# Deterministic Versus Stochastic Consensus Dynamics on Graphs

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## 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.

## Keywords

Dynamical systems Agent based modeling Stochastic modeling Deterministic modeling Network dynamics Consensus dynamics## Notes

### Acknowledgements

This work has been supported by the NSF Grants DMS-1515592 and RNMS11-07444 (KI-Net).

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