How robust are reconstruction thresholds for community detection?

The stochastic block model is one of the oldest and most ubiquitous models for studying clustering and community detection. In an exciting sequence of developments, motivated by deep but non-rigorous ideas from statistical physics, Decelle et al. conjectured a sharp threshold for when community detection is possible in the sparse regime. Mossel, Neeman and Sly and Massoulié proved the conjecture and gave matching algorithms and lower bounds. Here we revisit the stochastic block model from the perspective of semirandom models where we allow an adversary to make 'helpful' changes that strengthen ties within each community and break ties between them. We show a surprising result that these 'helpful' changes can shift the information-theoretic threshold, making the community detection problem strictly harder. We complement this by showing that an algorithm based on semidefinite programming (which was known to get close to the threshold) continues to work in the semirandom model (even for partial recovery). This suggests that algorithms based on semidefinite programming are robust in ways that any algorithm meeting the information-theoretic threshold cannot be. These results point to an interesting new direction: Can we find robust, semirandom analogues to some of the classical, average-case thresholds in statistics? We also explore this question in the broadcast tree model, and we show that the viewpoint of semirandom models can help explain why some algorithms are preferred to others in practice, in spite of the gaps in their statistical performance on random models.National Science Foundation (U.S.). Faculty Early Career Development Program (Award CCF-1453261)Google Faculty Research AwardNihon Denki Kabushiki Kaish

Inference on graphs via semidefinite programming

Inference problems on graphs arise naturally when trying to make sense of network data. Oftentimes, these problems are formulated as intractable optimization programs. This renders the need for fast heuristics to find adequate solutions and for the study of their performance. For a certain class of problems, Javanmard et al. (1) successfully use tools from statistical physics to analyze the performance of semidefinite programming relaxations, an important heuristic for intractable problems.National Science Foundation (U.S.) (Grant DMS- 1317308

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