Convergence Properties of Dynamic Processes on Graphs
Timothy Horscroft
TL;DR
This work analyzes convergence properties of two dynamic graph processes modeling opinion dynamics: the load-balancing model, where adjacent node values move toward each other by one unit, and the synchronous maximum model, where nodes adopt the maximum among their neighbors. It develops a rigorous framework based on graph valuations, Markov chains, and SCC decompositions to prove absorbing-state convergence, characterizes the final states via division of sums and gcd-based cycle analysis, and derives bounds on convergence time and period. The study combines theoretical proofs with extensive experiments on Erdős–Rényi, Barabási–Albert, and real-world social-network graphs to validate worst-case and average-case behaviours, including special valuations (binomial and gambler’s ruin) that demonstrate tightness or near-tightness of bounds. The results provide a structured understanding of how network structure governs convergence speed and cycles, with implications for modeling information diffusion and consensus formation in social and technological networks. The work also outlines future directions, including tightening bounds for directed graphs, formalizing worst-case scenarios, and extending the models to asynchronous settings and broader graph families.
Abstract
Theoretical computer science plays an important role in the understanding of social networks and their properties. We can model information rippling throughout social networks, or the opinions of social media users for example, using graph theory and Markov chains. In this thesis, we model social networks as graphs, and consider two such processes: 1. Nodes talk to other nodes and find middle ground, causing their opinions to come closer to consensus (the load balancing model) 2. All nodes take the maximum value of their neighbours in lockstep (the synchronous maximum model) We study the convergence behaviours of each process, such as the eventual state of the graph, the convergence time and the period. We provide proofs of the eventual states and periods for each of the above models, and theoretical bounds for the worst case convergence times. We verify these with experiments, and explore further questions such as the average case convergence time of various special classes of graphs, or the convergence times when the model is altered slightly.