The Probabilistic Structure of Discrete Agent-Based Models

TL;DR

This paper formalizes agent-based models as random walks on regular graphs, revealing symmetries that enable systematic, lossless reduction of the model's state space while preserving its Markovian properties.

Contribution

It introduces a formal framework relating ABMs to graph automorphisms, allowing for symmetry-based state space reduction without loss of Markovian dynamics.

Findings

ABMs can be represented as random walks on regular graphs.

Graph automorphisms reveal symmetries that simplify ABMs.

Symmetry-based reduction preserves the Markov property.

Abstract

This paper describes a formalization of agent-based models (ABMs) as random walks on regular graphs and relates the symmetry group of those graphs to a coarse-graining of the ABM that is still Markovian. An ABM in which $N$ agents can be in $\delta$ different states leads to a Markov chain with $\delta^N$ states. In ABMs with a sequential update scheme by which one agent is chosen to update its state at a time, transitions are only allowed between system configurations that differ with respect to a single agent. This characterizes ABMs as random walks on regular graphs. The non-trivial automorphisms of those graphs make visible the dynamical symmetries that an ABM gives rise to because sets of micro configurations can be interchanged without changing the probability structure of the random walk. This allows for a systematic loss-less reduction of the state space of the model.