加权复杂网络的拓扑层次
Topological strata of weighted complex networks
摘要：
复杂网络的统计力学方法是描述自然和社会复杂系统的主导范式。网络性质的研究，以及它们对动力学过程的影响，主要集中在局部定义的节点和边的数量，如节点度，边权重和-最近-相邻节点之间的相关性。然而，统计方法在处理多体性质时很快变得繁琐，并且无法捕捉复杂网络的精确介观结构。在这里，我们介绍了一种新的方法，基于持久的同源性，检测特定的非局部结构，类似于加权孔内的链接权重网络结构，这是不可见的现有方法。它们的性质将加权网络分为两大类：一类是以小的层次嵌套洞为特征，而第二类则显示出更大更长的不均匀性。这些类不能被减少到已知的本地或准本地网络属性，因为固有的非局部性的同调属性，从而产生一个新的分类建立在高阶协调模式。我们的研究结果表明，拓扑结构可以提供新的见解相关的多体相互作用的社会和空间网络。此外，这种新方法创建了网络理论和代数拓扑之间的第一座桥梁，这将允许将代数方法的工具集导入复杂系统。
英文摘要：
The statistical mechanical approach to complex networks is the dominant paradigm in describing natural and societal complex systems. The study of network properties, and their implications on dynamical processes, mostly focus on locally defined quantities of nodes and edges, such as node degrees, edge weights and --more recently-- correlations between neighboring nodes. However, statistical methods quickly become cumbersome when dealing with many-body properties and do not capture the precise mesoscopic structure of complex networks. Here we introduce a novel method, based on persistent homology, to detect particular non-local structures, akin to weighted holes within the link-weight network fabric, which are invisible to existing methods. Their properties divide weighted networks in two broad classes: one is characterized by small hierarchically nested holes, while the second displays larger and longer living inhomogeneities. These classes cannot be reduced to known local or quasilocal network properties, because of the intrinsic non-locality of homological properties, and thus yield a new classification built on high order coordination patterns. Our results show that topology can provide novel insights relevant for many-body interactions in social and spatial networks. Moreover, this new method creates the first bridge between network theory and algebraic topology, which will allow to import the toolset of algebraic methods to complex systems.
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