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Many complex systems can be represented as networks, and how a network breaksup into subnetworks or communities is of wide interest. However, thedevelopment of a method to detect nodes important to communities that is bothfast and accurate is a very challenging and open problem. In this manuscript,we introduce a new approach to characterize the node importance to communities.First, a centrality metric is proposed to measure the importance of networknodes to community structure using the spectrum of the adjacency matrix. Wedefine the node importance to communities as the relative change in theeigenvalues of the network adjacency matrix upon their removal. Second, we alsopropose an index to distinguish two kinds of important nodes in communities,i.e., "community core" and "bridge". Our indices are only relied on thespectrum of the graph matrix. They are applied in many artificial networks aswell as many real-world networks. This new methodology gives us a basicapproach to solve this challenging problem and provides a realistic result.

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