Modeling Dynamic Processes on Complex Networks

The dissertation studies dynamic processes driven by neighbor-to-neighbor interactions defined on graphs. The first portion establishes a mathematical framework for Markov processes operating on directed graphs, where the values at each vertex are influenced by the values of their inbound neighbors and, potentially, by independent noises. These noises may originate either within the vertices themselves or along the inbound edges. Typical examples include the Generalized DeGroot model, the PageRank algorithm, and various information diffusion processes. Assuming the existence of a stationary distribution for this Markov chain, the primary goal is to characterize the marginal distribution of a uniformly chosen vertex in the graph. For a meaningful characterization, the study assumes that the underlying graph converges in the local weak sense to a marked Galton-Watson tree, such as a directed configuration graph or an inhomogeneous random digraph. We then show that the stationary distribution we study on the graph converges in some Wasserstein metric to a function of i.i.d. copies of the attracting endogenous solution to a branching distributional fixed-point equation. Given the heightened interest in political polarization, the second segment aims to develop models that capture how opinions evolve within social networks. To study how people process information, we propose an opinion model built upon the well-known DeGroot and Friedkin-Johnsen framework by introducing additional complexity, allowing for the inclusion of various vertex attributes and the utilization of vertex-dependent media signals. This integration provides a versatile modeling tool to describe social phenomena, including the framing effect, selective exposure, and the impact of social bots, all of which make contribution to polarization. Our analysis explores the model within directed random graphs, which have the capacity to approximate real-world networks with degree inhomogeneity and community structure. The main findings of this research offer insights into the stationary distribution of opinions within the network, providing metrics for conditional means and variances across different communities to better understand opinion dynamics.