Characterizing Flow Complexity in Transportation Networks using Graph Homology

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 notio...

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Bibliographic Details
Published inarXiv.org
Main Authors Deshpande, Shashank A, Balakrishnan, Hamsa
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 09.03.2024
Subjects
Combinatorial analysis
Complexity
Deviation
Homology
Network topologies
Robustness
Transportation networks
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Summary: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.
ISSN:2331-8422