基于矩阵运算的超网络构建方法研究及特性分析

摘要

We study supernetwork building based on the Khatri-Rao product operation and the Khatri-Rao sum operation on adjacency matrices.In addition,the marginal-and joint-node degrees are introduced to investigate the mechanism of a supernetwork.The Khatri-Rao product operation is iteratively applied to a simple initial network to form the adjacent supernetwork matrix and obtain a self-similarity supernetwork with fractal dimensions of no longer than 3.If all initial networks are connected with nonbipartite graphs,the obtained supernetwork has a diameter that does not exceed twice the summation of all initial networks.Furthermore,the Khatri-Rao sum operation is sequentially applied to multiple simple initial networks to form adjacency matrices of supernetwork and obtain a random supernetwork with one marginal node degree,with one-dimensional Gaussian distribution,and a joint node degree,with a high-dimensional Gaussian distribution.Finally,several properties of the proposed supernetwork building based on matrix operation are presented.%基于邻接矩阵Khatri-Rao积运算及Khatri-Rao和运算,研究了构建超网络的方法,并通过边际节点度及联合节点度来研究超网络的内在机理.将Khatri-Rao积运算迭代地应用于一个初始图序列组成超网络的邻接矩阵,得到一个分形维数不超过3的自相似超网络.若所有初始图均是连通非二分图,则得到的超网络同时具有小世界特性,其直径不超过所有初始图直径和的两倍.此外,将Khatri-Rao和运算顺次应用于多个初始图序列组成超网络的邻接矩阵,得到一个边际节点度呈一维高斯分布而联合节点度呈高维高斯分布的随机超网络.最后,给出了基于矩阵运算的超网络构建方法的若干性质.