Rigorous Continuum Limit for the Discrete Network Formation Problem

TL;DR

This paper establishes a rigorous mathematical connection between a discrete network formation model inspired by biological transport networks and its continuum limit, using $ ext{Gamma}$-convergence in a 2D setting.

Contribution

It provides the first rigorous proof of the continuum limit for a discrete network formation model constrained by a linear system, extending understanding of such models in 2D.

Findings

Proves $ ext{Gamma}$-convergence of the discrete energy to a continuum functional

Shows the continuum limit as the number of nodes increases and edge lengths decrease

Provides a mathematical foundation for modeling biological transport networks

Abstract

Motivated by recent physics papers describing the formation of biological transport networks we study a discrete model proposed by Hu and Cai consisting of an energy consumption function constrained by a linear system on a graph. For the spatially two-dimensional rectangular setting we prove the rigorous continuum limit of the constrained energy functional as the number of nodes of the underlying graph tends to infinity and the edge lengths shrink to zero uniformly. The proof is based on reformulating the discrete energy functional as a sequence of integral functionals and proving their $\Gamma$-converge towards a continuum energy functional.