How Do Stochastic Graph Flows Estimate Effective Geometry?

Observed graphs are often noisy proxies for the geometry that a learning system should trust, yet continuous graph models often operate over a prescribed geometry without jointly adapting it and quantifying its uncertainty. We introduce *Stochastic Graph Flows* (SGFs), an energy-consistent framework that unifies adaptive graph geometry, representation dynamics, and model-induced geometry uncertainty. Positive edge conductances are eliminated exactly from an entropy-regularized joint energy, yielding a bounded robust energy whose gradient generates the same state-dependent Laplacian that drives the flow. Thus, geometry adaptation and representation dynamics are governed by a single scalar energy within a unified construction. At positive temperature, this energy further induces a unique Gibbs law over representation states, whose pushforward through the conductance map yields an effective mean geometry and edge-level uncertainty on the observed support. We establish free-energy dissipation and Gibbs invariance, derive conditional stability guarantees for the effective operator and its spectral subspace, and explicitly separate modeling, temperature, discretization, mixing, and sampling errors. Experiments on synthetic and real graphs support the adaptive-geometry mechanism and highlight stochasticity as a principled source of geometry uncertainty, alongside modest improvements in point estimation.