Exploring the Node Importance Based on von Neumann Entropy

When analyzing the statistical and topological characteristics of complex networks, an effective and convenient way is to compute the centralities for recognizing influential and significant nodes or structures, yet most of them are restricted to local environment or some specific configurations. In this paper we propose a new centrality for nodes based on the von Neumann entropy, which allows us to investigate the importance of nodes in the view of spectrum eigenvalues distribution. By presenting the performances of this centrality with network examples in reality, it is shown that the von Neumann entropy node centrality is an excellent index for selecting crucial nodes as well as classical ones. Then to lower down the computational complexity, an approximation calculation to this centrality is given which only depends on its first and second neighbors. Furthermore, in the optimal spreader problem and reducing average clustering coefficients, this entropy centrality presents excellent efficiency and unveil topological structure features of networks accurately. The entropy centrality could reduce the scales of giant connected components fastly in Erdos-Renyi and scale-free networks, and break down the cluster structures efficiently in random geometric graphs. This new methodology reveals the node importance in the perspective of spectrum, which provides a new insight into networks research and performs great potentials to discover essential structural features in networks.