Research on Centralities Based on von Neumann Entropy for Nodes and Motifs

Abstract

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. Centralities for nodes are widely researched to depict the networks from a certain perspective and perform great efficiency, yet most of them are restricted to local environment or some specific configurations and hard to be generalized to structural patterns of networks. In this paper we propose a new centrality for nodes and motifs by the von Neumann entropy, which allows us to investigate the importance of nodes or structural patterns in the view of structural complexity. By calculating and comparing similarities of this centrality with classical ones, it is shown that the von Neumann entropy node centrality is an all-round index for selecting crucial nodes, and able to evaluate and summarize the performance of other centralities. Furthermore, when the analysis is generalized to motifs to achieve the von Neumann entropy motif centrality, the all-round property is kept, the structural information is sufficiently reflected by integrating the nodes and connections, and the high-centrality motifs found by this mechanism perform greater impact on the networks than high-centrality single nodes found by classical node centralities. This new methodology reveals the influence of various structural patterns on the regularity and complexity of networks, which provides us a fresh perspective to study networks and performs great potentials to discover essential structural features in networks.