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.

中文翻译：

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完全同构的网络


在网络科学中，节点度的非同质性一直是一个值得关注的研究问题。然而，随着现代网络技术的发展，传统的社交传播拓扑已经从以节点为中心的结构演变为基于在线循环的社区，迫切需要新的网络理论和工具。将焦点从节点度切换到网络循环可以从复杂网络中完全同质网络或子网络的角度揭示许多有趣的属性，特别是基本单纯形（派系），例如链接和三角形。显然，与节点度相比，处理网络循环更具挑战性。为了研究后者，本文引入了一种新的团向量空间框架，其中以链接组成的基的向量空间的维度等于链接的数量，以三角形组成的基的向量空间的维度等于三角形的数量等。这两个向量空间通过边界运算符相关，例如将一个空间中的三角形边界映射到另一空间中的三个链接的总和。在新的框架下，代数拓扑中的一些重要概念和方法，如特征数、同调群和贝蒂数等，将在网络科学中发挥作用，带来可预见的新的研究方向。作为直接应用，本文阐述了影响复杂网络集体行为的一些重要特征、节点的一些新的依赖于周期的重要性指数以及对网络同步和脑网络分析的影响。