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arXiv · 2403.05749

Characterizing Flow Complexity in Transportation Networks using Graph Homology

Shashank A Deshpande·Hamsa Balakrishnan
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Abstract

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.

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Shashank A Deshpande, Hamsa Balakrishnan. 2024-03-09. Characterizing Flow Complexity in Transportation Networks using Graph Homology. https://arxiv.org/abs/2403.05749

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