A measure of identifying influential community based on the state of critical functionality

As an open issue, the measure of community influential is no uniform standard, how to measure the influence of community has attracted extensive attention. This paper proposed a quantitative measure to identify the influence of community. Based on the state of critical functionality (SCF), a new function, which names as the weighted state of critical functionality (WSCF), is defined. For the WSCF, not only the connections among communities but also the topology within community is considered. When the community structure of the complex network is divided, each community is renormalized as a node by the renormalization method. Then, the influence of community is measured by the values of WSCF, the greater the value of WSCF, the less the influence of the corresponding community. The influence of community of three classic constructed networks (i.e., a Erodös-Rényi (ER) random network, a BA scale free network and a small-word (SW) network) is measured by the proposed method. To further verify the feasibility of the method, two community detection algorithms are used to divide community structure in the real networks. The influence of the community of the 9/11 terrorist network, a US Air network and a PolBooks network is measured by the proposed method. The influence of each community could be measured and the most influential community in each network is identified by the proposed method. The results reveal that the proposed method is a feasible measure to identify the influence of community, its recognition effect is better than SCF, and accuracy is higher. The SCF is a special case of WSCF, when the weights and cluster coefficients are equal to 1. For the proposed method, once the community structure of the network is divided, the corresponding community influence is identified by the proposed method, which is not affected by the community division algorithms.