A Single--Index Theory of Optimal Branching: Murray Laws, Gilbert Networks, and Young--Herring Junctions
Justin Bennett
TL;DR
This work introduces a minimal quadratic, scale-free ledger for branched networks and proves that Murray-type flux–radius laws, Gilbert-type concave transport costs, and Young–Herring junction balances are different faces of the same structure governed by a single dimensionless index $\chi = \frac{m}{m+p}$. Through an Euler homogeneity argument, any admissible ledger on a scale-free cone reduces to the two-term form $P(Q,r)=a\frac{Q^2}{r^{p}}+b r^{m}$, from which a generalized Murray law, a radius-weighted Young–Herring vector balance, and a flux-only concave cost $\propto |Q|^{\beta}$ with $\beta=\frac{2m}{m+p}$ follow. The theory provides a one-parameter dictionary linking observable exponents and angles: $\alpha=(m+p)/2$, $\beta=2m/(m+p)$, $\theta$ with $\cos(\theta/2)=2^{\frac{m-p}{m+p}}$, all governed by $\chi=\beta/2=1- rac{p}{2\alpha}=\frac{1}{2}(1+\log_2\cos(\tfrac{\theta}{2}))$. A rigidity theorem shows that exact power-law behavior of branchwise minimizers and flux-only costs enforces the two-term form, making Murray, Gilbert, and Young–Herring descriptions co-emergent. The index $\chi$ thus provides a falsifiable, geometry-to-thermodynamics bridge for scale-free branching in microfluidic, diffusive, and geophysical contexts, with concrete diagnostics offered for experimental tests.
Abstract
Murray-type flux-radius laws, Gilbert-type concave transport costs, and Young-Herring triple-junction angle balances are usually treated as separate theories. This work shows that, within a natural class of quadratic, scale-free ledgers for branched networks, all three are different faces of a single structure controlled by one dimensionless index chi. Each edge carries a flux Q, an effective radius r, and a per-length ledger P(Q,r) encoding transport dissipation and structural burden. Under locality, evenness in Q, linear-response (quadratic) dependence, and an exact homogeneity ansatz in (Q,r), any admissible ledger reduces in the scale-free regime to the two-term form P(Q,r) = a Q^2 r^{-p} + b r^m. Local optimality then implies simultaneously: (i) a flux-radius power law with generalized Murray closures at degree-3 nodes; (ii) a Young-Herring-type vector balance with radius weights r^m and a fixed symmetric Y-junction angle; and (iii) an effective flux-only cost of Gilbert/branched-transport type with exponent beta. The exponents alpha and beta, the symmetric angle, and the split between transport and structural cost are all set by chi = m/(m+p) = beta/2. A rigidity theorem shows conversely that any quadratic ledger that yields power-law optimal radii and power-law flux-only cost on an open scaling cone must belong to this two-term family and obey the same Murray-Gilbert-Young dictionary. Examples for Poiseuille, diffusive, and geophysical trees illustrate how chi can be inferred from geometry and used as a falsifiable order parameter for scale-free branching architectures.