## Department Colloquium

### Friday, November 20th, 2020

**Time:** 2:30 p.m. **Place:** Online (via Zoom)

**Speaker:** Nicolas Fraiman (University of North Carolina)

**Title:** Stochastic Recursions on Random Graphs.

**Abstract: ** We study a family of Markov processes on directed graphs where the values at each vertex are inﬂuenced by the values of its inbound neighbors and by independent ﬂuctuations either on the vertices themselves or on the edges connecting them to their inbound neighbors. Typical examples include PageRank and other information propagation processes. Assuming a stationary distribution exists for this Markov chain, our goal is to characterize the marginal distribution of a uniformly chosen vertex in the graph. In order to obtain a meaningful characterization, we assume that the underlying graph is either a directed conﬁguration graph or an inhomogeneous random digraph, both of which are known to converge, in the local weak sense, to a marked Galton-Watson process. We prove that the stationary distribution on the graph converges in a Wasserstein metric to a function of i.i.d. copies of the special endogenous solution to a branching distributional ﬁxed-point equation. This is joint work with Mariana Olvera-Cravioto and Tzu-Chi Lin.

**Nicolas Fraiman** is an Assistant Professor in the Department of Statistics and Operations Research at the University of North Carolina at Chapel Hill. He obtained his Ph.D. from McGill University in 2013. He was a Postdoctoral Fellow at the University of Pennsylvania and Harvard University. He works on the probabilistic analysis of random structures, stochastic dynamics, randomized algorithms and combinatorial statistics.