Bayesian model selection of stochastic block models

A central problem in analyzing networks is partitioning them into modules or communities. One of the best tools for this is the stochastic block model, which clusters vertices into blocks with statistically homogeneous pattern of links. Despite its flexibility and popularity, there has been a lack of principled statistical model selection criteria for the stochastic block model. Here we propose a Bayesian framework for choosing the number of blocks as well as comparing it to the more elaborate degree-corrected block models, ultimately leading to a universal model selection framework capable of comparing multiple modeling combinations. We will also investigate its theoretic connection to the minimum description length principle.

I. Introduction

An important task in network analysis is community detection, or finding groups of similar vertices which can then be analyzed separately [1]. Community structures offer clues to the processes which generated the graph, on scales ranging from face-to-face social interaction [2] through social-media communications [3] to the organization of food webs [4]. While previous work often defines a “community” as a group of vertices with high density of connections, we are interested in a more general definition of functional community—a group of vertices that connect to the rest of the network in similar ways. A set of similar predators form a functional group in a food web, not because they eat each other, but because they feed on similar prey. In English, nouns often follow adjectives, but seldom follow other nouns.