Nonequilibrium systems are supposed to be complicated. Drive a system away from thermal equilibrium and you lose the powerful tools — partition functions, free energies, fluctuation-dissipation relations — that make equilibrium tractable. The steady state of a driven system is generally a mess: nonlinear dependencies, broken detailed balance, currents that cycle without settling.
Bebon and Speck found something clean hiding in the mess.
Take any continuous-time Markov network in steady state. Perturb it along a single edge — change one transition rate. Watch how the probabilities of different states respond. The probabilities of any two states are linearly related to each other. Not approximately. Exactly. Arbitrarily far from equilibrium.
The proof uses spanning trees. The steady-state probability of any state in a Markov network can be written as a sum over directed spanning trees rooted at that state. When you perturb a single edge rate, these tree polynomials change in a specific way: they share a common factor that depends on the perturbation parameter. Dividing out that common factor reveals the linear relationship. The spanning tree structure imposes a geometric constraint on how state probabilities can co-vary.
This extends beyond probabilities. Currents — the net flow between states — and generic counting observables also satisfy mutual linearity. If you measure any two quantities that can be expressed as sums over states, their responses to a single-edge perturbation trace out a line, not a curve.
The result is stronger than it looks. Most exact results in nonequilibrium statistical mechanics require specific models or specific limits. This one requires nothing except the Markov property and a single perturbed edge. It holds for any graph topology, any rate values, any distance from equilibrium. The constraint is topological, not dynamical.
What it means practically: if you can measure how two observables respond to a perturbation, and the response isn't linear, then either the system isn't Markovian or the perturbation touches more than one edge. The linearity test becomes a diagnostic for hidden structure.
Far from equilibrium, where nearly everything is complicated, the spanning trees still hold things in line.