Probabilistic Community Detection With Unknown Number of Communities

ABSTRACT

A fundamental problem in network analysis is clustering the nodes into groups which share a similar connectivity pattern. Existing algorithms for community detection assume the knowledge of the number of clusters or estimate it a priori using various selection criteria and subsequently estimate the community structure. Ignoring the uncertainty in the first stage may lead to erroneous clustering, particularly when the community structure is vague. We instead propose a coherent probabilistic framework for simultaneous estimation of the number of communities and the community structure, adapting recently developed Bayesian nonparametric techniques to network models. An efficient Markov chain Monte Carlo (MCMC) algorithm is proposed which obviates the need to perform reversible jump MCMC on the number of clusters. The methodology is shown to outperform recently developed community detection algorithms in a variety of synthetic data examples and in benchmark real-datasets. Using an appropriate metric on the space of all configurations, we develop nonasymptotic Bayes risk bounds even when the number of clusters is unknown. Enroute, we develop concentration properties of nonlinear functions of Bernoulli random variables, which may be of independent interest in analysis of related models. Supplementary materials for this article are available online.

Keywords:

• Bayesian nonparametrics
• Clustering consistency
• MCMC
• Mixture models
• Model selection
• Network analysis

Supplementary Materials

Additional simulations exploring sensitivity, convergence diagnostics, and robustness, and proofs of all technical results, are provided in the supplemental materials. The supplemental material additionally contains a second real data example.

Additional information

Funding

Dr. Bhattacharya acknowledges NSF CAREER (DMS 1653404), NSF DMS 1613156, and National Cancer Institute's R01 CA158113, and Dr. Pati acknowledges NSF DMS 1613156 for supporting this research.