Estimation of dense stochastic block models visited by random walks

TL;DR

This paper develops methods to infer the structure of dense stochastic block models from subgraphs sampled by random walks, addressing biases and unobserved types using likelihood-based and de-biasing strategies.

Contribution

It introduces a likelihood framework for subgraph sampling by random walks and proposes a de-biasing approach to improve estimation in dense stochastic block models.

Findings

Likelihood of sampled subgraph accounts for sampling biases.

SAEM algorithm effectively estimates unobserved types.

De-biasing improves accuracy of community detection.

Abstract

We are interested in recovering information on a stochastic block model from the subgraph discovered by an exploring random walk. Stochastic block models correspond to populations structured into a finite number of types, where two individuals are connected by an edge independently from the other pairs and with a probability depending on their types. We consider here the dense case where the random network can be approximated by a graphon. This problem is motivated from the study of chain-referral surveys where each interviewee provides information on her/his contacts in the social network. First, we write the likelihood of the subgraph discovered by the random walk: biases are appearing since hubs and majority types are more likely to be sampled. Even for the case where the types are observed, the maximum likelihood estimator is not explicit any more. When the types of the vertices is unobserved, we use an SAEM algorithm to maximize the likelihood. Second, we propose a different estimation strategy using new results by Athreya and Roellin. It consists in de-biasing the maximum likelihood estimator proposed in Daudin et al. and that ignores the biases.