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

In a paper that initiated the modern study of the stochastic block model, Decelle, Krzakala, Moore and Zdeborov\'a made a fascinating conjecture: Denote by $k$ the number of balanced communities, $a/n$ the probability of connecting inside clusters and $b/n$ across clusters, and set $\mathrm{SNR}=(a-b)^2/(k(a+(k-1)b))$; for any $k \geq 2$, it is possible to detect efficiently communities whenever $\mathrm{SNR}>1$ (the KS bound), whereas for $k\geq 5$, 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 bound is efficiently achievable for $k=2$, while Mossel et al.\ proved that the KS bound cannot be crossed information-theoretically for $k=2$. The above conjecture remained open beyond two communities. This paper proves this conjecture. Specifically, an acyclic belief propagation (ABP) algorithm is developed and proved to detect communities for any $k$ whenever $\mathrm{SNR}>1$, i.e., down the KS bound, with a complexity of $O(n \log n)$. The parameters are also learned efficiently down the KS bound. Thus ABP improves upon the state-of-the-art both in terms of complexity and universality in achieving the KS bound. This also gives a rigorous instance where message passing succeeds with optimal guarantees while handling cycles and a random initialization. Further, a non-efficient algorithm sampling a typical clustering is shown to break down the KS bound at $k=5$. The gap between the KS bound and the information-theoretic bound is shown to be large in some cases; if $a=0$, the KS bound reads $b \gtrsim k^2$ whereas the information-theoretic bound reads $b \gtrsim k \ln(k)$. The results extend to general SBMs.