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In this work we propose a centrality measure for networks, which we refer to as Laplacian centrality, that provides a general framework for the centrality of a vertex based on the idea that the importance (or centrality) of a vertex is related to the ability of the network to respond to the deactivation or removal of that vertex from the network. In particular, the Laplacian centrality of a vertex is defined as the relative drop of Laplacian energy caused by the deactivation of this vertex. The Laplacian energy of network G with n vertices is defined as , where  is the eigenvalue of the Laplacian matrix of G. Other dynamics based measures such as that of Masuda and Kori and PageRank compute the importance of a node by analyzing the way paths pass through a node while our measure captures this information as well as the way these paths are “redistributed” when the node is deleted. The validity and robustness of this new measure are illustrated on two different terrorist social network data sets and 84 networks in James Moody’s Add Health in school friendship nomination data, and is compared with other standard centrality measures.

Terrorist Networks, Network Energy and Node Removal: A New Measure of Centrality Based on Laplacian Energy