  Despite substantial progress in non-equilibrium physics, steady-state (s.s.)
probabilities remain intractable to analysis. For a Markov process, s.s.
probabilities can be expressed in terms of transition rates using the
Matrix-Tree theorem (MTT) in the graph-based linear framework. The MTT reveals
that, away from equilibrium, s.s. probabilities become globally dependent on
all rates, with expressions growing exponentially in the system size. This
overwhelming complexity and lack of thermodynamic interpretation have greatly
impeded analysis. Here, we show that s.s. probabilities are proportional to the
average of $\exp(-S(P))$, where $S(P)$ is the entropy generated along minimal
paths, $P$, in the graph, and the average is taken over a probability
distribution on spanning trees. Assuming Arrhenius rates, this "arboreal"
distribution becomes Boltzmann-like, with the energy of a tree being its total
edge barrier energy. This reformulation offers a thermodynamic interpretation
that smoothly generalises equilibrium statistical mechanics and reorganises the
expression complexity: the number of distinct minimal-path entropies depends on
the entropy production index, a new graph-theoretic measure of non-equilibrium
complexity, not on graph size. We demonstrate the power of this reformulation
by extending Hopfield's analysis of discrimination by kinetic proofreading to
any graph with index 1. We derive a general formula for the error ratio and use
it to show that local energy dissipation can yield optimal discrimination
through global synergy.
