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Approximate lumpability for Markovian agent-based models using local symmetries

  • Wasiur R. KhudaBukhsh (a1), Arnab Auddy (a2), Yann Disser (a3) and Heinz Koeppl (a3)

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.

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Corresponding author

*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

References

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[1] 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.
[2] Apers, S., Ticozzi, F. and Sarlette, A. (2017). Lifting Markov chains to mix faster: limits and opportunities. Available at arXiv:1705.08253.
[3] Arvind, V., Köbler, J., Rattan, G. and Verbitsky, O. (2016). Graph isomorphism, color refinement, and compactness. Comput. Complexity 26, 627685.
[4] 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.
[5] Babai, L., Erdős, P. and Selkow, S. M. (1980). Random graph isomorphism. SIAM J. Comput . 9, 628635.
[6] Banisch, S. (2016). Markov Chain Aggregation for Agent-Based Models. Springer.
[7] 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. 145156. Springer.
[8] Boldi, P., Lonati, V., Santini, M. and Vigna, S. (2006). Graph fibrations, graph isomorphism, and PageRank. RAIRO Theoret. Inform. Appl. 40, 227253.
[9] Boldi, P. and Vigna, S. (2002). Fibrations of graphs. Discrete Math . 243, 2166.
[10] Buchholz, P. (1994). Exact and ordinary lumpability in finite Markov chains. J. Appl. Prob. 31, 5975.
[11] 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.
[12] Chatterjee, S. and Durrett, R. (2009). Contact processes on random graphs with power law degree distributions have critical value 0. Ann. Prob. 37, 23322356.
[13] 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. 275281. ACM, New York.
[14] Courtois, P. J. (1977). Decomposability: Queueing and Computer System Applications. Academic Press, New York.
[15] 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, 27932808.
[16] Elbert Simões, J., Figueiredo, D. R. and Barbosa, V. C. (2016). Local symmetry in random graphs. Available at arXiv:1605.01758.
[17] Feret, J., Henzinger, T., Koeppl, H. and Petrov, T. (2012). Lumpability abstractions of rule-based systems. Theoret. Comput. Sci. 431, 137164.
[18] França, G. and Bento, J. (2017). Markov chain lifting and distributed ADMM. IEEE Signal Process. Lett . 24, 294298.
[19] Ganguly, A., Petrov, T. and Koeppl, H. (2014). Markov chain aggregation and its applications to combinatorial reaction networks. J. Math. Biol. 69, 767797.
[20] Geiger, B. C., Petrov, T., Kubin, G. and Koeppl, H. (2015). Optimal Kullback–Leibler aggregation via information bottleneck. IEEE Trans. Automat. Control 60, 10101022.
[21] Godsil, C. and Royle, G. F. (2013). Algebraic Graph Theory (Graduate Texts in Mathematics 207). Springer, New York.
[22] 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.
[23] Hendrickx, J. M. (2014). Views in a graph: to which depth must equality be checked? IEEE Trans. Parallel Distrib. Systems 25, 19071912.
[24] Katehakis, M. N. and Smit, L. C. (2012). A successive lumping procedure for a class of Markov chains. Probab. Engrg Inform. Sci. 26, 483508.
[25] Kemeny, J. G. and Snell, J. L. (1960). Finite Markov Chains. Van Nostrand, Princeton, NJ.
[26] 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. 386394. IEEE.
[27] Kim, J. H., Sudakov, B. and Vu, V. H. (2002). On the asymmetry of random regular graphs and random graphs. Random Structures Algorithms 21, 216224.
[28] 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.
[29] Kratochvíl, J., Proskurowski, A. and Telle, J. A. (1998). Complexity of graph covering problems. Nordic J. Comput . 5, 173195.
[30] 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.
[31] Łuczak, T. (1988). The automorphism group of random graphs with a given number of edges. Math. Proc. Cambridge Phil. Soc. 104, 441449.
[32] McKay, B. D. and Wormald, N. C. (1984). Automorphisms of random graphs with specified vertices. Combinatorica 4, 325338.
[33] Nijholt, E., Rink, B. and Sanders, J. (2016). Graph fibrations and symmetries of network dynamics. J. Diff. Equations 261, 48614896.
[34] 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.
[35] Norris, N. (1995). Universal covers of graphs: isomorphism to depth ${n-1}$ implies isomorphism to all depths. Discrete Appl. Math . 56, 6174.
[36] Riordan, O. and Wormald, N. (2010). The diameter of sparse random graphs. Combin. Probab. Comput. 19, 835926.
[37] Rubino, G. and Sericola, B. (1989). On weak lumpability in Markov chains. J. Appl. Prob. 26, 446457.
[38] Rubino, G. and Sericola, B. (1993). A finite characterization of weak lumpable Markov processes, II: The continuous time case. Stochastic Process. Appl. 45, 115125.
[39] 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, 945964.
[40] Simon, P. L., Taylor, M. and Kiss, I. Z. (2011). Exact epidemic models on graphs using graph-automorphism driven lumping. J. Math. Biol. 62, 479508.
[41] 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.
[42] 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.
[43] Yamashita, M. and Kameda, T. (1996). Computing on anonymous networks, I: Characterizing the solvable cases. IEEE Trans. Parallel Distrib. Systems 7, 6989.

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Approximate lumpability for Markovian agent-based models using local symmetries

  • Wasiur R. KhudaBukhsh (a1), Arnab Auddy (a2), Yann Disser (a3) and Heinz Koeppl (a3)

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