Abstract
Most existing community discovery methods focus on partitioning all nodes of the network into communities. However, many real networks contain background nodes that do not belong to any community. In such a situation, typical methods tend to artificially split the background nodes and group them together with communities with relatively stronger connection, hence lead to distorted results. To avoid this, some community extraction methods have been developed to achieve community discovery with background nodes, which are based on searching algorithms, hence have difficulties in handling large-scale networks due to high computational complexity. To this end, in this paper we propose some algorithms with polynomial complexity to achieve community extraction of large-scale networks. We rigorously show that the proposed algorithms have attractive theoretical properties. In particular, the estimators of the community labels using the proposed algorithms reaches the asymptotic minimax risk under the community extraction model, a specific stochastic block model. Then, we illustrate the advantages and feasibility of the proposed algorithms via extensive simulated networks and a political blog network.