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

Inference in the Stochastic Block Model with a Markovian assignment of the communities

We tackle the community detection problem in the Stochastic Block Model (SBM) when the communities of the nodes of the graph are assigned with a Markovian dynamic. To recover the partition of the nodes, we adapt the relaxed K-means SDP program presented in [11]. We identify the relevant signal-to-noise ratio (SNR) in our framework and we prove that the misclassification error decays exponentially fast with respect to this SNR. We provide infinity norm consistent estimation of the parameters of our model and we discuss our results through the prism of classical degree regimes of the SBMs' literature. MSC 2010 subject classifications: Primary 68Q32; secondary 68R10, 90C35