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, hierarchical structure , and metadata . The benefits of an SBM approach include statistical defensibility, theoretical tractability, asymptotic consistency under certain conditions, definable transitions between solvable and unsolvable regimes, and theoretically optimal algorithms |
. As reviewed in , there are numerous other approaches for community detection, and statistical inference using SBMs is a method of choice among many people in the network-science community. A recent empirical study compared several types of SBMs and other community-detection approaches on a variety of examples |
Recently, Newman showed that one can interpret modularity maximization , which is still among the most popular approaches for community detection, as a special case of an SBM |
. In another paper , it was shown that one can also interpret modularity maximization in terms of graph cuts and total-variation (TV) minimization. The latter connection allows the application of methods from geometric partial differential equations (PDEs) and [MATH] minimization to community detection. This relationsh... |
In this paper, we develop such a formulation, and we also incorporate substantial new ingredients to do so. The principal one is the notion of surface tension as a generalization of total variation. Additionally, we need to examine an energy landscape that requires a novel splitting–merging heuristic to navigate it, wh... |
, in that our modified Allen–Cahn (AC) and Merriman–Bence–Osher (MBO) schemes involve diffusion with all-to-all coupling in addition to coupling that arises from a potential well, balance terms, or thresholding. |
The main result of the present work is the establishment of an equivalence between SBMs and surface-tension models from the literature on PDEs that model crystal growth. Crystal growth is an important aspect of certain annealing processes in metallurgy |
. It is a consolidation process, wherein the many crystals in a metal grow and absorb each other to reduce the surface-tension energy that is associated to the interfaces between them. The various processes involved have been modeled from many perspectives, including molecular dynamics |
, front tracking , vertex models , and many others. (See for a much more extensive set of references.) It has been observed experimentally that the interface between any two grains evolves according to motion by mean curvature |
. Because mean-curvature flow is related to gradient descent (in the [MATH] inner product) of the TV energy , this leads naturally to formulations in terms of level sets |
, phase fields , and threshold dynamics . Although the interfaces follow mean-curvature flow, each different interface can evolve at a different rate, as there are different surface-tension densities between each pair of crystals. In realistic cases, surface tensions are both inhomogeneous and anisotropic, and they req... |
, especially for dealing with the topological challenges that arise at crystal junctions, which routinely form and disappear. Recently, Jacobs showed how to apply techniques from models of crystal growth to graph-cut problems from semisupervised learning |
. (See for additional related work.) Several other recent papers, which do not directly involve surface tension, have used ideas from perimeter minimization and/or TV minimization for graph cuts and clustering in machine learning |
. Three of those papers are concerned explicitly with ideas from network science Each community in a network is analogous to a crystal, and the set of edges between nodes from a pair of communities is akin to the topological boundary between a pair of crystals. The surface-tension densities correspond to the differing ... |
Our paper proceeds as follows. In Section , we present background information about stochastic block models, total variation, and surface tension. In Section |
, we state and prove our main result, which establishes an equivalence between discrete surface tension and maximum-likelihood estimation via an SBM. In Section |
, we discuss three numerical approaches for performing SBM MLE: mean-curvature flow, [MATH] -convergence, and threshold dynamics. We discuss our results on both synthetic and real-world networks in Section |
. In Section , we conclude and discuss our results. We give additional technical details in appendices. Background 2.1 Stochastic Block Models (SBMs) |
The most basic type of SBM has [MATH] nodes and an assignment [MATH] that associates each node with one of [MATH] sets. It also has an associated [MATH] symmetric, nonnegative matrix [MATH] that encodes the affinities between pairs of communities. One generates an undirected, unweighted graph as follows: for each pair ... |
