自聚集、吸引核与聚集量

研究复杂系统的自聚集演化过程和聚集量.文中给出两个类似生长网络的模型.第一个模型比较简单,每一时间步长只有一条新边进入网中,但概括面较广,例如可描写选举、科学论文引用、食物源对蚁群蜂群的吸引、某种商品或股票、堤坝渗漏处,等等.第二个模型比较一般,每次可有m条新边进入网络.文中引用BA网络模型给出的“优先连接”的概念,研究上面两个网络中各点的聚集量.结果表明：对于这两个模型,各点可能的聚集量均可用一个数学期望的简单公式描述,即Et^s=ks/t0t .其中,s表示网中某点,t0是初始时间,ks是t0时点s的顶点度,t是任何时间,t也是此时网的总度数,或总聚集量.ks/t0表征点s的初始优势或初始吸引能力,点可称为吸引核,ks/t0可称为吸引系数.文中解释了对于不同情况下 Et^s=k/t0t的意义.

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.