A semidefinite program for unbalanced multisection in the stochastic block model

We propose a semidefinite programming (SDP) algorithm for community detection in the stochastic block model, a popular model for networks with latent community structure. We prove that our algorithm achieves exact recovery of the latent communities, up to the information-theoretic limits determined by Abbe and Sandon (2015). Our result extends prior SDP approaches by allowing for many communities of different sizes. By virtue of a semidefinite approach, our algorithms succeed against a semirandom variant of the stochastic block model, guaranteeing a form of robustness and generalization. We further explore how semirandom models can lend insight into both the strengths and limitations of SDPs in this setting.Comment: 29 page

A note on Probably Certifiably Correct algorithms

Many optimization problems of interest are known to be intractable, and while there are often heuristics that are known to work on typical instances, it is usually not easy to determine a posteriori whether the optimal solution was found. In this short note, we discuss algorithms that not only solve the problem on typical instances, but also provide a posteriori certificates of optimality, probably certifiably correct (PCC) algorithms. As an illustrative example, we present a fast PCC algorithm for minimum bisection under the stochastic block model and briefly discuss other examples

Exact Clustering of Weighted Graphs via Semidefinite Programming

As a model problem for clustering, we consider the densest k-disjoint-clique problem of partitioning a weighted complete graph into k disjoint subgraphs such that the sum of the densities of these subgraphs is maximized. We establish that such subgraphs can be recovered from the solution of a particular semidefinite relaxation with high probability if the input graph is sampled from a distribution of clusterable graphs. Specifically, the semidefinite relaxation is exact if the graph consists of k large disjoint subgraphs, corresponding to clusters, with weight concentrated within these subgraphs, plus a moderate number of outliers. Further, we establish that if noise is weakly obscuring these clusters, i.e, the between-cluster edges are assigned very small weights, then we can recover significantly smaller clusters. For example, we show that in approximately sparse graphs, where the between-cluster weights tend to zero as the size n of the graph tends to infinity, we can recover clusters of size polylogarithmic in n. Empirical evidence from numerical simulations is also provided to support these theoretical phase transitions to perfect recovery of the cluster structure