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Approximate lumpability for Markovian agent-based models using local symmetries

Part of: Markov processes Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  01 October 2019

Wasiur R. KhudaBukhsh ,
Arnab Auddy ,
Yann Disser  and
Heinz Koeppl
Wasiur R. KhudaBukhsh*
Affiliation:
The Ohio State University
Arnab Auddy*
Affiliation:
Columbia University
Yann Disser*
Affiliation:
Technische Universität Darmstadt
Heinz Koeppl*
Affiliation:
Technische Universität Darmstadt
*
*Postal address: Mathematical Biosciences Institute, The Ohio State University, Jennings Hall, 3rd Floor, 1735 Neil Avenue, Columbus, Ohio 43210, USA. Email address: khudabukhsh.2@osu.edu
**Postal address: Department of Statistics, Columbia University, Room 1005 SSW, MC 4690, 1255 Amsterdam Avenue, New York, NY 10027, USA. Email address: arnab.auddy@columbia.edu
***Postal address: Department of Mathematics, Technische Universität Darmstadt, Dolivostrasse 15, 64293 Darmstadt, Germany. Email address: disser@mathematik.tu-darmstadt.de
****Postal address: Department of Electrical Engineering and Information Technology, Technische Universität Darmstadt, Rundeturmstrasse 12, 64283 Darmstadt, Germany. Email address: heinz.koeppl@bcs.tu-darmstadt.de
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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. Recently, Simon, Taylor, and Kiss [40] 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 the 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.

Keywords

Lumpabilityaggregationagent-based modelslocal symmetryfibration

MSC classification

Primary: 60J28: Applications of continuous-time Markov processes on discrete state spaces
Secondary: 60J22: Computational methods in Markov chains 60J27: Continuous-time Markov processes on discrete state spaces 65C50: Other computational problems in probability
Type
Research Papers
Information
Journal of Applied Probability , Volume 56 , Issue 3 , September 2019 , pp. 647 - 671
Copyright
© Applied Probability Trust 2019 

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