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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2507.15829 |
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| _version_ | 1866915745314635776 |
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| author | Alves, Nuno J. Haskovec, Jan |
| author_facet | Alves, Nuno J. Haskovec, Jan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_15829 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rigorous dense graph limit of a model for biological transportation networks Alves, Nuno J. Haskovec, Jan Optimization and Control 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. |
| title | Rigorous dense graph limit of a model for biological transportation networks |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2507.15829 |