Cost convexity controls the architecture of resistance-optimal networks
Abstract
A conducting network improves access to its interior only by spending material, so architectures compare only at equal budget. We redistribute conductance under a power-law resource law Cα=∑e ae weα and measure access by the reciprocal of the mean pairwise effective resistance. When the cost is strictly convex the allocation is unique and every admissible edge keeps positive conductance; at α=1 uniqueness survives but an edge can be priced out. The marginal worth of an edge is then its squared biharmonic distance, a Murray-type balance whose balanced quantity is global rather than junction-local. The same optimum scores any weighted network at its own budget, separating an allocation factor from a topological one. Under concave cost the marginal price of an incipient channel diverges, the feasible set is nonconvex, and conductance condenses onto sparse backbones. The uniform star and the uniform complete graph cross at αc(n)=log(n/2)/log(n−1), near which intermediate states beat both. Up to n=64 the best states found grow approximately linearly in n, so channels per node change much less than the active fraction does. Loops are governed instead by a collapse exponent α*(n), resolved to $0.005$ at every size: it rises from $0.555$ at n=8 to $0.675$ at n=64, and the collapse across it is abrupt, the support at n=64 falling from $133$ active edges to the $63$ of a spanning tree. A fixed exponent therefore changes character with size by crossing this boundary. One prescribed source and sink instead gives a loopless optimum.
Related
- How PNIPAM Microgel Architecture Controls Pickering Foam Formation
- Optimal quantum metrology with restricted controls
- Lie Algebra-Based Quantum Optimal Controls Interpolation
- Interaction order controls stochastic-resonance enhancement and redistribution in delayed multiplex neural networks
- Nanoscale Stray Fields from Micromagnets for Optimal Spin Qubit Architecture
BibTeX
@misc{tovar2026cost,
title = {{Cost convexity controls the architecture of resistance-optimal networks}},
author = {Hugo Tovar and Genaro Vázquez-Victorio and Enrique Hernández-Lemus and Alejandra Cervera and Daniel Pérez-Calixto},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03627},
eprint = {2610.03627},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03627},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Cost convexity controls the architecture of resistance-optimal networks
AU - Hugo Tovar
AU - Genaro Vázquez-Victorio
AU - Enrique Hernández-Lemus
AU - Alejandra Cervera
AU - Daniel Pérez-Calixto
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.03627
UR - https://arxiv.org/abs/2610.03627
AB - A conducting network improves access to its interior only by spending material, so architectures compare only at equal budget. We redistribute conductance under a power-law resource law C_α=∑_e a_e w_e^α and measure access by the reciprocal of the mean pairwise effective resistance. When the cost is strictly convex the allocation is unique and every admissible edge keeps positive conductance; at α=1 uniqueness survives but an edge can be priced out. The marginal worth of an edge is then its squared biharmonic distance, a Murray-type balance whose balanced quantity is global rather than junction-local. The same optimum scores any weighted network at its own budget, separating an allocation factor from a topological one. Under concave cost the marginal price of an incipient channel diverges, the feasible set is nonconvex, and conductance condenses onto sparse backbones. The uniform star and the uniform complete graph cross at α_c(n)=log(n/2)/log(n−1), near which intermediate states beat both. Up to n=64 the best states found grow approximately linearly in n, so channels per node change much less than the active fraction does. Loops are governed instead by a collapse exponent α^*(n), resolved to $0.005$ at every size: it rises from $0.555$ at n=8 to $0.675$ at n=64, and the collapse across it is abrupt, the support at n=64 falling from $133$ active edges to the $63$ of a spanning tree. A fixed exponent therefore changes character with size by crossing this boundary. One prescribed source and sink instead gives a loopless optimum.
ER -