超网络模型构建及特性分析

Hypernetwork Model and Its Properties

关联矩阵是超网络的一种表述形式,节点度、节点超度和超边度是度量超网络的一种方法。从关联矩阵出发对超网络进行研究,重点研究了自相似超网络及随机超网络,并给出了基于矩阵运算的超网络构建方法的若干性质。自相似超网络可通过对一个简单初始超图的关联矩阵进行迭代的Tracy-Singh积运算得到,而随机超网络可通过对多个简单初始超图的关联矩阵进行顺次的Tracy-Singh和运算得到。自相似超网络的分形维数不超过2,且当初始超图是连通的且非二分超图时,自相似超网络的直径不超过初始超图直径的两倍,即同时具有小世界特性。随机超网络的节点度、节点超度和超边度均呈正态分布。仿真实验证实了所构建的超网络的各项特性。

Correlation matrix describes hypernetwork briefly and intuitively.Hypernetwork can be characterized bynode degree,node hyperdegree and hyperedge degree.This paper studies hypernetwork especially self-similar hypernetworkand random hypernetwork from the perspective of correlation matrix,and shows several properties ofapproaches for constructing hypernetwork based on matrix operation.Self-similar hypernetwork can be obtained byTracy-Singh product on the correlation matrix of a simple initial hypergraph iteratively,and random hypernetwork canbe obtained by Tracy-Singh sum on the correlation matrixes of multiple simple initial hypergraphs sequentially.The fractal dimension of self-similar hypernetworks is no larger than2.When the initial hypergraph is a connected and nonbipartitehypergraph,the diameter of self-similar hypernetwork does not exceed twice of that of the initial hypergraph,namely,it also shares a small-world property.The distributions of node degrees,node hyperdegrees and hyperedgedegrees of random hypernetworks are normal.The results of simulation experiments validate the properties of the constructedhypernetwork.