A Proof of the Block Model Threshold Conjecture
Combinatorica(2017)
摘要
We study a random graph model called the “stochastic block model” in statistics and the “planted partition model” in theoretical computer science. In its simplest form, this is a random graph with two equal-sized classes of vertices, with a within-class edge probability of q and a between-class edge probability of q ′. A striking conjecture of Decelle, Krzkala, Moore and Zdeborová [9], based on deep, non-rigorous ideas from statistical physics, gave a precise prediction for the algorithmic threshold of clustering in the sparse planted partition model. In particular, if q = a / n and q ′= b / n , s =( a − b )/2 and d =( a + b )/2, then Decelle et al. conjectured that it is possible to efficiently cluster in a way correlated with the true partition if s 2 > d and impossible if s 2 < d . By comparison, until recently the best-known rigorous result showed that clustering is possible if s 2 > Cd ln d for sufficiently large C . In a previous work, we proved that indeed it is information theoretically impossible to cluster if s 2 ≤ d and moreover that it is information theoretically impossible to even estimate the model parameters from the graph when s 2 < d . Here we prove the rest of the conjecture by providing an efficient algorithm for clustering in a way that is correlated with the true partition when s 2 > d . A different independent proof of the same result was recently obtained by Massoulié [20].
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05C80
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