Abstract: Correlation matrix describes hypernetwork briefly and intuitively. Hypernetwork can be characterized by node degree, node hyperdegree and hyperedge degree. This paper studies hypernetwork especially self-similar hypernetwork and random hypernetwork from the perspective of correlation matrix, and shows several properties of  approaches for constructing hypernetwork based on matrix operation. Self-similar hypernetwork can be obtained by Tracy-Singh product on the correlation matrix of a simple initial hypergraph iteratively, and random hypernetwork can be 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 than 2. When the initial hypergraph is a connected and non-bipartite hypergraph, 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 hyperedge    degrees of random hypernetworks are normal. The results of simulation experiments validate the properties of the constructed hypernetwork.

Key words: hypernetwork, matrix operations, self-similar hypernetwork, fractal dimension, random hypernetwork

摘要： 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。

关键词: 超网络, 矩阵运算, 自相似超网络, 分形维数, 随机超网络