Detection in the stochastic block model with multiple clusters: proof of the achievability conjectures, acyclic BP, and the information-computation gap

TL;DR

This paper proves the conjecture that community detection in the stochastic block model is achievable at the KS threshold for any number of communities, using an acyclic belief propagation algorithm and spectral methods, and explores the information-computation gap.

Contribution

It extends the detection threshold proof to non-symmetrical SBMs with a generalized detection notion, and introduces an efficient ABP algorithm that achieves the KS threshold for any number of communities.

Findings

ABP detects communities down to the KS threshold in O(n log n) time.

The paper establishes the optimality of ABP in the presence of cycles.

It demonstrates a significant information-computation gap in the SBM, especially when a=0.

Abstract

In a paper that initiated the modern study of the stochastic block model, Decelle et al., backed by Mossel et al., made the following conjecture: Denote by $k$ the number of balanced communities, $a/n$ the probability of connecting inside communities and $b/n$ across, and set $\mathrm{SNR}=(a-b)^2/(k(a+(k-1)b)$; for any $k \geq 2$, it is possible to detect communities efficiently whenever $\mathrm{SNR}>1$ (the KS threshold), whereas for $k\geq 4$, it is possible to detect communities information-theoretically for some $\mathrm{SNR}<1$. Massouli\'e, Mossel et al.\ and Bordenave et al.\ succeeded in proving that the KS threshold is efficiently achievable for $k=2$, while Mossel et al.\ proved that it cannot be crossed information-theoretically for $k=2$. The above conjecture remained open for $k \geq 3$. This paper proves this conjecture, further extending the efficient detection to non-symmetrical SBMs with a generalized notion of detection and KS threshold. For the efficient part, a linearized acyclic belief propagation (ABP) algorithm is developed and proved to detect communities for any $k$ down to the KS threshold in time $O(n \log n)$. Achieving this requires showing optimality of ABP in the presence of cycles, a challenge for message passing algorithms. The paper further connects ABP to a power iteration method with a nonbacktracking operator of generalized order, formalizing the interplay between message passing and spectral methods. For the information-theoretic (IT) part, a non-efficient algorithm sampling a typical clustering is shown to break down the KS threshold at $k=4$. The emerging gap is shown to be large in some cases; if $a=0$, the KS threshold reads $b \gtrsim k^2$ whereas the IT bound reads $b \gtrsim k \ln(k)$, making the SBM a good study-case for information-computation gaps.