. In the present paper, we use the SBM from There is considerable flexibility in the choice of [MATH] , which leads in turn to flexibility in the SBMs themselves |
. Three examples of [MATH] , using [MATH] , will help illustrate the diversity of possible block structures. 1. If [MATH] , one obtains traditional assortative community structure, in which nodes have a larger probability to be adjacent to nodes in the same community, instead of ones in different communities. |
2. If [MATH] , nodes tend to associate more with nodes that are in other communities. As [MATH] the graph becomes increasingly bipartite. |
3. If [MATH] , there is a core–periphery (CP) structure: nodes from set [MATH] are connected densely to many nodes, but nodes from set [MATH] are connected sparsely to other nodes |
We illustrate these three examples in Fig. . To simplify our presentation, we refer to latent block structures as “community structure,” regardless of the form of the matrix [MATH] |
The above SBM is not realistic enough for many applications, largely because each node has the same expected degree . To address this issue, one can suppose that one knows the degree sequence [MATH] and then define connection probabilities to take this information into account. The easiest approach (see the discussion ... |
) is to model the adjacency-matrix elements [MATH] (which is [MATH] , where [MATH] is the number of edges in the associated network and [MATH] is now allowed to take any value in [MATH] . This allows both multi-edges and self-edges. Although such edges can have important effects (including in configuration models) |
, we neglect them for simplicity. Observe that the parameters [MATH] [MATH] , and [MATH] are necessary and sufficient to specify [MATH] as a random variable. In the present paper, we focus on the SBM that we described in this paragraph; it is known as a “degree-corrected” SBM |
Given an observed network, one can attempt to infer some sort of underlying community structure by statistical fitting methods. There are several ways to do this, including maximum-likelihood estimation (MLE), maximum a posteriori (MAP) estimation, and maximum marginal likelihood (MML) estimation. In MLE, one chooses t... |
. We use MLE, which is the simplest approach. In mathematical terms, the problem is to determine [EQUATION] where [MATH] is the probability density function. Because we determine the edges independently, [MATH] is given by |
[EQUATION] We use a Poisson distribution, so [EQUATION] where the need for cases arises from our convention that [MATH] if a self-edge is present. To solve Eq. |
, one can equivalently maximize the logarithm of [MATH] . Conveniently, this changes the multiplicative structure into additive structure and allows us to drop irrelevant constants. The resulting problem is |
[EQUATION] If [MATH] , the quantity [MATH] is understood to be [MATH] if [MATH] and [MATH] otherwise. Common optimization heuristics for solving Eq. |
include greedy ascent , Kernighan–Lin (KL) node swapping , and coordinate descent . As far as we are aware, the theory of these approaches has not received much attention. |
In light of the extreme nonconvexity of the modularity objective function (which is known to be related to the planted-partition form of SBMs |
), we expect that it is necessary to use multiple random initializations for any local algorithm. (Ideas from consensus clustering may also be helpful |
.) Ways to elaborate the SBM of interest include incorporating overlapping and hierarchical communities , generalizing to structures such as time-dependent and multilayer networks |
, and incorporating metadata . There are also Bayesian models and pseudo-likelihood-based methods . We do not consider such embellishments in this paper, although we conjecture that it is possible to generalize our approach to some (and perhaps all) of these settings. |
2.2 Total Variation Consider a smooth function [MATH] for some [MATH] . The total variation (TV) of [MATH] is [EQUATION] For [MATH] , equation ( ) describes the total amount of increase and decrease of the function [MATH] . If [MATH] is smooth except for jump discontinuities along a smooth hypersurface [MATH] , one can... |
[EQUATION] where [MATH] is the height of the jump across the discontinuity. The first integral uses a [MATH] -dimensional measure, and the second one uses a [MATH] -dimensional measure. In the particular case in which [MATH] and [MATH] is the characteristic function of some set [MATH] , we see that [MATH] is the perime... |
Total variation is an important regularizer in machine learning. It is worth contrasting it with the Dirichlet energy [MATH] , which has minimizers that satisfy [MATH] , a condition that guarantees smoothness. However, minimizers of TV need not be smooth, as they can admit jump discontinuities. In image denoising, for ... |
