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National Science Review 2019年第5期6卷 962-969页

作者： Dinghua Shi Linyuan Lü Guanrong Chen Department of Mathematics College of Science Shanghai University Institute of Fundamental and Frontier Sciences University of Electronic Science and Technology of China Department of Electronic Engineering City University of Hong Kong

In network science, the non-homogeneity of node degrees has been a concerning issue for ***, 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 *** 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 ***, compared with node degrees, it is much more challenging to deal with network *** 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 *** 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 *** 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 *** 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

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A Generalized Discrete Morse-Floer Theory

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Communications in Mathematics and Statistics 2019年第3期7卷 225-252页

作者： Jürgen Jost Sylvia Yaptieu Max Planck Institute for Mathematics in the Sciences Inselstrasse 2204103 LeipzigGermany Santa Fe Institute for the Sciences of Complexity Santa FeNM 87501USA

Forman has developed a version of discrete Morse theory that can be understood in terms of arrow patterns on a(simplicial,polyhedral or cellular)complex without closed orbits,where each cell may either have no arrows,receive a single arrow from one of its facets,or conversely,send a single arrow into a cell of which it is a *** following arrows,one can then construct a natural Floer-type boundary ***,we develop such a construction for arrow patterns where each cell may support several outgoing or incoming arrows(but not both),again in the absence of closed *** main technical achievement is the construction of a boundary operator that squares to 0 and therefore recovers the homology of the underlying complex.

关键词： CW complex boundary operator Floer theory Poincarépolynomial Betti number Discrete Morsetheory Discrete Morse-Floertheory Conleytheory

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