Evaluation methods of node importance in undirected weighted networks based on complex network dynamics models

Identifying the most important nodes is significant for investigating the robustness and vulnerability of complex network. A lot of methods based on network structure have been proposed, such as degree, K-shell and betweenness, etc. In order to identify the important nodes in a more reasonable way, both the network topologies and the characteristics of nodes should be taken into account. Even at the same location, the nodes with different characteristics have different importance. The topological structures and the characteristics of the nodes are considered in the complex network dynamics model. However, such methods are rarely explored and their applications are restricted. In order to identify the important nodes in undirected weighted networks, in this paper we propose a method based on dynamics model. Firstly, we introduce a way to construct the corresponding dynamics model for any undirected weighted network, and the constructed model can be flexibly adjusted according to the actual situation. It is proved that the constructed model is globally asymptotic stable. To measure the changes of the dynamic model state, the mean deviation and the variance are presented, which are the criteria to evaluate the importance of the nodes. Finally, disturbance test and destructive test are proposed for identifying the most important nodes. Each node is tested in turn, and then the important nodes are identified. If the tested node can recover from the damaged state, the disturbance test is used. If the tested node is destroyed completely, the destructive test is used. The method proposed in this paper is based on the dynamics model. The node importance is influenced by the network topologies and the characteristics of nodes in these two methods. In addition, the disturbance test and destructive test are used in different situations, forming a complementary advantage. So the method can be used to analyze the node importance in a more comprehensive way. Experiments are performed on the advanced research project agency networks, the undirected networks with symmetric structures, the social network, the Dobbs-Watts-Sabel networks and the Barrat-Barthelemy-Vespignani networks. If the nodes in the network have the same dynamic model, the network is considered to be the homogeneous network; otherwise, the network is heterogeneous network. And experiments can be divided into four categories, namely, the disturbance test, the destructive test on the homogeneous network, the disturbance test and the destructive test on the heterogeneous network. The experimental results show that the methods proposed in this paper are effective and credible.