Relaxational Dynamics and the Constrained Free Energy of an Oriented, Weighted Exponential Random Graph Model

This paper develops the statistical mechanics of relaxational dynamics for a valued, oriented ERGM-type Hamiltonian: continuous edge weights under reflected Langevin dynamics coupled to discrete edge orientations, with degree and orientation-frustration interactions. The constrained free energy of the mean edge weight is computed exactly in the local and orientation sectors and to controlled cumulant order in the interacting ones; a three-line Dirichlet-form argument proves that the order-parameter mobility is the bare kinetic coefficient, exactly and at finite size. A replica-symmetric cavity solution, extended to the full orientation-coupling sector, reproduces every simulated observable within statistical error and locates first-order coexistence sixfold more accurately than perturbative landscapes. The program is extended to projected observables, irregular degrees, metastability (validated against Eyring–Kramers), continuum interfaces, two-temperature steady states, and cavity-calibrated inference, and the gradient structure itself is derived via Mori–Zwanzig bath elimination. Finally, deforming the orientations into continuous flow directions preserves every exact identity while breaking the Markovian closures until scale separation is restored — separating the paper's theorems from its scale-separation phenomena within a single model family.