Detecting Important Nodes to Community Structure Using the Spectrum of the Graph

Community structure analysis is a powerful tool for complex networks, which can simplify their functional analysis considerably. Many approaches have recently been proposed to the communities in complex networks, but a method to characterize the node importance to communities is still lacking. In this paper a centrality metric is proposed to measure the importance of network nodes to community structure using the spectrum of the adjacency matrix. We define the node importance to communities as the relative change in the eigenvalues of the network adjacency matrix upon their removal. Besides that, we also propose an index to distinguish two kind of important nodes in communities, ie. "community core" and "bridge". The method has been tested in many artificial and real-world networks. The results show that our method preforms well in many cases, including artificial networks and many real-world networks.