks for each algorithm, and we report the best result in each case. |
For comparison, we also report the results of a Kernighan–Lin (KL) algorithm, which was reported in to be effective. We summarize our results in Table |
, and we highlight that we consistently recover the underlying structure in the synthetic examples. For the real networks, we compare our results with a reference partition based on metadata that is thought to be correlated with the community structure. We find that the MCF scheme performs the best among our three sche... |
We implement our methods in Matlab , so one should interpret computation times in Table as indicative that the run time is reasonable for networks with millions of edges. Given a careful implementation in a compiled language, it is possible to study even larger networks. |
For an example of code for a similar problem that was solved by an MBO scheme at large scale (including a weighted graph with almost [MATH] million nodes and [MATH] edges), see |
We briefly describe the three families of SBM-related networks that we use in our numerical experiments. Planted partition (PP) is a 16,000-node graph that consists of [MATH] equal-size communities. It is produced by the method that was described in |
. It builds a degree-corrected SBM with a truncated power-law degree distribution with exponent [MATH] . The parameter [MATH] from Equation (27) in |
is [MATH] , indicating a fairly clear separation between communities. Lancichinetti–Fortunato–Radicchi (LFR) is a standard benchmark SBM network |
. We construct 1000-node LFR graphs with a power-law degree distribution (with exponent [MATH] ), mean degree [MATH] , maximum degree [MATH] , power-law-distributed community sizes (with exponent [MATH] ), community sizes between [MATH] and [MATH] nodes, and mixing parameter [MATH] |
Multiscale SBM (MS). To construct such a graph, we take a sequence of disjoint components; in order, these are a [MATH] -clique, a [MATH] -clique, and a sequence of Erdős–Rényi (ER) graphs (drawn from the [MATH] model with [MATH] nodes and [MATH] ) of sizes [MATH] [MATH] [MATH] , …, [MATH] . Each of these graphs has a ... |
The hyperspectral video is a recording of a gas plume as it was released at the Dugway Proving Ground . A hyperspectral video is different from an RGB video, in that each pixel in the former encodes the intensity of light at a large number (e.g., [MATH] , in this case) of different wavelengths rather than at only [MATH... |
. Specifically, we use the following construction. For each pixel [MATH] and in each of [MATH] frames, we construct a vector [MATH] by concatenating the data in a [MATH] window that is centered at [MATH] . We then use a weighted cosine similarity measure (which is a common choice for hyperspectral imaging applications)... |
, we build an unweighted 10-nearest-neighbor graph using the similarity measure and a [MATH] -dimensional tree (with [MATH] . We see from Fig. |
that partitions with small values of Eq. correspond to meaningful segmentations of the image. In Table , we show an example of a [MATH] matrix that we obtain from an MS network to illustrate that we recover different surface tensions between different pairs of communities. |
Conclusions and Discussion We have shown that a particular stochastic block model (SBM) maximum-likelihood estimation (MLE) problem is equivalent to a discrete version of a well-known surface-tension problem. This equivalence, which associates graph cuts to surface areas and SBM parameters to physical surface tensions,... |
We also proved a [MATH] -convergence result that gives theoretical justification for our algorithms. Although our paper has focused on a specific form of an SBM and an associated MLE problem, our techniques should also be insightful for other studies of SBMs and their applications. One straightforward adaption is to co... |
). Another viable extension is to incorporate a small amount of supervision into the community-inference process using techniques (such as quadratic fidelity terms) from image processing. A similar idea was used for modularity maximization in |
and was tested further in Introducing supervision helps alleviate severe nonconvexity by penalizing local minima that are inconsistent with the (ideally) ground-truth classifications from which one draws the supervision. It is also important to generalize our approach to more complicated types of networks, such as mult... |
and temporal networks , and to incorporate metadata into our inference methodology. For example, given our successful results on the hyperspectral video, it may be particularly interesting to use temporal network clustering to analyze time-dependent communities in the video. |
Approaches such as inference using SBMs and modularity maximization are also related to other approaches for community detection, and the results in the present paper may help further illuminate those connections. These include recent work that relates SBMs to local methods for community detection that are based on per... |
and very recent work that established new connections between modularity maximization and several other approaches . We expect that further mapping of the relations between the diverse available perspectives for community detection (and other problems in network clustering) will yield many new insights for network theo... |
Acknowledgements ZMB and ALB were funded by NSF grants DMS-1737770 and DMS-1417674, as well as ONR grant N00014-16-1-2119. ZMB was also supported by the Department of Defense (DoD) through the National Defense Science & Engineering Graduate Fellowship (NDSEG) Program and the Eunice Kennedy Shriver National Institute of... |
