Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-21T13:43:33.007Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Angluin, D. (1980). Local and global properties in networks of processors (extended abstract). In Proceedings of the Twelfth Annual ACM Symposium on Theory of Computing (STOC ’80), pp. 82–93. ACM.CrossRefGoogle Scholar
Apers, S., Ticozzi, F. and Sarlette, A. (2017). Lifting Markov chains to mix faster: limits and opportunities. Available at arXiv:1705.08253.Google Scholar
Arvind, V., Köbler, J., Rattan, G. and Verbitsky, O. (2016). Graph isomorphism, color refinement, and compactness. Comput. Complexity 26, 627–685.CrossRefGoogle Scholar
Babai, L. (2016). Graph isomorphism in quasipolynomial time (extended abstract). In Proceedings of the Forty-Eighth Annual ACM Symposium on Theory of Computing (STOC ’16), pp. 684–697. ACM.CrossRefGoogle Scholar
Babai, L., Erdős, P. and Selkow, S. M. (1980). Random graph isomorphism. SIAM J. Comput . 9, 628–635.CrossRefGoogle Scholar
Banisch, S. (2016). Markov Chain Aggregation for Agent-Based Models. Springer.CrossRefGoogle Scholar
Berkholz, C., Bonsma, P. and Grohe, M. (2013). Tight lower and upper bounds for the complexity of canonical colour refinement. In Proceedings of the 21st Annual European Symposium on Algorithms (ESA 2013) (Lecture Notes in Computer Science 8125), pp. 145–156. Springer.CrossRefGoogle Scholar
Boldi, P., Lonati, V., Santini, M. and Vigna, S. (2006). Graph fibrations, graph isomorphism, and PageRank. RAIRO Theoret. Inform. Appl. 40, 227–253.CrossRefGoogle Scholar
Boldi, P. and Vigna, S. (2002). Fibrations of graphs. Discrete Math . 243, 21–66.CrossRefGoogle Scholar
Buchholz, P. (1994). Exact and ordinary lumpability in finite Markov chains. J. Appl. Prob. 31, 59–75.CrossRefGoogle Scholar
Buchholz, P. and Kemper, P. (2004). Kronecker based matrix representations for large Markov models. In Validation of Stochastic Systems (Lecture Notes in Computer Science 2925), pp. 256–295. Springer.CrossRefGoogle Scholar
Chatterjee, S. and Durrett, R. (2009). Contact processes on random graphs with power law degree distributions have critical value 0. Ann. Prob. 37, 2332–2356.CrossRefGoogle Scholar
Chen, F., Lovász, L. and Pak, I. (1999). Lifting Markov chains to speed up mixing. In Proceedings of the Thirty-First Annual ACM Symposium on Theory of Computing (STOC ’99), pp. 275–281. ACM, New York.Google Scholar
Courtois, P. J. (1977). Decomposability: Queueing and Computer System Applications. Academic Press, New York.Google Scholar
Deng, K., Mehta, P. G. and Meyn, S. P. (2011). Optimal Kullback–Leibler aggregation via spectral theory of Markov chains. IEEE Trans. Automat. Control 56, 2793–2808.CrossRefGoogle Scholar
Elbert Simões, J., Figueiredo, D. R. and Barbosa, V. C. (2016). Local symmetry in random graphs. Available at arXiv:1605.01758.Google Scholar
Feret, J., Henzinger, T., Koeppl, H. and Petrov, T. (2012). Lumpability abstractions of rule-based systems. Theoret. Comput. Sci. 431, 137–164.CrossRefGoogle Scholar
França, G. and Bento, J. (2017). Markov chain lifting and distributed ADMM. IEEE Signal Process. Lett . 24, 294–298.CrossRefGoogle Scholar
Ganguly, A., Petrov, T. and Koeppl, H. (2014). Markov chain aggregation and its applications to combinatorial reaction networks. J. Math. Biol. 69, 767–797.CrossRefGoogle ScholarPubMed
Geiger, B. C., Petrov, T., Kubin, G. and Koeppl, H. (2015). Optimal Kullback–Leibler aggregation via information bottleneck. IEEE Trans. Automat. Control 60, 1010–1022.CrossRefGoogle Scholar
Godsil, C. and Royle, G. F. (2013). Algebraic Graph Theory (Graduate Texts in Mathematics 207). Springer, New York.Google Scholar
Hemberg, M. and Barahona, M. (2008). A Dominated Coupling From The Past algorithm for the stochastic simulation of networks of biochemical reactions. BMC Systems Biology 2, 42.CrossRefGoogle ScholarPubMed
