Identification of important nodes based on dynamic evolution of inter-layer isomorphism rate in temporal networks

The identification of important nodes can not only improve the research about the structure and function of the network, but also encourage people to widely promote the application fields such as in infectious disease prevention, power grid fault detection, information dissemination control, etc. Currently, numerous conclusions have been proved on the identification of important nodes based on the static-network, which may lead the general property to be weakened as resistivity and conductivity experience the dynamic evolution of the relationship between network nodes with time. Temporal network analysis can more accurately describe the change of interaction order and interaction relationship of network nodes in the process of spatio-temporal evolution, and establish an appropriate temporal network model, as well as provide scientific theoretical support for the identification of important nodes. In this paper, we pay attention to considering the intensity of adjacent and cross-layer coupling, and propose a super-adjacency matrix (ISAM) method based on inter-layer isomorphism rate to represent the temporal networks and measure the importance of nodes. And at the same time, it is given that the temporal network G has N nodes and T time layers, and the ISAM is a super adjacency matrix composed of intra-layer and inter-layer relationships of adjacent and cross-layer networks, and its size is NT × NT. We focus on the study of the coupling between adjacent and cross-layer networks. The traditional method (SAM) considers the isomorphism rate of adjacent layers as a constant. In the improved method (SSAM), the connection between layers is described by a neighbor topological overlap coefficient. In this paper, the concept of the compatible similarity between cross-layer networks is given first, and then, by combining the projection value of vectors in n-dimensional real space and the contribution value of node neighbors, the inter-layer approximation relation coefficient of temporal network is inferred and analyzed. Generally speaking, it ensures the difference in coupling degree among different nodes in the inter-layer relationship. We calculate the importance of nodes based on eigenvector centrality in temporal network, which presents the importance of node i progressing with time. Simultaneously, the robustness of temporal network is studied by making use of the difference in temporal global efficiency. In the end, the operator of Kendall correlation coefficient is used to evaluate the node ranking effect of different time layers between the eigenvector-based centrality and the difference of temporal global efficiency. According to the experimental results of ISAM, SSAM and SAM on Workspace and Email-eu-core data sets, the average Kendall τ of both ISAM methods considering adjacent and cross-layer network isomorphism rate can be increased by 8.37% and 2.99% respectively. The conclusions show that the measurement method of temporal network inter-layer isomorphism rate is reliable and effective.