Exploring the heterogeneity for node importance byvon Neumann entropy

Abstract When analyzing and describing the statistical and topological characteristics of complex networks, the heterogeneity can provide profound and systematical recognition to illustrate the difference of individuals, and many node significance indices have been investigated to describe heterogeneity in different perspectives. In this paper a new node heterogeneity index based on the von Neumann entropy is proposed, which allows us to investigate the differences of nodes features in the view of spectrum eigenvalues distribution, and examples in reality networks present its great performance in selecting crucial individuals. Then to lower down the computational complexity, an approximation calculation to this index is given which only depends on its first and second neighbors. Furthermore, in reducing the network heterogeneity index by Estrada, this entropy heterogeneity presents excellent efficiency in Erdos-Renyi and scale-free networks compared to other node significance measurements; in reducing the average clustering coefficient, this node entropy index could break down the cluster structures efficiently in random geometric graphs, even faster than clustering coefficient itself. This new methodology reveals the node heterogeneity and significance in the perspective of spectrum, which provides a new insight into networks research and performs great potentials to discover essential structural features in networks.