Characterizing Flow Complexity in Transportation Networks Using Graph Homology

Series-parallel networks generally exhibit simplified dynamics, and lend themselves to computationally tractable optimization problems. We are interested in a systematic analysis of the flow complexity that emerges as a network deviates from a series-parallel topology. This letter introduces the notion of a robust p-path on a directed acyclic graph to localize and quantify this complexity. We develop a graph homology with robust p-paths as the bases of its p-chain spaces. We expect that this association between the collection of robust p-paths within a graph and an algebraic structure will provide a framework for the analysis of flow networks. To this end, we show that the simplicity of the series-parallel class corresponds to triviality of high-order chain spaces
$(p\gt 2)$
. Consequently, the susceptibility of a flow network to the Braess Paradox is associated with the space of 3-chains. Moreover, the computational complexity of decision problems on a network can be related to the order of chains within the proposed homology.