Totally homogeneous networks  

作　　者：Dinghua Shi Linyuan Lu Guanrong Chen 

机构地区：[1]Department of Mathematics,College of Science,Shanghai University,Shanghai 200444,China [2]Institute of Fundamental and Frontier Sciences,University of Electronic Science and Technology of China.Chengdu 610054,China [3]Department of Electronic Engineering,City University of Hong Kong,Hong Kong,China

出　　处：《National Science Review》2019年第5期962-969,共8页国家科学评论（英文版）

基　　金：supported by the National Natural Science Foundation of China(61174160,11622538 and 61673150);the Zhejiang Provincial Natural Science Foundation of China(LR16A050001);the Hong Kong Research Grant Council under GRF Grant City U11200317

摘　　要：In network science,the non-homogeneity of node degrees has been a concerning issue for study.Yet,with today’s modern web technologies,the traditional social communication topologies have evolved from node-central structures into online cycle-based communities,urgently requiring new network theories and tools.Switching the focus from node degrees to network cycles could reveal many interesting properties from the perspective of totally homogenous networks or sub-networks in a complex network,especially basic simplexes(cliques)such as links and triangles.Clearly,compared with node degrees,it is much more challenging to deal with network cycles.For studying the latter,a new clique vector-space framework is introduced in this paper,where the vector space with a basis consisting of links has a dimension equal to the number of links,that with a basis consisting of triangles has the dimension equal to the number of triangles and so on.These two vector spaces are related through a boundary operator,for example mapping the boundary of a triangle in one space to the sum of three links in the other space.Under the new framework,some important concepts and methodologies from algebraic topology,such as characteristic number,homology group and Betti number,will play a part in network science leading to foreseeable new research directions.As immediate applications,the paper illustrates some important characteristics affecting the collective behaviors of complex networks,some new cycle-dependent importance indexes of nodes and implications for network synchronization and brain-network analysis.In network science, the non-homogeneity of node degrees has been a concerning issue for study.Yet, with today’s modern web technologies, the traditional social communication topologies have evolved from node-central structures into online cycle-based communities, urgently requiring new network theories and tools.Switching the focus from node degrees to network cycles could reveal many interesting properties from the perspective of totally homogenous networks or sub-networks in a complex network, especially basic simplexes(cliques) such as links and triangles.Clearly, compared with node degrees, it is much more challenging to deal with network cycles.For studying the latter, a new clique vector-space framework is introduced in this paper, where the vector space with a basis consisting of links has a dimension equal to the number of links, that with a basis consisting of triangles has the dimension equal to the number of triangles and so on.These two vector spaces are related through a boundary operator, for example mapping the boundary of a triangle in one space to the sum of three links in the other space.Under the new framework,some important concepts and methodologies from algebraic topology, such as characteristic number,homology group and Betti number, will play a part in network science leading to foreseeable new research directions.As immediate applications, the paper illustrates some important characteristics affecting the collective behaviors of complex networks, some new cycle-dependent importance indexes of nodes and implications for network synchronization and brain-network analysis.

关 键 词：BOUNDARY OPERATOR CLIQUE vector space CYCLE HOMOLOGY group totally HOMOGENOUS network 

分 类 号：O157.5[理学—数学]