Abstract
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.
Highlights
Network science has gained popularity due to its great achievements in the past twenty years, where small-world networks 1 are built from nearest-neighbor regular networks through rewiring, presenting two significant characteristics of short average path-length and large clustering coefficient, while scale-free networks 2 are modeled based on random networks 3, possessing a scale-free power-law node-degree distribution
This paper presents the mathematical description of a new framework of clique vector spaces, firstly introducing related concepts of various cycles, secondly defining a sequence of clique vector spaces associated with boundary operators, and discussing chain group, cycle group, boundary group and homology group
Using a sequence of clique vector spaces along with boundary operators to describe complex networks has well demonstrated that cliques, simplexes and fully-connected sub-networks are the backbones of various networks
Summary
Network science has gained popularity due to its great achievements in the past twenty years, where small-world networks 1 are built from nearest-neighbor regular networks through rewiring, presenting two significant characteristics of short average path-length and large clustering coefficient, while scale-free networks 2 are modeled based on random networks 3, possessing a scale-free power-law node-degree distribution. The three fundamental concepts in network science—average path-length, degree distribution and clustering coefficient—correspond to three basic structures: chain, star and cycle. Clustering coefficient is calculated based on triangles, but cycle
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
