Community Detection and Stochastic Block Models: Recent Developments; Emmanuel Abbe

The stochastic block model (SBM) is a random graph model with
planted clusters. It is widely employed as a canonical model to
study clustering and community detection, and provides generally
a fertile ground to study the statistical and computational
tradeoffs that arise in network and data sciences. This note
surveys the recent developments that establish the fundamental
limits for community detection in the SBM, both with respect to
information-theoretic and computational thresholds, and for
various recovery requirements such as exact, partial and weak
recovery (a.k.a., detection). The main results discussed are the
phase transitions for exact recovery at the Chernoff-Hellinger
threshold, the phase transition for weak recovery at the Kesten-
Stigum threshold, the optimal distortion-SNR tradeoff for
partial recovery, the learning of the SBM parameters and the gap
between information-theoretic and computational thresholds. The
note also covers some of the algorithms developed in the quest
of achieving the limits, in particular two-round algorithms via
graph-splitting, semi-definite programming, linearized belief
propagation, classical and nonbacktracking spectral methods. A
few open problems are also discussed.

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