Deep Community Detection
A deep community in a graph is a connected component that can only be seen after removal of nodes or edges from the rest of the graph. This paper formulates the problem of detecting deep communities as multi-stage node removal that maximizes a new centrality measure, called the local Fiedler vector centrality (LFVC), at each stage. The LFVC is associated with the sensitivity of algebraic connectivity to node or edge removals. We prove that a greedy node/edge removal strategy, based on successive maximization of LFVC, has bounded performance loss relative to the optimal, but intractable, combinatorial batch removal strategy. Under a stochastic block model framework, we show that the greedy LFVC strategy can extract deep communities with probability one as the number of observations becomes large. We apply the greedy LFVC strategy to real-world social network datasets. Compared with conventional community detection methods we demonstrate improved ability to identify important communities and key members in the network.