Dynamical Cavity Method for Continuous-Time Dynamics on Sparse Random Graphs
This work introduces a continuous-time dynamical cavity method for analyzing stochastic dynamics on sparse random graphs with pairwise interactions. The method extends dynamical mean-field theory (DMFT) by providing exact path-level descriptions on trees and locally tree-like graphs, explicitly accounting for how edge directionality (directed, reciprocal, bidirected) affects dynamical closures. Ensemble averaging yields message-passing laws, and the framework clarifies the relationship between sparse and dense network descriptions.