Self-Clustering, Attraction Kernel and Quantity of Clustering

In this paper,the self-clustering and evolving processes and quantity of clustering for complex systems are investigated. Two models similar to growing networks are given. One is more simpler,at every time step,only one new edge is entering the network. But this model describes various problems in different fields,such as election,citations between academic papers,food resources for ants and bees,commodities or shares,seepage spots at dikes,etc. Another is comparatively general. At every time step m new edges are entering the network. By using the idea of "preferential attachment" in the BA model,we explore the quantity of clustering for every vertex in the networks for both models. Results show that the possible quantity of clustering for every vertex in both models is expressed by a simple formula of mathematical ex- k, pectation Est =ks/t0t, where s-vertex of networks, t0-initial time,ks-the degree of s at t0,t-time , also the number of total degrees in the network at t, or the total quantity of clustering,ks/t0-the initial superiori- ty or attractive ability. The vertices of the networks may be called "attractive kernel" ,and ks/t0-coefficient of attraction. The meanings of Est = ks/t0t are explained for different problems in different cases.