Murray's Law as an Entropy-per-Information-Cost Extremum

TL;DR

This paper introduces a new theoretical framework for transport network design based on an entropy-per-information-cost extremum, deriving geometric and energetic principles validated by retinal bifurcation data.

Contribution

It formulates a novel entropy-based optimization principle for transport networks, linking Murray's law with information costs and providing testable geometric predictions.

Findings

Derived a generalized Murray scaling law for flow and radius.

Predicted bifurcation angles and power partitioning consistent with biological data.

Validated the model with retinal bifurcation measurements, showing high accuracy.

Abstract

Transport networks must balance viscous pumping losses with the energetic cost of maintaining an operative architecture. This paper formulates that trade-off as an entropy-per-information-cost (EPIC) extremum that prices structural upkeep in calibrated units (joules per bit). An upkeep law r^m distinguishes volume-priced (m = 2) from surface-priced (m = 1) maintenance. In laminar Poiseuille flow, stationarity yields (i) a generalized Murray scaling Q proportional to r^alpha with alpha = (m + 4)/2; (ii) a tariff-weighted vector balance that fixes bifurcation geometry and predicts near-symmetric daughter openings of about 75 degrees for m = 2 and about 97 degrees for m = 1; and (iii) a universal partition of power between pumping and upkeep. Eliminating radii gives a strictly concave flux cost proportional to Q^gamma with gamma = 2m/(m + 4), favoring mergers and deep tree hierarchies, and defines a routing index that induces Snell-like refraction of optimal paths across spatial tariff contrasts. A preregistered, held-out test on retinal bifurcations from the High-Resolution Fundus dataset (N = 19,126) shows sharp vector closure: the median residual is R = 0.232 with a nonparametric 95 percent bootstrap interval [0.229, 0.236], 91 percent of junctions fall under the pre-specified strict threshold, and structure-preserving nulls shift decisively to larger residuals. These results render classical branching relations explicitly unit-bearing (J/bit) and provide falsifiable geometric targets and quantitative design rules for transport networks.