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

Justin Bennett

TL;DR

This work reframes Murray’s vascular branching principle as an entropy‑per‑information‑cost (EPIC) extremum, pricing structural upkeep in units of $\mathrm{J/bit}$ and unifying energy and morphology through a calibrated information ledger. By deriving a node‑level variational framework, it yields a generalized Murray scaling $Q \propto r^{\alpha}$ with $\alpha=(m+4)/2$, a tariff‑weighted vector closure that fixes junction angles, and a concave, flow‑cost edge function with exponent $\gamma=2m/(m+n)$, all under a growth‑cone bound on the rate of reliably maintained structure. The theory predicts Snell‑like refraction of optimal routes in heterogeneous price fields via a dimensionless routing index $\mathfrak n(x)$, and it delivers falsifiable geometric targets that are tested on HRF retinal bifurcations. Empirical results show sharp vector closure with a held‑out median residual $R(m)\approx0.232$ and strong rejection of nulls, establishing EPIC as a unit‑bearing design principle with broad applicability to biological and engineered branched networks.

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.

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

TL;DR

This work reframes Murray’s vascular branching principle as an entropy‑per‑information‑cost (EPIC) extremum, pricing structural upkeep in units of and unifying energy and morphology through a calibrated information ledger. By deriving a node‑level variational framework, it yields a generalized Murray scaling with , a tariff‑weighted vector closure that fixes junction angles, and a concave, flow‑cost edge function with exponent , all under a growth‑cone bound on the rate of reliably maintained structure. The theory predicts Snell‑like refraction of optimal routes in heterogeneous price fields via a dimensionless routing index , and it delivers falsifiable geometric targets that are tested on HRF retinal bifurcations. Empirical results show sharp vector closure with a held‑out median residual and strong rejection of nulls, establishing EPIC as a unit‑bearing design principle with broad applicability to biological and engineered branched networks.

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.

Paper Structure

This paper contains 21 sections, 101 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. EPIC ledger: tariffs, structural load, and information throughput
  3. Classical benchmark: pumping versus upkeep at fixed daughter demands
  4. EPIC extremum at a node
  5. Domain program to node objective
  6. Branch invariant and weighted Murray (Poiseuille)
  7. Angle selection by junction translation
  8. Generalizations and rheology
  9. Empirical tests
  10. Analysis methods
  11. Results
  12. Conclusions and future steps
  13. Growth‑cone bound: rate limits for reliable structure
  14. Design corollaries: equipartition, concavity, refraction
  15. Conclusion
  16. ...and 6 more sections

Figures (3)

  • Figure 1: From image to analyzable geometry (HRF). (a) Original color fundus image (HRF, file 13_g). (b) Binary vessel map from the normalized green channel after multiscale vesselness filtering, Otsu thresholding, and light morphological cleanup. (c) Skeletonized vessel graph with per–junction tariffs and R‑band (interpretive) classes overlaid. All images are drawn from the High--Resolution Fundus (HRF) Image Database (Pattern Recognition Lab, FAU), distributed under a Creative Commons Attribution 4.0 International (CC BY 4.0) license HRF_web; see Refs. Budai2013IJBIOdstrcilik2013IETIP for dataset and label details.
  • Figure 2: Held‑out closure residuals on HRF (sato+otsu). Histogram of $R(m)$ over HRF junctions; the vertical line marks the observed median with a 95% bootstrap CI ($B{=}5000$) EfronTibshirani1993. Colored outlines show structure‑preserving null controls. Lower values indicate tighter vector closure; shaded bands mark preregistered R‑band (interpretive) thresholds.
  • Figure 3: Positive control: recovery of $m$ from analytic junctions. Estimated upkeep exponents $m_{\mathrm{est}}$ versus ground‑truth $m_{\mathrm{true}}$ for $N=1266$ synthetic Y‑junctions with directional and radius noise and a $20\%$ parent‑mislabel fraction. Blue points denote correctly labelled nodes; orange points denote mislabelled nodes. The grey line shows the identity $m_{\mathrm{est}}=m_{\mathrm{true}}$.