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Characterizing Flow Complexity in Transportation Networks using Graph Homology

Shashank A Deshpande, Hamsa Balakrishnan

TL;DR

It is shown that the simplicity of the series-parallel class corresponds to triviality of high-order chain spaces $(p\gt 2)$ .

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.

Characterizing Flow Complexity in Transportation Networks using Graph Homology

TL;DR

It is shown that the simplicity of the series-parallel class corresponds to triviality of high-order chain spaces .

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 -path on a directed acycylic graph, with increasing values of the length reflecting increasing deviations. We propose a graph homology with robust -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.
Paper Structure (15 sections, 4 theorems, 28 equations, 3 figures)

This paper contains 15 sections, 4 theorems, 28 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Series-parallel graphs and the Braess embedding
  3. Notation
  4. Graph Homology of Robust Paths
  5. Two-terminal DAGs and colored route simplices
  6. Development of the robust path homology
  7. Robust Paths in Series Parallel Graphs
  8. Series-parallel two-terminal DAGs
  9. Path complexes of series-parallel graphs
  10. Robust 3-paths and the Braess paradox
  11. Conclusion
  12. Proof of Proposition \ref{['prop:homologysetup']} continued.
  13. Proof of Theorem \ref{['thm:spbifurcation']} continued
  14. Case 1, Part (ii)
  15. Case 2, Part (ii)

Key Result

Proposition 2.1

For all $p\geq 1$, we have (i) $\partial_{p-1}\circ\partial_{p}=0$. (ii) $\partial \Omega_p(\mathsf{R}({\cal G}))\subseteq \Omega_{p-1}(\mathsf{R}({\cal G}))$. Consequently, we obtain the following chain complex

Figures (3)

  • Figure 1: (a) Series and Parallel Combination (b) The Braess Embedding (c) Robust 2-paths combine into a robust 4-path
  • Figure 2: ${\cal G}=\bigcup_\alpha {\cal R}_\alpha$; $\mathsf{R}({\cal G}): \mathop{\hbox{\rm dim}}\Omega_3(\mathsf{R}({\cal G}))\neq0$
  • Figure 3: Representative Topologies for each case in the proof of the Proposition \ref{['prop:homologysetup']}

Theorems & Definitions (16)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Proposition 2.1
  • proof
  • Definition 3.1
  • ...and 6 more