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A discrete Benamou-Brenier formulation of Optimal Transport on graphs

Kieran Morris, Oliver Johnson

TL;DR

This work develops a discrete transport framework on graphs by introducing a transport equation for triples $(f,v,g)$ that couple vertex and edge quantities, and it derives a Benamou–Brenier-type formulation for the Wasserstein-1 distance. By first solving the tree case via tail distributions and then extending to general graphs through a reduced formulation and incidence-inversion arguments, it proves that $W_1$ on graphs can be expressed as an infimum of a kinetic-energy-like functional over time-dependent edge flows. The results show that minimizing $(v,g)$ yields constant-speed $W_1$ geodesics and that convex interpolations along these minimizing flows reproduce $W_1$ paths, providing a unifying discrete-geometric perspective and potential computational tools for discrete optimal transport on networks. This framework connects to Beckmann-type formulations and generalizes existing discrete OT approaches, enabling exact characterizations of geodesics and efficient interpolations on graphs.

Abstract

We propose a discrete transport equation on graphs which connects distributions on both vertices and edges. We then derive a discrete analogue of the Benamou-Brenier formulation for Wasserstein-$1$ distance on a graph and as a result classify all $W_1$ geodesics on graphs.

A discrete Benamou-Brenier formulation of Optimal Transport on graphs

TL;DR

This work develops a discrete transport framework on graphs by introducing a transport equation for triples that couple vertex and edge quantities, and it derives a Benamou–Brenier-type formulation for the Wasserstein-1 distance. By first solving the tree case via tail distributions and then extending to general graphs through a reduced formulation and incidence-inversion arguments, it proves that on graphs can be expressed as an infimum of a kinetic-energy-like functional over time-dependent edge flows. The results show that minimizing yields constant-speed geodesics and that convex interpolations along these minimizing flows reproduce paths, providing a unifying discrete-geometric perspective and potential computational tools for discrete optimal transport on networks. This framework connects to Beckmann-type formulations and generalizes existing discrete OT approaches, enabling exact characterizations of geodesics and efficient interpolations on graphs.

Abstract

We propose a discrete transport equation on graphs which connects distributions on both vertices and edges. We then derive a discrete analogue of the Benamou-Brenier formulation for Wasserstein- distance on a graph and as a result classify all geodesics on graphs.
Paper Structure (15 sections, 13 theorems, 65 equations)

This paper contains 15 sections, 13 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Notation and the Discrete Transport Equation
  3. Matrices, Operators, Equations and Integrals
  4. The Discrete Transport Equation
  5. Main Results
  6. Benamou-Brenier on the Vertices of a Tree
  7. The Reduced Formulation on a Graph
  8. Benamou-Brenier on the Vertices of a graph
  9. Conclusion
  10. Proof of Lemma \ref{['lem: const']} for $q \geq 1$
  11. Proof that Convex Interpolation is a Geodesic on a Tree
  12. Proof of Theorem \ref{['thm: benamoutree']} for $q \geq 1$
  13. Proof of Lemma \ref{['lem: const2']}
  14. Proof of Theorem \ref{['thm: benamou']}
  15. Supplementary Results

Key Result

Proposition 1

Given two distributions $f(0),f(1) \in \mathcal{P}(\mathbb{R}^d)$, we can then express where $(f_t,v_t)$ are constrained by the transport equation - also known as the continuity equation: where $\nabla \cdot$ is the divergence operator, (see benamou2000computational).

Theorems & Definitions (45)

  • Definition 1: Kantorovich Formulation of optimal transport
  • Proposition 1: Benamou-Brenier Formulation
  • Example 1
  • Definition 3: Incidence Matrix and Arrow Shorthand
  • Definition 4: Floor and Ceiling Notation
  • Definition 5: The Gradient and Divergence Operator
  • Definition 6: Discrete Transport Equation
  • Example 2
  • Definition 7: Integral Formulation
  • Definition 8: Tails of Distributions
  • ...and 35 more