Stochastic Block Models are a Discrete Surface Tension

Abstract

Networks, which represent agents and interactions between them, arise in myriad applications throughout the sciences, engineering, and even the humanities. To understand large-scale structure in a network, a common task is to cluster a network's nodes into sets called "communities", such that there are dense connections within communities but sparse connections between them. A popular and statistically principled method to perform such clustering is to use a family of generative models known as stochastic block models (SBMs). In this paper, we show that maximum likelihood estimation in an SBM is a network analog of a well-known continuum surface-tension problem that arises from an application in metallurgy.To illustrate the utility of this relationship, we implement network analogs of three surface-tension algorithms, with which we successfully recover planted community structure in synthetic networks and which yield fascinating insights on empirical networks that we construct from hyperspectral videos.(MBO) scheme, geometric partial differential equations AMS subject classifications. 65K10, 49M20, 35Q56, 62H30, 91C20, 91D30, 94C151. Introduction. The study of networks, in which nodes represent entities and edges encode interactions between entities [66], can provide useful insights into a wide variety of complex systems in myriad fields, such as granular materials [73], disease spreading [74], criminology [37], and more. In the study of such applications, the analysis of large data sets -from diverse sources and applications -continues to grow ever more important.The simplest type of network is a graph, and empirical networks often appear to exhibit a complicated mixture of regular and seemingly random features [66]. Additionally, it is increasingly important to study networks with more complicated features, such as time-dependence [39], multiplexity [50], annotations [67], and connections that go beyond a pairwise paradigm [72]. One also has to worry about "features" such as missing information and false positives [48]. Nevertheless, it is convenient in the present paper to restrict our attention to undirected, unweighted graphs for simplicity.To try to understand the large-scale structure of a network, it can be very insightful to coarse-grain it in various ways [27,79,81,83,84]. The most popular type of clustering is the detection of assortative "communities," in which dense sets of nodes are connected sparsely to other dense sets of nodes [27,81]. A statistically principled approach is to treat community detection as a statistical inference problem using a model such * Submitted to the editors DATE.

Generative hypergraph clustering: From blockmodels to modularity

“…Another direction concerns the properties of the dyadic modularity objective that extend to the hypergraph modularity objectives discussed here. In addition to its role as a comparison against null models (75) and as a term in the DCSBM likelihood (48), the dyadic modularity also expresses the stability of diffusion processes on graphs (76) and the energy of discrete surface tensions defined on graphs (77). Extensions of these properties, or explanations of why they fail to generalize, would be helpful for both theorists and practitioners.…”