Equilibrium Index and Core Node Set of New Social Networks

In the last fifteen years, the network-based new media has greatly promoted the development of social networks, therefore it becomes the main form of social composition. Research on social networks commenced a long time ago and revealed meaningful features of the old social network. However, in the new social networks, group members are almost 24-h online and can exchange various information that cannot be dynamically handled or described by the most of the existing studies. In this paper, we construct a formula to calculate the amount of the data dissemination of a node. Thereafter, we present participation equilibrium index formula and maximum equilibrium index formula that can measure the dynamic equilibrium of social networks even for different topics. With these formulas, we define λ-set and core node set of a social group and also present algorithms to work out the two sets. To evaluate our theory and method, we have developed SN-Python-I that can filter and calculate the amount of data exchanged by any node in any group. Based on a contract signed in 2013 with Ali and Tencent, we are permitted to track 300 actual groups for the past three years. Real data of those group reveals that equilibrium index as well as λ-set and core node set of social network change with the time, ages, topics, and classification of the groups. What's more, monthly equilibrium index can also indicate life stages of a group. Those significant properties and rules have not yet been revealed by traditional social network models.