  Stochastic dynamics on sparse graphs and disordered systems often lead to
complex behaviors characterized by heterogeneity in time and spatial scales,
slow relaxation, localization, and aging phenomena. The mathematical tools and
approximation techniques required to analyze these complex systems are still
under development, posing significant technical challenges and resulting in a
reliance on numerical simulations. We introduce a novel computational framework
for investigating the dynamics of sparse disordered systems with continuous
degrees of freedom. Starting with a graphical model representation of the
dynamic partition function for a system of linearly-coupled stochastic
differential equations, we use dynamic cavity equations on locally tree-like
factor graphs to approximate the stochastic measure. Here, cavity marginals are
identified with local functionals of single-site trajectories. Our primary
approximation involves a second-order truncation of a small-coupling expansion,
leading to a Gaussian form for the cavity marginals. For linear dynamics with
additive noise, this method yields a closed set of causal integro-differential
equations for cavity versions of one-time and two-time averages. These
equations provide an exact dynamical description within the local tree-like
approximation, retrieving classical results for the spectral density of sparse
random matrices. Global constraints, non-linear forces, and state-dependent
noise terms can be addressed using a self-consistent perturbative closure
technique. The resulting equations resemble those of dynamical mean-field
theory in the mode-coupling approximation used for fully-connected models.
However, due to their cavity formulation, the present method can also be
applied to ensembles of sparse random graphs and employed as a message-passing
algorithm on specific graph instances.
