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Rigorous dense graph limit of a model for biological transportation networks

Nuno J. Alves, Jan Haskovec

TL;DR

This work rigorously derives the dense graph (graphon) limit of a discrete biological transport network model with pressure-driven flows and a convex energy combining pumping and metabolic costs under Kirchhoff mass conservation. By introducing a regularized, rescaled semi-discrete formulation and proving $\Gamma$-convergence to a continuum energy $\mathcal{F}$ defined on a graphon, the authors establish convergence of global minimizers from the discrete to the continuum problem via a nonlocal Poisson-type equation for the pressure. The analysis hinges on a careful treatment of the Kirchhoff law, the Poisson equation, and the kinetic/metabolic energy components, including convexity, lower semicontinuity, and strong-weak convergence arguments. The results provide a rigorous mathematical foundation for continuum descriptions of biological transport networks emerging from dense, discrete structures, clarifying the role of connectivity, graphon structure (0-1 valued), and scaling in the limiting process.

Abstract

We rigorously derive the dense graph limit of a discrete model describing the formation of biological transportation networks. The discrete model, defined on undirected graphs with pressure-driven flows, incorporates a convex energy functional combining pumping and metabolic costs. It is constrained by a Kirchhoff law reflecting the local mass conservation. We first rescale and reformulate the discrete energy functional as an integral `semi-discrete' functional, where the Kirchhoff law transforms into a nonlocal elliptic integral equation. Assuming that the sequence of graphs is uniformly connected and that the limiting graphon is 0-1 valued, we prove two results: (1) rigorous Gamma-convergence of the sequence of the semi-discrete functionals to a continuum limit as the number of graph nodes and edges tends to infinity; (2) convergence of global minimizers of the discrete functionals to a global minimizer of the limiting continuum functional. Our results provide a rigorous mathematical foundation for the continuum description of biological transport structures emerging from discrete networks.

Rigorous dense graph limit of a model for biological transportation networks

TL;DR

This work rigorously derives the dense graph (graphon) limit of a discrete biological transport network model with pressure-driven flows and a convex energy combining pumping and metabolic costs under Kirchhoff mass conservation. By introducing a regularized, rescaled semi-discrete formulation and proving -convergence to a continuum energy defined on a graphon, the authors establish convergence of global minimizers from the discrete to the continuum problem via a nonlocal Poisson-type equation for the pressure. The analysis hinges on a careful treatment of the Kirchhoff law, the Poisson equation, and the kinetic/metabolic energy components, including convexity, lower semicontinuity, and strong-weak convergence arguments. The results provide a rigorous mathematical foundation for continuum descriptions of biological transport networks emerging from dense, discrete structures, clarifying the role of connectivity, graphon structure (0-1 valued), and scaling in the limiting process.

Abstract

We rigorously derive the dense graph limit of a discrete model describing the formation of biological transportation networks. The discrete model, defined on undirected graphs with pressure-driven flows, incorporates a convex energy functional combining pumping and metabolic costs. It is constrained by a Kirchhoff law reflecting the local mass conservation. We first rescale and reformulate the discrete energy functional as an integral `semi-discrete' functional, where the Kirchhoff law transforms into a nonlocal elliptic integral equation. Assuming that the sequence of graphs is uniformly connected and that the limiting graphon is 0-1 valued, we prove two results: (1) rigorous Gamma-convergence of the sequence of the semi-discrete functionals to a continuum limit as the number of graph nodes and edges tends to infinity; (2) convergence of global minimizers of the discrete functionals to a global minimizer of the limiting continuum functional. Our results provide a rigorous mathematical foundation for the continuum description of biological transport structures emerging from discrete networks.

Paper Structure

This paper contains 8 sections, 16 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Main assumptions
  4. Reformulation of the model and main results
  5. The Kirchhoff law and the Poisson equation
  6. The energy functionals
  7. Proof of Theorem \ref{['thm:1']}
  8. Proof of Theorem \ref{['thm:2']}

Key Result

Theorem 4.1

Fix $\gamma>1$, $\nu > 0$, $\lambda>0$, and $r>0$, and let the assumptions ass:S--ass:L2 hold. For any $N\in\mathbb{N}$ and $B\in \mathbb{B}^N_r$ we have $\mathcal{F}^N[\mathcal{Q}^N[B]] = F^N[B].$ Moreover, the sequence of functionals $(\mathcal{F}^N)_{N\in\mathbb{N}}$, given by Fn:N--eq:Poissonsc,

Theorems & Definitions (30)

  • Theorem 4.1
  • Theorem 4.2
  • Lemma 5.1
  • proof
  • Lemma 5.2
  • proof
  • Lemma 5.3
  • proof
  • Lemma 5.4
  • proof
  • ...and 20 more