Community Detection in Hypergraphs, Spiked Tensor Models, and Sum-of-Squares

We study the problem of community detection in hypergraphs under a stochastic block model. Similarly to how the stochastic block model in graphs suggests studying spiked random matrices, our model motivates investigating statistical and computational limits of exact recovery in a certain spiked tensor model. In contrast with the matrix case, the spiked model naturally arising from community detection in hypergraphs is different from the one arising in the so-called tensor Principal Component Analysis model. We investigate the effectiveness of algorithms in the Sum-of-Squares hierarchy on these models. Interestingly, our results suggest that these two apparently similar models exhibit significantly different computational to statistical gaps.Comment: In proceedings of 2017 International Conference on Sampling Theory and Applications (SampTA

Exact Recovery for a Family of Community-Detection Generative Models

Generative models for networks with communities have been studied extensively for being a fertile ground to establish information-theoretic and computational thresholds. In this paper we propose a new toy model for planted generative models called planted Random Energy Model (REM), inspired by Derrida's REM. For this model we provide the asymptotic behaviour of the probability of error for the maximum likelihood estimator and hence the exact recovery threshold. As an application, we further consider the 2 non-equally sized community Weighted Stochastic Block Model (2-WSBM) on $h$-uniform hypergraphs, that is equivalent to the P-REM on both sides of the spectrum, for high and low edge cardinality $h$. We provide upper and lower bounds for the exact recoverability for any $h$, mapping these problems to the aforementioned P-REM. To the best of our knowledge these are the first consistency results for the 2-WSBM on graphs and on hypergraphs with non-equally sized community

SIS epidemic propagation on hypergraphs

Mathematical modeling of epidemic propagation on networks is extended to hypergraphs in order to account for both the community structure and the nonlinear dependence of the infection pressure on the number of infected neighbours. The exact master equations of the propagation process are derived for an arbitrary hypergraph given by its incidence matrix. Based on these, moment closure approximation and mean-field models are introduced and compared to individual-based stochastic simulations. The simulation algorithm, developed for networks, is extended to hypergraphs. The effects of hypergraph structure and the model parameters are investigated via individual-based simulation results

Consistency of Spectral Hypergraph Partitioning under Planted Partition Model

Hypergraph partitioning lies at the heart of a number of problems in machine learning and network sciences. Many algorithms for hypergraph partitioning have been proposed that extend standard approaches for graph partitioning to the case of hypergraphs. However, theoretical aspects of such methods have seldom received attention in the literature as compared to the extensive studies on the guarantees of graph partitioning. For instance, consistency results of spectral graph partitioning under the stochastic block model are well known. In this paper, we present a planted partition model for sparse random non-uniform hypergraphs that generalizes the stochastic block model. We derive an error bound for a spectral hypergraph partitioning algorithm under this model using matrix concentration inequalities. To the best of our knowledge, this is the first consistency result related to partitioning non-uniform hypergraphs.Comment: 35 pages, 2 figures, 1 tabl