Quickest Detection of the Change of Community via Stochastic Block Models

Community detection is a fundamental problem in network analysis and has important applications in sensor networks and social networks. In many cases, the community structure of the network may change at some unknown time and thus it is desirable to come up with efficient monitoring procedures that can detect the change as quickly as possible. In this work, we use the Erdős-Rényi model and the bisection stochastic block model (SBM) to model the pre-change and post-change distributions of the network, respectively. That is, initially, we assume there is no community in the network. However, at some unknown time, a change occurs, and two communities are formed in the network. We then propose an efficient monitoring procedure by using the number of k-cycles in the graph. The asymptotic detection properties of our proposed procedure are derived when all parameters are known. A generalized likelihood ratio (GLR) type detection procedure and an adaptive CUSUM type detection procedure are constructed to address the problem when parameters are unknown.