Another use of TV energy is in relaxations, in which one can transform a nonconvex problem involving piecewise-constant constraints into a convex problem with the same minimizers |
A common heuristic explanation for this phenomenon (see Fig. ) uses the shape of the [MATH] -norm unit ball. The simplest case is in two dimensions, where the [MATH] -norm ball is diamond-shaped, and minimizing the [MATH] -norm over certain domains (e.g., a line) gives a sparse solution, in the sense that most componen... |
In the context of TV minimization, we take the [MATH] -norm of a function’s gradient, rather than of the function itself. Therefore, instead of promoting sparsity of the function values, we promote sparse gradients, thereby incentivizing piecewise-constant minimizers for TV. Although our discussion is heuristic, note t... |
Algorithmically, one can minimize TV using approaches such as phase-field models or threshold dynamics , both of which rely on the fact that the gradient descent (in the [MATH] inner product) of TV is related to mean-curvature flow |
. The alternating-directions method of multipliers (ADMM) and graph-cut methods, such as the one in , are also very effective at solving such problems. |
Thus far, we have restricted our discussion of TV to a continuum setting. There exist graph analogs of the mathematical objects — gradients, measures, integrals, tangent spaces, divergences, and so on — that one uses to define TV in a continuum setting. For instance, for any function [MATH] on the nodes of a graph and ... |
[EQUATION] Using the inner products [EQUATION] on the spaces of functions on the nodes and edges, respectively, gives the divergence as the adjoint of the gradient: |
[EQUATION] In a continuum, an alternative definition of TV is [EQUATION] where the supremum is over an appropriate set of test functions. For a graph, ( ) is equivalent to |
[EQUATION] See for a detailed justification of these definitions. Some methods for graph clustering (e.g., see ) rely on the combinatorial graph Laplacian [MATH] , which is a discrete analog of the continuum Laplacian [MATH] . The continuum Laplacian arises in solutions to constrained optimization problems that involve... |
, which do not require thresholding to give classification information. Several previous papers have exploited this property of TV on graphs |
2.3 Surface Tension Very roughly, one can consider a metal object as being composed of a large number of crystals that range in size from microscopic to macroscopic |
. Each crystal is a highly-ordered lattice; and there is a thin, disordered interface between crystals. The sizes and orientations of these crystals affect material properties, and one goal of annealing processes is to allow crystals to reorganize to produce a useful metal. |
The potential energy of a crystal configuration is roughly [EQUATION] where [MATH] is the interface between crystals [MATH] and [MATH] , and [MATH] is the surface-tension energy density between these crystals. Each [MATH] is different, based on physical considerations that involve the exact offset between the orientati... |
by growing to consume their neighboring crystals. See for further background information. In the study of SBMs, one can use TV to express Eq. |
, but we find a more natural formulation in terms of surface-tension energy (a related notion). Specifically, we exploit the appearance of surface area in Eq. |
to cast it as a TV problem. Mathematically, we model the metal as a region of space that is partitioned into [MATH] regions, corresponding to the crystals in the metal. Let [MATH] and [MATH] , respectively, denote the characteristic functions of the regions [MATH] and [MATH] . Therefore, |
[EQUATION] Each interface between two regions evolves according to mean-curvature flow. Consequently, the surface-tension flow is locally mean-curvature flow, except at the junction of three or more crystals |
Because of this connection, one can use some of the ideas (such as phase-field and threshold-dynamics methods from TV minimization to perform surface-tension minimization. When using threshold dynamics, it is possible to do theoretical analysis in the form of Lyapunov functionals, [MATH] -convergence, and descent condi... |
An Equivalence Between SBM MLE and Discrete Surface Tension We now present a mathematical result that connects SBM MLE and discrete surface tension. |
Proposition 3.1 Maximizing the likelihood of the parameters [MATH] (i.e., node assignments) and [MATH] (i.e., affinities) in the degree-corrected SBM (see Eq. |
) is equivalent to minimizing [EQUATION] where [MATH] , the volume term is [MATH] , and [MATH] (so [MATH] ). One immediately has the following well-posedness results. For a fixed [MATH] , the expression ( ) has a solution, because the state space over which one is minimizing is of finite cardinality. Furthermore, for f... |
