Network Structure Entropy Based On Node Degree Relationship And Its Application

Since the theory of entropy was introduced into complex networks,it has become a hot spot in the research field of complex networks to measure the complexity of networks by using the structure entropy.The existing literature has rich research results on degree structure entropy and betweenness structure entropy,but they all measure complexity based on some isolated characteristics of node.This paper focuses on the definition of structure entropy from the perspective of the relationship between node degrees,and discusses its application in node importance identification based on the newly defined structural entropy.The main work is as follows:(1)The structure entropy of complex networks is defined from the "exclusion" relationship between node degrees.Specifically,a complex network structure entropy is constructed based on Coulomb's law.The nodes of complex network are regarded as charges,and the degree of each node is compared to the amount of charge.Using Coulomb theorem for reference,the mutual "repulsive force" between nodes is considered,and this "repulsive force" is used as the strength of nodes,so the weighted strength of each node in the network is obtained.Finally,considering the inconsistency of node weighted strength,the structure entropy of complex networks is defined.Firstly,the new structure entropy is applied to several special networks such as global coupling network,nearest neighbor network,symmetric network and spindle network.It is found that the structure entropy defined based on exclusion relationship can not only reflect the network characteristics shown by degree structure entropy,betweenness structure entropy and Tsallis structure entropy,but also have a better effect on discrimination.Then we measure the small world networks randomly generated with different number of nodes and real networks: Graph and digraph glossary network,Central literature Network,US air lines network and Yeast network.The results show that for the same network,the order of entropy is degree structure entropy,betweenness structure entropy,Tsallis structure entropy and the structure entropy defined in this paper.This reflects that the more node degree factors are considered,the more inconsistencies are reflected,and the smaller the entropy is.(2)The structure entropy of complex networks is defined from the "similarity" relationship of node degrees.Specifically,a complex network structure entropy is constructed based on the degree-degree distance theory.The node degree is compared to the attribute value measuring the "capability size" of the node.The closer the "capability size" of the two nodes is,the greater the "similarity" is.Using the degreedegree distance theory for reference,the "similarity" between nodes is considered as the strength of nodes,so the weighted strength of each node in the network is obtained.Finally,considering the inconsistency of node weighted strength,the structure entropy of complex network is defined.Firstly,the new structure entropy is applied to several scale-free networks.It is found that the structure entropy defined based on the node degree similarity relationship can not only reflect the network characteristics shown by degree structure entropy,betweenness structure entropy and Tsallis structure entropy,but also have a better effect on the discrimination degree.Then we measure the real networks: Zachary karate club network,Graph and digraph glossary network,Central literature network and US air lines network.The results show that for the same structure entropy,according to the order of network scale,the entropy value also changes from small to large.This reflects that the larger the scale of the network has,the stronger the heterogeneity of the structure is,and the greater the entropy is.(3)Based on the structure entropy given in this paper,the importance identification of nodes is discussed by using the entropy weight method.The structure entropy constructed in this paper is based on the relationship between node degrees,which has the function of identifying nodes.In the entropy weight method,the weighted degree based on Coulomb's law is combined with Kshell,and the locality and globality of nodes are comprehensively considered,which not only eliminates the degeneracy of Kshell method,but also avoids the limitation of weighted degree.The structure entropy constructed in this paper is applied to an example network with 16 nodes.It is found that this method can well identify the important nodes in the network.Then the real networks: Zachary karate club network,Dolphin network,Iceland network,Similarities network,Yeast network and BA network are used as experimental data,and the node importance is verified through "survivability".The experimental results show that among all networks,the network with node importance ranking based on entropy weight method has the most obvious downward trend of polar pass coefficient,Moreover,in most networks,the initial stage of node attack shows better attack effect than other indicator.In addition,the same method is used to conduct experiments to observe the changes of network efficiency.It is found that deleting the top nodes of the proposed method leads to the greatest decline in network efficiency.The experimental results show that the structural entropy constructed in this paper shows a good effect in the ranking of important nodes.