Series-parallel network topologies generally exhibit simplified dynamical behavior and avoid high combinatorial complexity. A comprehensive analysis of how flow complexity emerges with a graph's deviation from series-parallel topology is therefore of fundamental interest. We introduce the notion of a robust $k$-path on a directed acycylic graph, with increasing values of the length $k$ reflecting increasing deviations. We propose a graph homology with robust $k$-paths as the bases of its chain spaces. In this framework, the topological simplicity of series-parallel graphs translates into a triviality of higher-order chain spaces. We discuss a correspondence between the space of order-three chains and sites within the network that are susceptible to the Braess paradox, a well-known phenomenon in transportation networks. In this manner, we illustrate the utility of the proposed graph homology in sytematically studying the complexity of flow networks.

翻译：串并联网络拓扑通常表现出简化的动态行为，并避免高组合复杂度。因此，全面分析流复杂度如何随图偏离串并联拓扑而涌现具有根本性意义。我们引入了有向无环图上稳健$k$路径的概念，其中长度$k$的增加反映了偏差程度的增大。我们提出了一种以稳健$k$路径为链空间基的图同调。在此框架下，串并联图的拓扑简单性转化为高阶链空间的平凡性。我们讨论了三阶链空间与网络中易出现布雷斯悖论（交通网络中一个著名现象）的位置之间的对应关系。通过这种方式，我们展示了所提出的图同调在系统研究流网络复杂度方面的实用性。