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.

这篇研究使用 persistent homology 方法来检测比较难以用传统统计方法描述的网络拓扑结构，将加权网络基于这些结构分类为具有不同特征的两类，并将代数拓扑工具引入复杂系统中。