Extremum Characteristics Of Two Kinds Of Graph Entropy And Its Applications In Complex Network

Graph entropy is the combination of graph theory and information theory,and it is also an important research direction in graph theory.In recent years,the study of extremum of graph entropy is a hot topic.Because of the close relationship between invariant and entropy,graph entropy can accurately reflect the graph structure information based on the invariant.The graph entropy constructed by local and global invariants can reflect the relationship between the structural information of local and global invariants.When graph entropy is applied to complex networks as a decision strategy of network centrality,the structural information reflected by graph entropy can be regarded as network complexity.Therefore,graph entropy is not only one of the important research directions of graph theory,but also has important applications in computer science,communication information and network.In this paper,the distance von Neumann entropy of general graph and Shannon entropy based on Laplacian degree of hypergraph are studied from two aspects.Not only the maximum and minimum entropy of the graph under certain conditions are determined,but also the cor-responding extremum graph when the extremum is obtained.The extremum characteristic of distance von Neumann entropy is used to identify important nodes in complex networks,and an attack strategy(DV)based on distance von Neumann entropy is proposed.The static attack and the dynamic attack are used to distinguish the attack efficiency of the attack strategy based on the distance von Neumann entropy.The experimental results show that the attack strategy based on the distance von Neumann entropy can identify the center node of the network better.The main achievements of this paper are as follows:(1)In general graphs,the concept of distance density matrix of graphs is defined by the way of constructing density matrix of reference graphs.Based on the relationship between the von Neumann entropy and the density matrix of a graph,the concept of von Neumann entropy is extended to the distance von Neumann entropy.Then,according to the relationship between the number of vertices,point transfer and other invariants,the maximum and minimum of distance von Neumann entropy are calculated and deduced by using mathematical tools in many different cases.In addition,the corresponding extremum graphs are determined when the maximum and minimum of the distance von Neumann entropy are obtained.(2)In hypergraph,based on the definition of Shannon entropy in general graph and Lapla-cian degree of hypergraph,Shannon entropy based on Laplacian degree is defined.The ex-tremum results of Shannon entropy in some simple graphs are extended to k-uniform hyper-graphs.According to the structural characteristics of different graphs and a kind of edge shifting operation,the maximum and minimum values of Shannon entropy based on Laplacian degree in k-uniform hypergraph,unicyclic k-uniform hypergraph,bicyclic k-uniform hypergraph and k-uniform chemical hypertree are determined respectively,and the corresponding graph when the extremum value is obtained is determined.(3)Combined with the characteristics of complex network,the distance von Neumann entropy is used in the algorithm of important node identification.A new attack strategy based on distance von Neumann entropy is proposed,which is named——distance von Neumann entropy centrality(DV).The static attack and dynamic attack are used to distinguish the attack efficiency of the attack strategy based on distance von Neumann entropy in the Karate Club network,Dolphin Social network and American University Football Club network.The exper-imental results show that the attack strategy based on distance based von Neumann entropy can accurately and quickly identify the central nodes of the network,especially the attacks related to distance strategy——Closeness centrality(CC),with obvious effect advantages.The static attack and dynamic attack are used to distinguish the attack efficiency of distance based von Neumann entropy in three kinds of networks.The experimental results show that the attack strategy based on the distance von Neumann entropy can identify the center node of the net-work well,especially compared with the distance related attack strategy,the effect advantage is obvious.It shows that the attack strategy based on the distance von Neumann entropy can well reflect the importance of nodes in the relative center of the network.