The content is solely the responsibility of the authors and does not necessarily represent the official views of any of the agencies that supported this work. |
Appendix A Eliminating the Diagonal Elements of [MATH] It is difficult to interpret the parameters [MATH] in the context of Eq. and our surface-tension analogy, because they correspond to “internal” surface tensions of a single crystal. In this appendix, we use a change of variables to eliminate these diagonal terms an... |
We begin with the identity [EQUATION] and we compute [EQUATION] Combining Eq. 16 with Eq. 15 yields [EQUATION] assuming that [MATH] is finite for each [MATH] This formulation removes the diagonal from the double sum at the cost of introducing asymmetry into the subscripts of the coefficients. We can fix this new issue ... |
17 with [EQUATION] where [MATH] . The matrix [MATH] is symmetric and has [MATH] values on the diagonal. Finally, we expand a bit on the role of the volume terms in Eq. |
. The term [EQUATION] is the inner product of the vector of volumes with the diagonal of [MATH] . We minimize Eq. 19 , subject to the constraints [MATH] and [MATH] , by placing all of the nodes in the community that corresponds to the smallest entry in the diagonal of [MATH] . Therefore, these terms incentivize placing... |
Appendix B [MATH] -Convergence of the Ginzburg–Landau Approximation of ( The notion of [MATH] -convergence is defined as follows: |
Definition B.1 Let [MATH] be a metric space, and let [MATH] be a sequence of functionals that take values in [MATH] We say that [MATH] |
[MATH] -converges to another functional [MATH] if for all [MATH] , the following bounds hold: 1. (Lower bound) For every sequence [MATH] , we have [MATH] |
2. (Upper bound) For every [MATH] , there is a sequence [MATH] such that [MATH] We now prove Theorem 4.1 Proof B.2 We largely follow |
, although we generalize to account for the multiphase nature of our problem. The terms that do not involve the potential [MATH] are continuous and independent of [MATH] , so they cannot interfere with [MATH] -convergence |
Consequently, it suffices to prove that [MATH] [MATH] -converges to [EQUATION] To prove the lower bound, let [MATH] and [MATH] . (In this proof, the subscript [MATH] indexes the sequence, rather than the matrix columns.) If [MATH] corresponds to a partition, [MATH] , which is automatically less than or equal to [MATH] ... |
To prove the upper bound, let [MATH] be any [MATH] matrix. If [MATH] corresponds to a partition, then letting [MATH] for all [MATH] gives the required sequence. If [MATH] does not correspond to a partition, then [MATH] for all [MATH] still satisfies the upper bound. |
Therefore, both the upper and lower bound requirements hold, and we have proven [MATH] -convergence. Appendix C Additional Notes on the AC and MBO Schemes |
In this appendix, we discuss some practical details about our implementation of the AC and MBO solvers. The choice of [MATH] in AC is important, because it selects a characteristic scale of the transition between the [MATH] and [MATH] regions. If [MATH] is too small, the barrier to transition is large, and no evolution... |
4.1 asserts only that the minimizers of Eq. and Eq. 13 are related when [MATH] is sufficiently small. In our numerical experiments, we set [MATH] , a choice that we selected by hand-tuning using our synthetic networks. There is no reason to believe that the same value should work for all networks. For example, for the ... |
, we obtain much better results for [MATH] . A very interesting problem is to determine a correct notion of distance and accompanying quantitative estimates to allow an automated selection of [MATH] to obtain a transition layer with an appropriate width to give useful results. We discretize the AC equation via convex s... |
[EQUATION] where [MATH] . Using the constant [MATH] leads to an unconditionally stable scheme, which negates the stiffness caused by the [MATH] scale. |
It is necessary to solve a linear system of the form [EQUATION] many times. In a continuum setting, one can use a fast Fourier transform, but we do not know of a graph analog with comparable computational efficiency. Instead, we find the [MATH] eigenvectors that correspond to the smallest eigenvalues of [MATH] and the ... |
20 is then approximately equivalent to [EQUATION] Letting [MATH] and [MATH] , we write [EQUATION] which is easy to solve for [MATH] . We convert [MATH] to a solution using [MATH] . (See |
for a discussion of this method of recovering [MATH] from [MATH] .) One final detail that we wish to note is that we want the evolution of [MATH] to be restricted to have a row sum of [MATH] , so that we can interpret it in terms of probabilities. To do this, we use a modification of the projection algorithm from |
at each time step. The MBO solver uses a very similar pseudospectral scheme, although it does not include convex splitting. Unlike in the AC scheme, we need to estimate two time steps automatically in our code, instead of tuning them by hand. The first is the inner-loop step (i.e., the time step that we use for computi... |
# Source: arxiv 1806.02524 # Title: How nice are free completions of categories? # Sections: all # Downloaded: 2026-03-03T02:41:09.988615+00:00 |
How nice are free completions of categories? (Date: April 1, 2019) Abstract. Every category [MATH] has a free completion [MATH] under colimits and a free completion [MATH] under coproducts. A number of properties of [MATH] transfer to [MATH] and [MATH] (e.g., completeness or cartesian closedness). We prove that [MATH] ... |
