  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.
