随机派系网络的渗流相变研究

Percolation of random clique networks

现实的复杂网络往往具有小世界特征和模块化层次结构,随机派系(Clique)网络不仅有高的聚类系数和短的平均路径长度(小世界特征),而且具有模块化层次结构,可以较好地描述现实世界中许多复杂网络的结构特征.本文采用蒙特卡罗模拟方法和临界现象有限尺度标度理论对该网络的渗流相变进行了研究.在网络的演化过程中,其最大团簇单时间步的最大尺度跳跃及对应的时间步可以用来确立临界行为和临界点.我们的模拟研究表明:随机派系网络的渗流相变都为连续相变,不同派系大小k的相变点发生的时间步不同,但约化边数相同;网络最大团簇的最大尺度跳跃?与网络大小N呈幂律关系,幂指数对应于渗流相变的序参量临界指数,且不随派系大小k变化;与?相关的另外三个临界指数也不随k值变化,与ER随机网络的临界指数相等.这些结果表明随机派系网络的渗流相变与ER网络渗流相变同属一个普适类.我们的研究加深了对复杂网络模块化层次结构产生机理的认识,有助于进一步认识复杂网络的渗流相变.

We propose a random clique network （RCN）, which is constructed by adding cliques randomly. In a k-clique, there are k nodes which are connected each other completely. The RCN possesses some characters of the small world network and the modular hierarchical structure. At k=2, the RCN becomes the Erd0s-REnyi （ER） random network. In this paper, we study the percolation transition of RCN by investigating the biggest size gap A of the largest cluster in the network and the corresponding evolution step, which is taken as the transition point. From the Monte Carlo simulations of RCN at k=2, 3, 4, 5, we can calculate the mean values and the mean square root of fluctuations for A and the transition point. They all show a power-law dependence on the network size N. This leads to the conclusion that the percolation transitions of RCN at k=2, 3, 4, 5 are continuous. From the exponents of power-law behaviors, the critical exponents fib vl of A and the critical exponents ,β2, v2 of the transition points can be obtained. These critical exponents of different RCN are shown to be independent of the clique size k. The percolation transitions of RCN belong to the same universality class as the ER random network.