. (We obtain a minimum of because it is concave up.) Therefore, the full problem, in which we allow [MATH] to vary, also has a solution, because there are a finite number of candidate pairs [MATH] . Uniqueness is not guaranteed, because one can permute the community labels (and the corresponding entries in [MATH] ) to ... |
The analogy with continuum surface tension is as follows. Graph cuts are analogous to surface area: given a domain in [MATH] , one can superimpose a fine grid on space and count the number of edges that cross the boundary to estimate its surface area. In the limit of an infinitely fine grid, this estimate converges to ... |
. Similarly, graph volumes are analogous to continuum volumes. The quantities [MATH] play the role of surface tensions [MATH] , so the first set of terms is analogous to Eq. |
. One can view the second set of terms as a soft volume constraint. A constraint is “soft” if violating it adds a finite penalty on an objective function, so minimizers usually approximately satisfy the constraint. Volume-constrained versions of Eq. |
have received a great deal of attention We now prove Proposition 3.1. Proof 3.2 (Proposition 3.1) In , it was shown that maximizing the log-likelihood of the parameters [MATH] and [MATH] for a particular version of the degree-corrected SBM amounts to maximizing Eq. |
. Let [MATH] be the set of partitions of the nodes of a graph [MATH] (associated with an adjacency matrix [MATH] ) into at most [MATH] sets. Substituting [MATH] into Eq. |
gives [EQUATION] Rearranging the summations gives [EQUATION] where the inner sums are over all nodes [MATH] and [MATH] such that [MATH] and [MATH] . Rearranging again gives |
[EQUATION] Using the definition of [MATH] in the first set of terms and summing over the [MATH] index independently in the second set of terms gives |
[EQUATION] Finally, we sum over the [MATH] index in the second set of terms to obtain [EQUATION] One difference between Eq. and Eq. |
is that in Eq. , one performs optimization over the [MATH] , whereas in Eq. (i.e., in a continuum), one ordinarily treats the surface-tension densities as fixed by the choice of material that one is modeling. Another difference is that the surface-tension coefficients in the graph setting can be any element of [MATH] ,... |
). By contrast, for a continuum, one needs further restrictions to ensure well-posedness. Esedoglu and Otto proved the following sufficient conditions for well-posedness: |
(1) [MATH] (2) [MATH] (3) [MATH] In a graph setting, one can use a straightforward change of variables, [MATH] , to make [MATH] satisfy requirement (2). |
In general, however, at least one of requirements (1) and (3) is not necessarily satisfied for a graph. Requirement (1) is false whenever some component of [MATH] is negative; this occurs exactly when [MATH] has a component that is larger than [MATH] . In the continuum, requirement (3) has the interpretation of prevent... |
The analogy of Eq. with continuum surface tension is simplest for the case of assortative communities, although it is also relevant for other types of block structure. For disassortative blocks, rather than an energy cost from surface area, particles in one phase can achieve a lower energy by interacting with particles... |
then involve backward diffusion. In the continuum case, this is ill-posed; however, a graph does not include arbitrarily small length scales, so ill-posedness does not cause a problem. Backward diffusion on graphs also appeared recently in |
in the context of image processing, and it would be interesting to see if their techniques would be insightful in our context. For core–periphery structure, one phase has an energy penalty from interacting with itself but lowers its energy by interacting with the other phase. The other phase, however, prefers to intera... |
as a generalization of discrete surface tension, rather than as an analog. In the present paper, we emphasize applications to assortative community structure. |
Mean-Curvature Flow (MCF), [MATH] -Convergence, and Threshold Dynamics We now outline three algorithmic approaches that illustrate how one can use tools from surface-tension theory to solve SBM MLE problems. Our three algorithms are graph versions of mean-curvature flow (MCF), Allen–Cahn (AC) evolution, and Merriman–Be... |
, we will conduct several numerical experiments to demonstrate that these algorithms can effectively solve Eq. . We expect the performance of these algorithms to be good relative to other algorithms for SBM MLE, although a full evaluation of this claim is beyond the scope of our paper. We have posted our code at github... |