Hendrickx, J. M. (2014). Views in a graph: to which depth must equality be checked? IEEE Trans. Parallel Distrib. Systems 25, 1907–1912.CrossRefGoogle Scholar
Katehakis, M. N. and Smit, L. C. (2012). A successive lumping procedure for a class of Markov chains. Probab. Engrg Inform. Sci. 26, 483–508.CrossRefGoogle Scholar
Kemeny, J. G. and Snell, J. L. (1960). Finite Markov Chains. Van Nostrand, Princeton, NJ.Google Scholar
KhudaBukhsh, W. R., Rückert, J., Wulfheide, J., Hausheer, D. and Koeppl, H. (2016). Analysing and leveraging client heterogeneity in swarming-based live streaming. In 2016 IFIP Networking Conference (IFIP Networking) and Workshops, pp. 386–394. IEEE.CrossRefGoogle Scholar
Kim, J. H., Sudakov, B. and Vu, V. H. (2002). On the asymmetry of random regular graphs and random graphs. Random Structures Algorithms 21, 216–224.CrossRefGoogle Scholar
Kiss, I. Z., Miller, J. C. and Simon, P. L. (2017). Mathematics of Epidemics on Networks: From Exact to Approximate Models (Interdisciplinary Applied Mathematics 46). Springer.Google Scholar
Kratochvíl, J., Proskurowski, A. and Telle, J. A. (1998). Complexity of graph covering problems. Nordic J. Comput . 5, 173–195.Google Scholar
Kuntz, J., Thomas, P., Stan, G.-B. and Barahona, M. (2017). Rigorous bounds on the stationary distributions of the chemical master equation via mathematical programming. Available at arXiv:1702.05468.Google Scholar
Łuczak, T. (1988). The automorphism group of random graphs with a given number of edges. Math. Proc. Cambridge Phil. Soc. 104, 441–449.CrossRefGoogle Scholar
McKay, B. D. and Wormald, N. C. (1984). Automorphisms of random graphs with specified vertices. Combinatorica 4, 325–338.CrossRefGoogle Scholar
Nijholt, E., Rink, B. and Sanders, J. (2016). Graph fibrations and symmetries of network dynamics. J. Diff. Equations 261, 4861–4896.CrossRefGoogle Scholar
Nodelman, U., Shelton, C. R. and Koller, D. (2002). Continuous time Bayesian networks. In Proceedings of the Eighteenth Conference on Uncertainty in Artificial Intelligence, pp. 378–387. Morgan Kaufmann.Google Scholar
Norris, N. (1995). Universal covers of graphs: isomorphism to depth ${n-1}$ implies isomorphism to all depths. Discrete Appl. Math . 56, 61–74.CrossRefGoogle Scholar
Riordan, O. and Wormald, N. (2010). The diameter of sparse random graphs. Combin. Probab. Comput. 19, 835–926.CrossRefGoogle Scholar
Rubino, G. and Sericola, B. (1989). On weak lumpability in Markov chains. J. Appl. Prob. 26, 446–457.CrossRefGoogle Scholar
Rubino, G. and Sericola, B. (1993). A finite characterization of weak lumpable Markov processes, II: The continuous time case. Stochastic Process. Appl. 45, 115–125.CrossRefGoogle Scholar
Simon, P. L. and Kiss, I. Z. (2012). From exact stochastic to mean-field ODE models: a new approach to prove convergence results. IMA J. Appl. Math . 78, 945–964.CrossRefGoogle Scholar
Simon, P. L., Taylor, M. and Kiss, I. Z. (2011). Exact epidemic models on graphs using graph-automorphism driven lumping. J. Math. Biol. 62, 479–508.CrossRefGoogle ScholarPubMed
Stewart, W. J. (2000). Numerical methods for computing stationary distributions of finite irreducible Markov chains. In Computational Probability (International Series in Operations Research & Management Science 24), pp. 81–111. Springer.CrossRefGoogle Scholar
Takahashi, Y. (1975). A lumping method for numerical calculations of stationary distributions of Markov chains. B-18, Department of Information Sciences, Tokyo Institute of Technology.Google Scholar
Yamashita, M. and Kameda, T. (1996). Computing on anonymous networks, I: Characterizing the solvable cases. IEEE Trans. Parallel Distrib. Systems 7, 69–89.CrossRefGoogle Scholar