Non-Equilibrium Steady State & Flux Cycles — A Markov jump process on a network reaches a NESS when detailed balance is broken by thermodynamic forces (affinities) A = ln(k₊/k₋). The Schnakenberg cycle decomposition identifies independent cycles — each carries a net current Jᵢ, generating entropy at rate σ = Σᵢ Jᵢ·Aᵢ ≥ 0 (Clausius inequality). At equilibrium (A=0): all fluxes vanish, σ=0. Arrow widths show net probability flux through each edge of the 3-state cycle.