第13卷第3期 智能系统学报 Vol.13 No.3 2018年6月 CAAI Transactions on Intelligent Systems Jun.2018 D0:10.11992/tis.201706055 网络出版t地址：http:/kns.cnki.net/cms/detail/23.1538.TP.20180412.0858.002.html 基于矩阵运算的超网络构建方法研究及特性分析 刘胜久2，李天瑞2，洪西进2，王红军2，珠杰24 (1.西南交通大学信息科学与技术学院，四川成都611756,2.西南交通大学四川省云计算与智能技术高校重点实 验室，四川成都611756：3.台湾科技大学资讯工程系，台湾台北10607：4.西藏大学计算机系，西藏拉萨850000) 摘要：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。 关键词：矩阵运算；复杂网络；超网络：模型构建；分形维数：自相似超网络；随机超网络；特性分析 中图分类号：TP393文献标志码：A文章编号：1673-4785(2018)03-0359-07 中文引用格式：刘胜久，李天瑞，洪西进，等.基于矩阵运算的超网络构建方法研究及特性分析J.智能系统学报，2018,13(3)： 359-365. 英文引用格式：LIU Shengjiu,.LI Tianrui,,HORNG Xijin,etal.Supernetwork building based on matrix operation and property analysis[J.CAAI transactions on intelligent systems,2018,13(3):359-365. Supernetwork building based on matrix operation and property analysis LIU Shengjiu,LI Tianrui2,HORNG Xijin',WANG Hongjun'2,ZHU Jie3 (1.School of Information Science and Technology,Southwest Jiaotong University,Chengdu 611756,China;2.Sichuan Key Lab of Cloud Computing and Intelligent Technique,Southwest Jiaotong University,Chengdu 611756,China;3.Department of Computer Science and Information Engineering,National Taiwan University of Science and Technology,Taipei 10607,China;4.Department of Computer Science,Tibetan University,Lhasa 850000,China) Abstract:We study supernetwork building based on the Khatri-Rao product operation and the Khatri-Rao sum opera- tion on adjacency matrices.In addition,the marginal-and joint-node degrees are introduced to investigate the mechan- ism of a supernetwork.The Khatri-Rao product operation is iteratively applied to a simple initial network to form the ad- jacent 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 mul- tiple 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. Keywords:matrix operation;complex network;supernetwork;model building;fractal dimension;self-similarity super- network;random supernetwork;property analysis 收稿日期：2017-06-14.网络出版日期：2018-04-12 复杂网络可较好地描述并刻画复杂系统，对其 基金项目：国家自然科学基金项目(61573292,61262058)】 通信作者：李天瑞.E-mail:tri@swjtu.cdu.cn. 系统性研究起源于20世纪Erdos与Renyi二者合DOI: 10.11992/tis.201706055 网络出版地址: http://kns.cnki.net/kcms/detail/23.1538.TP.20180412.0858.002.html 基于矩阵运算的超网络构建方法研究及特性分析 刘胜久1,2，李天瑞1,2，洪西进1,2,3，王红军1,2，珠杰1,2,4 （1. 西南交通大学 信息科学与技术学院，四川 成都 611756; 2. 西南交通大学 四川省云计算与智能技术高校重点实 验室，四川 成都 611756; 3. 台湾科技大学 资讯工程系，台湾 台北 10607; 4. 西藏大学 计算机系，西藏 拉萨 850000） 摘 要：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。 关键词：矩阵运算；复杂网络；超网络；模型构建；分形维数；自相似超网络；随机超网络；特性分析 中图分类号：TP393 文献标志码：A 文章编号：1673−4785(2018)03−0359−07 中文引用格式：刘胜久, 李天瑞, 洪西进, 等. 基于矩阵运算的超网络构建方法研究及特性分析[J]. 智能系统学报, 2018, 13(3): 359–365. 英文引用格式：LIU Shengjiu, LI Tianrui, HORNG Xijin, et al. Supernetwork building based on matrix operation and property analysis[J]. CAAI transactions on intelligent systems, 2018, 13(3): 359–365. Supernetwork building based on matrix operation and property analysis LIU Shengjiu1,2 ，LI Tianrui1,2 ，HORNG Xijin1,2,3 ，WANG Hongjun1,2 ，ZHU Jie1,2,4 (1. School of Information Science and Technology, Southwest Jiaotong University, Chengdu 611756, China; 2. Sichuan Key Lab of Cloud Computing and Intelligent Technique, Southwest Jiaotong University, Chengdu 611756, China; 3. Department of Computer Science and Information Engineering, National Taiwan University of Science and Technology, Taipei 10607, China; 4. Department of Computer Science, Tibetan University, Lhasa 850000, China) Abstract: We study supernetwork building based on the Khatri-Rao product operation and the Khatri-Rao sum opera￾tion on adjacency matrices. In addition, the marginal-and joint-node degrees are introduced to investigate the mechan￾ism of a supernetwork. The Khatri-Rao product operation is iteratively applied to a simple initial network to form the ad￾jacent 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 mul￾tiple 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. Keywords: matrix operation; complex network; supernetwork; model building; fractal dimension; self-similarity super￾network; random supernetwork; property analysis 复杂网络可较好地描述并刻画复杂系统，对其 系统性研究起源于 20 世纪 Erdos 与 Renyi 二者合 收稿日期：2017−06−14. 网络出版日期：2018−04−12. 基金项目：国家自然科学基金项目 (61573292，61262058). 通信作者：李天瑞. E-mail：trli@swjtu.edu.cn. 第 13 卷第 3 期 智 能 系 统 学 报 Vol.13 No.3 2018 年 6 月 CAAI Transactions on Intelligent Systems Jun. 2018