Asymptotic Mutual Information for the Two-Groups Stochastic Block Model

TL;DR

This paper provides an explicit information-theoretic characterization of the two-group stochastic block model, revealing a phase transition in community detection and establishing the limits of label estimation based on mutual information.

Contribution

It introduces a single-letter formula for the mutual information in the symmetric two-group model, linking it to estimation limits and phase transitions.

Findings

Mutual information matches independent edges below the critical point.

Above the threshold, better-than-random estimation is possible.

Identifies a phase transition in community detection performance.

Abstract

We develop an information-theoretic view of the stochastic block model, a popular statistical model for the large-scale structure of complex networks. A graph $G$ from such a model is generated by first assigning vertex labels at random from a finite alphabet, and then connecting vertices with edge probabilities depending on the labels of the endpoints. In the case of the symmetric two-group model, we establish an explicit `single-letter' characterization of the per-vertex mutual information between the vertex labels and the graph. The explicit expression of the mutual information is intimately related to estimation-theoretic quantities, and --in particular-- reveals a phase transition at the critical point for community detection. Below the critical point the per-vertex mutual information is asymptotically the same as if edges were independent. Correspondingly, no algorithm can estimate the partition better than random guessing. Conversely, above the threshold, the per-vertex mutual information is strictly smaller than the independent-edges upper bound. In this regime there exists a procedure that estimates the vertex labels better than random guessing.