# Stochastic versus deterministic consensus dynamics on graphs

## TL;DR

This paper compares stochastic and deterministic opinion formation models on graphs, analyzing how graph structure influences long-term consensus emergence and convergence rates through theoretical and numerical methods.

## Contribution

It provides a comprehensive analysis of both models' convergence conditions and highlights key differences in their dynamics based on graph topology.

## Key findings

- Necessary and sufficient conditions for consensus in both models
- Distinct convergence rates between stochastic and deterministic models
- Graph structure critically influences long-term opinion dynamics

## Abstract

We study two agent based models of opinion formation - one stochastic in nature and one deterministic. Both models are defined in terms of an underlying graph; we study how the structure of the graph affects the long time behavior of the models in all possible cases of graph topology. We are especially interested in the emergence of a consensus among the agents and provide a condition on the graph that is necessary and sufficient for convergence to a consensus in both models. This investigation reveals several contrasts between the models - notably the convergence rates - which are explored through analytical arguments and several numerical experiments.