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 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 SNR=(a-b)2/(k(a+(k-1)b); for any k ≥ 2, it is possible to detect communities efficiently whenever SNR>1 (the KS threshold), whereas for k≥ 4, it is possible to detect communities information-theoretically for some 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 ≥ 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 k2 whereas the IT bound reads b gtrsim k ln(k), making the SBM a good study-case for information-computation gaps.