Both authors are supported by the Grant agency of the Czech republic under the grant 19-00902S. The second author was also supported by the Grant agency of the Czech republic under the grant P201/12/G028. |
1. Introduction The free completion [MATH] of a locally small category [MATH] under (small) colimits is well known: if [MATH] is small, then [MATH] is the presheaf category [MATH] . For large categories [MATH] can be described as the full subcategory of |
[MATH] on small functors, i.e., small colimits of hom-functors. We study several properties of this completion: is it (locally) cartesian closed?, a topos?, wellpowered or cowellpowered? |
For complete categories [MATH] cartesian closedness of [MATH] was fully characterized by the second author , but we present two improvements. One is that cartesian closedeness of [MATH] implies that [MATH] is locally cartesian closed and complete. And the other is that for [MATH] complete the completion [MATH] is local... |
(a) additive – example: [MATH] , or (b) cartesian closed – examples: [MATH] [MATH] , or (c) dual to an extensive category – example: |
[EQUATION] The last example is the category of all accessible set functors. We will see that this category is a locally cartesian closed pretopos, i.e., a category both exact and extensive. Moreover [MATH] is wellpowered, cowellpowered and concrete (i.e., a faithful functor to [MATH] exists). Another interesting case i... |
, that [MATH] is cartesian closed plays a role in Scott’s theory of equilogical spaces ; we will see that [MATH] is also a locally cartesian closed pretopos, but it is neither wellpowered nor cowellpowered nor concrete. |
Whereas [MATH] is a topos for every small category, we prove that it practically never is a topos for large ones: if [MATH] has copowers and finite intersections, then [MATH] is not a topos unless [MATH] is essentially small. However, [MATH] always is a pretopos and monomorphisms and epimorphisms in it are regular. |
The situation with the free completion [MATH] under coproducts is simpler. If [MATH] has a strict initial object [MATH] (i.e., [MATH] inplies [MATH] ) and is complete, then [MATH] is cartesian closed iff [MATH] is. The same holds for local cartesian closedness and for being a topos. However, [MATH] is also cartesian cl... |
[MATH] (abelian groups) or [MATH] (small categories), [MATH] denotes the full subcategory of finite sets or groups, resp. We shorten ”cartesian closed” to cc and ”locally cartesian closed” to lcc. |
[MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] topos topos cc [MATH] lcc topos lcc topos lcc lcc [MATH] topos cc (lcc?) cc The connection between the two completions, as proved in |
, is that [MATH] is the exact completion of [MATH] The question of when an exact completion is a topos was studied in , at the end of Section 4 we turn to the results of that paper. |
Finally, for ”set-like” categories we prove that [MATH] is wellpowered and cowellpowered. Whereas [MATH] is neither wellpowered nor cowellpowered for |
[MATH] and a number of other categories. Related Work Properties of the category [MATH] of accessible set functors are also studied by Barto |
: he proves that this category is concrete (using a method different from ours) and universal, i.e., all concrete categories can be fully embedded into [MATH] |
Cartesian closedness of exact completions is investigated by Clementino et al. . For example, the exact completion of metric spaces (in Lawevere’s sense) and Lowen’s approach spaces are cartesian closed. |
Acknowledgement The authors are very grateful to the referee and the editor whose suggestions have improved the presentation of our results. |
2. The Free Coproduct Completion For every category [MATH] the free completion under coproducts is denoted by [EQUATION] That is, the category [MATH] has coproducts, and there is a full embedding [MATH] such that every functor |
[MATH] where [MATH] has coproducts has an extension to a coproduct-preserving functor [MATH] unique up to natural isomorphism. This category [MATH] can be described as the category of all collections [MATH] of objects of [MATH] , where a morphism [MATH] consists of |
(a) a function [MATH] , and (b) a collection of morphisms [MATH] [MATH] Composition and identity morphisms are as expected. The embedding [MATH] is, by abuse of notation, denoted by [MATH] and [MATH] |
In the present section, properties of [MATH] are studied. Some of them are well known or easy to see: 2.1 Example (a) [MATH] is the free coproduct completion of the terminal (one-arrow) category. |
(b) The category of sets with a unary relation is the free coproduct completion of the two-element chain [MATH] . Indeed, we have an equivalence functor assigning to every collection [MATH] of binary values the set [MATH] with the relation [MATH] |
(c) The category [MATH] is the free coproduct completion of [MATH] (d) More generally, if a category [MATH] with coproducts has the property that every object is a coproduct of coproduct-indecomposable objects, then [MATH] for the full subcategory [MATH] on all indecomposable objects. See |
, Lemma 42. (e) [MATH] can be described as the full subcategory of [MATH] consisting of coproducts of hom-functors. 2.2 Remark (1) [MATH] has (finite) limits iff [MATH] has multi-limits of all (finite) diagrams, see |
. Recall that a multi-limit of a diagram [MATH] is a set of cones such that every cone of [MATH] factorizes through a unique member of that set and, moreover, the factorization is unique. In categories with an initial object limits and multi-limits are clearly the same. |
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