A betweenness structural entropy of complex networks

Abstract

The structural entropy of the complex networks quantifies the static network's topological structure complexity. The definition of structural entropy is based on the Shannon information entropy and the structural components of each node. The traditional structural entropy of the complex networks is based on the degree distribution of nodes in the network. However, the degree-structural entropy is not always effective, especially when the topology structure change is under the same degree distribution. The isotopic networks with the same node's degree distribution but different structural complexity show that the definition of the structural entropy needs to base on different structural components. In this work, we propose the betweenness structural entropy of complex networks to quantify the structural complexity of static networks. Simultaneously, several processes of network growth with different seed networks under different growth rules are built in this work. These processes offer a series of networks that can be used to check how the structural entropy of the networks changes in network growth, both the degree and betweenness structural entropy. We find that the betweenness structural entropy is always smaller than the degree structural entropy of the same network. We also defined the structural entropy ratio to quantify the relative difference between the betweenness structural entropy and the degree structural entropy. Surprisingly, we find that the difference between the networks' structural entropies (degree and betweenness structural entropy) gives a new measurement to quantify the network's structure stability. This finding inspired us that the difference in different structural entropy can be used as a new structural complexity measurement for the networks: the structural entropy ratio. When the structural entropy ratio for a network is big, the network's topology structure is not stable.