Large Deviations of Semi-supervised Learning in the Stochastic Block Model

TL;DR

This paper uses statistical physics to analyze the rare events and fluctuations in semi-supervised community detection within the stochastic block model, revealing how label choices influence inference accuracy.

Contribution

It introduces a large deviation framework for semi-supervised learning in the stochastic block model, capturing fluctuations and correlations in label selection.

Findings

Characterizes fluctuations around typical behavior in community detection

Identifies a non-monotonic relationship between accuracy and free energy

Provides insights into label informativeness and rarity

Abstract

In community detection on graphs, the semi-supervised learning problem entails inferring the ground-truth membership of each node in a graph, given the connectivity structure and a limited number of revealed node labels. Different subsets of revealed labels can in principle lead to higher or lower information gains and induce different reconstruction accuracies. In the framework of the dense stochastic block model, we employ statistical physics methods to derive a large deviation analysis for this problem, in the high-dimensional limit. This analysis allows the characterization of the fluctuations around the typical behaviour, capturing the effect of correlated label choices and yielding an estimate of their informativeness and their rareness among subsets of the same size. We find theoretical evidence of a non-monotonic relationship between reconstruction accuracy and the free energy associated to the posterior measure of the inference problem. We further discuss possible implications for active learning applications in community detection.