4.1 Mean-Curvature Flow Surface-tension dynamics are governed by mean-curvature flow except at junctions. Intuitively, each point on a surface moves in the direction normal to the surface at a speed given by the mean curvature at that point. In the two-phase case, such dynamics have been well-studied, and there exist n... |
. In the multiphase case, the situation is much more complicated, especially because of the topological changes that can occur and the issue of defining the behavior at the junction of three or more phases. In two-phase surface-tension dynamics, it was shown in |
that one can approximate the flow by solving a discrete-time minimizing-movements problem. Let [MATH] be one of the two regions at time [MATH] , where [MATH] is the time step. To update [MATH] , one uses |
[EQUATION] where [EQUATION] the operation [MATH] denotes the symmetric difference, and [MATH] is the topological boundary operator. The idea behind this approach is, at each time step, to shorten the curve as much as possible without straying too far from the curve location at the previous time step. |
In the setting of graphs, a similar approach was developed in , where the mean-curvature flow was given by [EQUATION] the operation [MATH] is again the symmetric difference, and [MATH] is the shortest-path distance from node [MATH] to the boundary of [MATH] . In this context, the boundary of a set of nodes is the set o... |
may still evolve, as opposed to the MBO scheme (which we use later), which becomes “stuck” when the time step is too small. Such evolution can still occur, because the penalty (associated with moving any node in [MATH] ) induced by the second set of terms in Eq. |
is [MATH] , regardless of the value of [MATH] . Conveniently, this implies for sufficiently small [MATH] that the only acceptable moves at each time step are ones that are allowed to change only the boundary nodes themselves. This makes it possible to drastically reduce the search space when solving Eq. |
Because careful studies in the spirit of are not yet available for multi-way graph partitioning, we resort to a heuristic approach based on what is known for bipartitioning. Specifically, we are motivated by the situation in which time steps are sufficiently small that only boundary nodes can change their community ass... |
. To save computation time and facilitate implementation, we instead decouple the computations in the following manner. During a single time step, for each boundary node, we compute an optimal assignment of that node, assuming that all other nodes keep their assignment from the beginning of the time step. After this (b... |
in our SBM-generated graphs, we find it both more efficient and easier to consider reassigning all nodes in each time step, rather than maintaining and referencing a separate data structure to track the boundary. For other networks and initialization techniques, such as in networks that arise from nearest-neighbor grap... |
, we give pseudocode for this graph MCF procedure. Input [MATH] [MATH] [MATH] Initialize [MATH] so that nodes are assigned to communities uniformly at random. |
Let [MATH] be the entry-wise exponential of [MATH] while not converged do Let [MATH] for each [MATH] , where [MATH] is the Kronecker delta. |
Let [MATH] . // Counts the number of neighbors that each node has in each community Let [MATH] for [MATH] to [MATH] do Let [MATH] be the set of nodes that are currently assigned to community. |
[MATH] for a=1 to [MATH] do Let [MATH] be the indices [MATH] aside from [MATH] and [MATH] Let [MATH] be given by the following formula: |
[EQUATION] end for end for for [MATH] to [MATH] do [MATH] . // [Choose uniformly at random in case of a tie.] end for end while Output [MATH] |
Algorithm 1 Modified graph mean-curvature flow (MCF) for SBM MLE Eq. 4.2 Allen–Cahn (AC) Evolution Another approach for studying MCF is approximation by a Ginzburg–Landau (GL) functional. This approach is popular due to its simple implementation and the existence of unconditionally-stable numerical methods |
In the two-phase case, the GL functional is [EQUATION] where [MATH] is a smooth function and [MATH] is a small parameter. The [MATH] gradient descent of the GL functional is |
[EQUATION] which is the Allen–Cahn (AC) equation. The minimizers of the GL energy are predominantly (piecewise) constant, with [MATH] -width transition layers between the constant regions. One can show that the GL energy [MATH] -converges to the TV energy as [MATH] , assuming that [MATH] |
. Consequently, if [MATH] is a minimizer of the constrained GL energy with parameter [MATH] and the minimizers converge in [MATH] as [MATH] , then the accumulation point is a minimizer of the TV energy. |
In the setting of graphs, the first use of AC schemes for TV minimization was in . One can invoke the combinatorial graph Laplacian [MATH] to obtain a graph GL functional |
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