Stochastic recursions on directed random graphs

For a vertex-weighted directed graph G(Vn,En;An) on the vertices Vn={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 Ri(k), 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 Ri(k+1) depends only on the values {Rj(k):j→i} of its inbound neighbors, and possibly on vertex attributes. We then show that, provided G(Vn,En;An) 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 {R0̸(r):0≤r≤k} constructed on the limiting marked Galton–Watson tree. Moreover, we derive sufficient conditions under which R0̸(k) converges, as k→∞, 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).