Recovering Asymmetric Communities in the Stochastic Block Model

Abstract

We consider the sparse stochastic block model in the case where the degrees are uninformative. The case where the two communities have approximately the same size has been extensively studied and we concentrate here on the community detection problem in the case of unbalanced communities. In this setting, spectral algorithms based on the non-backtracking matrix are known to solve the community detection problem (i.e. do strictly better than a random guess) when the signal is sufficiently large namely above the so-called Kesten Stigum threshold. In this regime and when the average degree tends to infinity, we show that if the community of a vanishing fraction of the vertices is revealed, then a local algorithm (belief propagation) is optimal down to Kesten Stigum threshold and we quantify explicitly its performance. Below the Kesten Stigum threshold, we show that, in the large degree limit, there is a second threshold called the spinodal curve below which, the community detection problem is not solvable. The spinodal curve is equal to the Kesten Stigum threshold when the fraction of vertices in the smallest community is above p * = 1 2 − 1 2 √ 3 , so that the Kesten Stigum threshold is the threshold for solvability of the community detection in this case. However when the smallest community is smaller than p * , the spinodal curve only provides a lower bound on the threshold for solvability. In the regime below the Kesten Stigum bound and above the spinodal curve, we also characterize the performance of best local algorithms as a function of the fraction of revealed vertices. Our proof relies on a careful analysis of the associated reconstruction problem on trees which might be of independent interest. In particular, we show that the spinodal curve corresponds to the reconstruction threshold on the tree.

Achieving the Bayes Error Rate in Synchronization and Block Models by SDP, Robustly

“…We then apply Lemma 16 to W up to bound W up op , and apply Lemma 14 to W up and (W up ) to bound W up − W up 1 . 16 Note that the assumption p ≥ C n of Lemma 14 is satisfied by the assumption of this proposition that nI * ≥ C I * for some large enough C I * > 0 (since Fact 2(b) implies I * (p−q) 2 p ≤ p). We conclude that with probability at least 1 − 1 n 2 − 2 √ n ,…”

The Geometry of Community Detection via the MMSE Matrix

“…Remark 2. Similar to previous work [3][4][5][6][7][8], our proofs of Theorems 1 and 2 use a channel universality argument to relate the community detection problem to a low-rank estimation problem. Assumption 2 is needed for the proof of Theorem 1, which leverages [5,Theorem 12].…”

Efficient Inference in Stochastic Block Models With Vertex Labels

“…Typically, it is assumed that the fraction of vertices that reveal their true membership tends to zero when the graph becomes large [21], [3]. In this setting, a variant of the belief propagation algorithm including the vertex labels seems to perform optimally in a symmetric stochastic block model [21], but may not perform optimally if the communities are not of equal size [3]. Another special case of the label distribution is when the observed labels are a noisy version of the community memberships, where a fraction of β vertices receives the label corresponding to their community, and a fraction of 1 − β vertices receives the label corresponding to the other community.…”

“…In particular, this implies that the running time of the algorithm is linear, allowing it to be used on large networks. The algorithm is a variant of the belief propagation algorithm, and a generalization of the algorithms provided in [16], [3] to include arbitrary label distributions and an asymmetric stochastic block model. • In a regime where the average vertex degrees are large, we obtain an expression for the probability that the community of a vertex is identified correctly.…”