Degree-and-Cycle-ratio program

In this paper, a new vital nodes identification method DC (Degree and Cycle ratio) is proposed by integrating degree centrality (weight ) and cycle ratio (weight 1−). The results show that the dynamic observations (such as robustness, infection efficiency and immunization efficiency) and weight  are nonlinear (i.e., there exists an optimal value * for ), and DC performs better than single index in most networks.

According to the value of *, networks are classified into degree-dominant networks (* > 0.5) and cycle-dominant networks (* < 0.5). Specifically, in most degree-dominant networks (such as Chengdu-BUS, Chongqing-BUS and Beijing-BUS), degree is dominant in the identification of vital nodes, but the identification effect can be improved by adding cycle structure information to nodes. In most cycle-dominant networks (such as Email, Wiki and Hamsterster), the cycle ratio is dominant in the identification of vital nodes, but the effect can be notably enhanced by additional node degree information.