Convergence Properties of Dynamic Processes on Graphs

TL;DR

This paper analyzes the convergence behavior of two social network models, providing theoretical bounds and experimental verification for their convergence times and eventual states.

Contribution

It introduces and studies two specific opinion dynamics models on graphs, offering new theoretical bounds and experimental insights into their convergence properties.

Findings

Proved convergence and periodicity for both models.

Established worst-case bounds for convergence times.

Validated results through experiments.

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.