The thermodynamic cost of staying connected: a maintenance-threshold model for the failure sequence of sustained biological networks

Network neuroscience explains connectome topology as a trade-off between wiring cost and topological efficiency. That is an account of how a network forms; it contains no model of failure and predicts no failure sequence. This paper proposes that sustaining a coupling between two regions requires a minimum continuous power, with a nonzero standing term paid independently of traffic, and that crossing below this threshold — rather than the death of the elements themselves — is the primary failure event. Three results are derived, with full proofs in Appendix A: 1. A critical node budget exists below which at least one coupling is underfunded, set by the standing cost. Selective failure of the most expensive edges precedes any measurable regional metabolic deficit. 2. Failure proceeds from the highest-degree nodes exactly when the product of two measurable scaling exponents (αγ) falls below one. 3. Coupling decays along closed-form trajectories with a computable failure time, giving functional disconnection while cells remain anatomically intact. Under an explicit rebuilding-cost asymmetry the loss is hysteretic, so restoring supply does not restore the network. No new data are collected. The central prediction is conditional on an exponent product estimable from existing metabolic and resting-state imaging; a fitted value at or above one falsifies it. The paper states plainly that this condition is not established here, and discusses why the narrow dynamic range of regional glucose metabolism in healthy cohorts makes the exponent weakly identified in normative data. This version supersedes the previous deposit: the abstract has been shortened, a typographical error corrected, and the manuscript reformatted for journal submission. No scientific content was changed. Status: under review at the Journal of the Royal Society Interface (manuscript ID rsif-2026-0958, submitted 19 August 2026).