Community Detection via Discriminant functions for Random Walks in the degree-corrected Stochastic Block Model

Recent work theoretically connected the problem of seed set expansions with personalized page rank by showing that the optimal weights of random walk landing probabilities when constructing a classifier for community detection of nodes in the stochastic block model followed the weights in personalized page rank [3]. One drawback of the stochastic block model is that it can be unrealistic as a model for empirical graphs. The degree-corrected stochastic block model is an extension of the stochastic block model that allows edge probabilities to differ both by which communities the nodes are in as well as by latent “friendliness” parameters that each node has. The goal of this project was to extend the analysis of [3] to the degree-corrected stochastic block model and determine (i) whether the optimal weights of a classifier using random walk probabilities follow some known node-ranking model (possibly even personalized page rank as with the standard stochastic block model), (ii) a closed form for the weights in terms of the model parameters, and (iii) to explore performance the practicality of such a classifier with experiments. With this project I was able to extend the analysis of [3] to the degree-corrected stochastic block model, deriving the optimal weights of a classifier in the space of normalized random walk probabilities in terms of the parameters of the model. My central result is a proof akin to the one shown in [3] and highlights that with high probability, random walk probabilities in the degree corrected stochastic block model concentrate around a distribution characterized by a smaller linear system on the blocks with a conditional distribution within blocks proportional to the “friendliness” of each user within that block. This is of independent theoretical interest in relating graph models and random walks (which in turn link to ranking models such as personalized page rank), and in conjunction with some simple parameter estimation was able to produce accurate classification on synthetic data as well as two empirical networks.