Approximate lumpability for Markovian agent-based models using local symmetries

TL;DR

This paper introduces a local symmetry-based lumping method for Markovian agent-based models, enabling approximate model reduction on large asymmetric graphs by quantifying the error with Kullback-Leibler divergence.

Contribution

It proposes a novel local symmetry approach for lumping in Markovian models, extending beyond automorphism-based methods to handle asymmetric graphs.

Findings

Approximate lumpability improves with local symmetry measures.

The approximation error decreases monotonically.

Connections to graph fibrations are explored.

Abstract

We study a Markovian agent-based model (MABM) in this paper. Each agent is endowed with a local state that changes over time as the agent interacts with its neighbours. The neighbourhood structure is given by a graph. In a recent paper [Simon et al. 2011], the authors used the automorphisms of the underlying graph to generate a lumpable partition of the joint state space ensuring Markovianness of the lumped process for binary dynamics. However, many large random graphs tend to become asymmetric rendering the automorphism-based lumping approach ineffective as a tool of model reduction. In order to mitigate this problem, we propose a lumping method based on a notion of local symmetry, which compares only local neighbourhoods of vertices. Since local symmetry only ensures approximate lumpability, we quantify the approximation error by means of Kullback-Leibler divergence rate between the original Markov chain and a lifted Markov chain. We prove the approximation error decreases monotonically. The connections to fibrations of graphs are also discussed.