Stochastic recursions on directed random graphs

For a vertex-weighted directed graph G(V-n, E-n; A(n)) on the vertices V-n = {1, 2, ... , n}, we study the distribution of a Markov chain {R(k) : k >= 0} on Rn such that the ith component of R(k), denoted R(k) i , corresponds to the value of the process on vertex i at time k. We focus on processes {R(k) : k > 0} where the value of R-j((k+1)) depends only on the values {R(k) i j : j -> i} of its inbound neighbors, and possibly on vertex attributes. We then show that, provided G(Vn, E-n; A(n)) converges in the local weak sense to a marked Galton-Watson process, the dynamics of the process for a uniformly chosen vertex in Vn can be coupled, for any fixed k, to a process {R ((r) )(& empty;) : 0 <= r <= k} constructed on the limiting marked Galton-Watson tree. Moreover, we derive sufficient conditions under which R (k) & empty; converges, as k -> infinity, to a random variable R* that can be characterized in terms of the attracting endogenous solution to a branching distributional fixed-point equation. Our framework can also be applied to processes {R(k) : k >= 0} whose only source of randomness comes from the realization of the graph G(Vn, En; An). (c) 2022 Elsevier B.V. All rights reserved.