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

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.

References in corpus (9)

• Asymptotic analysis of the stochastic block model for modular networks and its algorithmic applications
• Social Structure of Facebook Networks
• Statistical physics of inference: Thresholds and algorithms
• Probabilistic Reconstruction in Compressed Sensing: Algorithms, Phase Diagrams, and Threshold Achieving Matrices
• Model Selection for Degree-corrected Block Models
• Constrained Low-rank Matrix Estimation: Phase Transitions, Approximate Message Passing and Applications
• Phase transitions in semisupervised clustering of sparse networks
• Phase Transitions in Community Detection: A Solvable Toy Model
• Statistical mechanical analysis of sparse linear regression as a variable selection problem