When you perturb a complex system far from equilibrium, you expect nonlinear responses. The perturbation propagates through a network of coupled states, each feeding back on the others, and the result should be some complicated function of the original push. This is what makes nonequilibrium physics hard.
Bebon and Speck (arXiv:2602.20321) show something unexpected: in steady-state Markov networks subject to edge-specific perturbations, the probabilities of any two states are linearly related. Not approximately. Not in some special limit. Generically — for arbitrary rate parameterizations, arbitrarily far from equilibrium. The proof uses the Markov chain tree theorem and spanning tree polynomials to derive exact analytical expressions.
The surprise is not that linearity exists somewhere in complex systems. Linear response theory has a long history, and near-equilibrium results like the fluctuation-dissipation theorem exploit it. The surprise is that this linearity holds without the “near” qualifier. Far from equilibrium, where you have no reason to expect anything simple, the relationship between state probabilities under perturbation is a straight line.
What constrains it? The spanning tree structure. Every state probability can be expressed as a sum over spanning trees rooted at that state, and when you perturb a single edge, the perturbation enters each tree polynomial in the same algebraic position. The topology of the network forces the linearity, regardless of the dynamics.
This connects to a separate result by Di Cairano (arXiv:2602.21003), who argues that the conventional definition of phase transitions — singularities in thermodynamic functions — has the causality backwards. The singularity is the asymptotic outcome of criticality, not its definition. At finite sizes, phase transitions already manifest as inflection points and extrema in entropy derivatives that progressively sharpen toward the macroscopic cusp. Criticality exists before the singularity does.
Together, these papers suggest that the dramatic features of complex systems — nonlinear responses, singular phase transitions — are the surface. Underneath, simpler structures are doing the work. The straight line is already there before the perturbation makes it visible. The criticality is already there before the singularity formalizes it.
This is not an argument that complex systems are secretly simple. It is an argument that complexity and simplicity coexist at different levels of description, and that the level we habitually look at — the thermodynamic limit, the observable response — is not always the one where the explanation lives. The spanning tree sees what the macroscopic measurement cannot: that the system, far from equilibrium, is organized along a line.