基于矩阵运算的复杂网络构建方法

Complex network construction based on matrix operation

本文从邻接矩阵的视角分析复杂网络,研究了基于矩阵Kronecker积运算与Kronecker和运算的复杂网络构建方法.将Kronecker积运算迭代地应用于一个简单初始网络的邻接矩阵,得到了一个Kronecker积图,也是一个分形维数不超过2的自相似网络.当初始网络是连通非二分图时,则得到的Kronecker积图同时具有小世界特性,其直径不超过初始网络直径的两倍.其次,将Kronecker和运算顺次应用于多个简单初始网络的邻接矩阵,得到了一个Kronecker和图,也是一个度分布呈正态分布的随机网络.最后,给出了基于矩阵运算的复杂网络构建方法的若干性质.

In this study,we explore complex networks from the perspective of an adjacency matrix and investigate approaches for constructing these networks based on the Kronecker product and Kronecker sum.The Kronecker product is iteratively applied to the adjacency matrix of a simple initial network;then,a Kronecker product graph is obtained.It is a self-similar network with a fractal dimension no larger than 2.If the initial network is a connected and non-bipartite graph,the obtained complex network is also a small-world network,and its diameter does not exceed twice that of the initial network.Furthermore,the Kronecker sum is sequentially applied to the adjacency matrixes of multiple simple initial networks and then a Kronecker sum graph is obtained.It is a random network,which has the degree distribution as a normal distribution.Finally,several properties of approaches for constructing complex networks based on matrix operation are presented.