Community detection in general stochastic block models: fundamental limits and efficient recovery algorithms

TL;DR

This paper characterizes the fundamental limits and proposes efficient algorithms for community detection in general stochastic block models with multiple, potentially overlapping, communities, extending previous results beyond symmetric cases.

Contribution

It introduces a new divergence-based threshold for community recovery, and develops algorithms that achieve these thresholds efficiently in various regimes.

Findings

Explicit recovery threshold characterized by divergence D+

Algorithm achieves optimal threshold with quasi-linear complexity

Community detection in constant degree regime with high accuracy

Abstract

New phase transition phenomena have recently been discovered for the stochastic block model, for the special case of two non-overlapping symmetric communities. This gives raise in particular to new algorithmic challenges driven by the thresholds. This paper investigates whether a general phenomenon takes place for multiple communities, without imposing symmetry. In the general stochastic block model $\text{SBM}(n,p,Q)$, $n$ vertices are split into $k$ communities of relative size $\{p_i\}_{i \in [k]}$, and vertices in community $i$ and $j$ connect independently with probability $\{Q_{i,j}\}_{i,j \in [k]}$. This paper investigates the partial and exact recovery of communities in the general SBM (in the constant and logarithmic degree regimes), and uses the generality of the results to tackle overlapping communities. The contributions of the paper are: (i) an explicit characterization of the recovery threshold in the general SBM in terms of a new divergence function $D_+$, which generalizes the Hellinger and Chernoff divergences, and which provides an operational meaning to a divergence function analog to the KL-divergence in the channel coding theorem, (ii) the development of an algorithm that recovers the communities all the way down to the optimal threshold and runs in quasi-linear time, showing that exact recovery has no information-theoretic to computational gap for multiple communities, in contrast to the conjectures made for detection with more than 4 communities; note that the algorithm is optimal both in terms of achieving the threshold and in having quasi-linear complexity, (iii) the development of an efficient algorithm that detects communities in the constant degree regime with an explicit accuracy bound that can be made arbitrarily close to 1 when a prescribed signal-to-noise ratio (defined in term of the spectrum of $\diag(p)Q$) tends to infinity.