Inference via Message Passing on Partially Labeled Stochastic Block   Models
  We study the community detection and recovery problem in partially-labeled stochastic block models (SBM). We develop a fast linearized message-passing algorithm to reconstruct labels for SBM (with $n$ nodes, $k$ blocks, $p,q$ intra and inter block connectivity) when $\delta$ proportion of node labels are revealed. The signal-to-noise ratio ${\sf SNR}(n,k,p,q,\delta)$ is shown to characterize the fundamental limitations of inference via local algorithms. On the one hand, when ${\sf SNR}>1$, the linearized message-passing algorithm provides the statistical inference guarantee with mis-classification rate at most $\exp(-({\sf SNR}-1)/2)$, thus interpolating smoothly between strong and weak consistency. This exponential dependence improves upon the known error rate $({\sf SNR}-1)^{-1}$ in the literature on weak recovery. On the other hand, when ${\sf SNR}<1$ (for $k=2$) and ${\sf SNR}<1/4$ (for general growing $k$), we prove that local algorithms suffer an error rate at least $\frac{1}{2} - \sqrt{\delta \cdot {\sf SNR}}$, which is only slightly better than random guess for small $\delta$. 
T. Tony Cai, Tengyuan Liang and Alexander Rakhlin
math.ST stat.ML stat.TH