Local Algorithms for Block Models with Side Information

Abstract

There has been a recent interest in understanding the power of local algorithms for optimization and inference problems on sparse graphs. Gamarnik and Sudan (2014) showed that local algorithms are weaker than global algorithms for finding large independent sets in sparse random regular graphs thus refuting a conjecture by Hatami, Lovász, and Szegedy (2012). Montanari (2015) showed that local algorithms are suboptimal for finding a community with high connectivity in the sparse Erdős-Rényi random graphs. For the symmetric planted partition problem (also named community detection for the block models) on sparse graphs, a simple observation is that local algorithms cannot have non-trivial performance.In this work we consider the effect of side information on local algorithms for community detection under the binary symmetric stochastic block model. In the block model with side information each of the n vertices is labeled + or − independently and uniformly at random; each pair of vertices is connected independently with probability a/n if both of them have the same label or b/n otherwise. The goal is to estimate the underlying vertex labeling given 1) the graph structure and 2) side information in the form of a vertex labeling positively correlated with the true one. Assuming that the ratio between in and out degree a/b is Θ(1) and the average degree (a + b)/2 = n o(1) , we show that a local algorithm, namely, belief propagation run on the local neighborhoods, maximizes the expected fraction of vertices labeled correctly in the following three regimes:• |a − b| < 2 and all 0 < α < 1/2• (a − b) 2 > C(a + b) for some constant C and all 0 < α < 1/2• For all a, b if the probability that each given vertex label is incorrect is at most α * for some constant α * ∈ (0, 1/2).

The Power of Side-Information in Subgraph Detection

“…In this setting we show that the computational barrier disappears when sideInria information is available. We emphasize that our results cannot be obtained as a special case of the results in [12,[26][27][28]]. …”

Efficient Inference in Stochastic Block Models With Vertex Labels

“…This difference causes the increase in the success probability. Figures 3a and 3b show the success probability given by (16) as a function λ for unbalanced communities. Here we also see that there is a small range of λ < 1 where the success probability increases rapidly in λ.…”

“…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.…”

Message‐Passing Algorithms for Synchronization Problems over Compact Groups

“…One might also formulate a version of our model that allows node potentials, as seen for instance in image segmentation [31] and some community detection problems [52,63]: For true probabilistic models, our approach attempts Bayes-optimal inference (minimizing mean-squared error), while the NUG approach attempts maximum-likelihood estimation that may have higher expected error.…”

“…1 in (3.2). One might also formulate a version of our model that allows node potentials, as seen for instance in image segmentation [31] and some community detection problems [52,63]:…”