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Large-time behaviour and the second eigenvalue problem for finite-state mean-field interacting particle systems

Part of: Limit theorems Special processes Computer system organization General theory of linear operators

Published online by Cambridge University Press:  08 July 2022

Sarath Yasodharan  and
Rajesh Sundaresan
Sarath Yasodharan*
Affiliation:
Indian Institute of Science
Rajesh Sundaresan*
Affiliation:
Indian Institute of Science
*
*Postal address: Department of Electrical Communication Engineering, Indian Institute of Science, Bengaluru 560012, India.
*Postal address: Department of Electrical Communication Engineering, Indian Institute of Science, Bengaluru 560012, India.
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Abstract

This article examines large-time behaviour of finite-state mean-field interacting particle systems. Our first main result is a sharp estimate (in the exponential scale) of the time required for convergence of the empirical measure process of the N-particle system to its invariant measure; we show that when time is of the order $\exp\{N\Lambda\}$ for a suitable constant $\Lambda > 0$, the process has mixed well and it is close to its invariant measure. We then obtain large-N asymptotics of the second-largest eigenvalue of the generator associated with the empirical measure process when it is reversible with respect to its invariant measure. We show that its absolute value scales as $\exp\{{-}N\Lambda\}$. The main tools used in establishing our results are the large deviation properties of the empirical measure process from its large-N limit. As an application of the study of large-time behaviour, we also show convergence of the empirical measure of the system of particles to a global minimum of a certain ‘entropy’ function when particles are added over time in a controlled fashion. The controlled addition of particles is analogous to the cooling schedule associated with the search for a global minimum of a function using the simulated annealing algorithm.

Keywords

Mean-field interactionMcKean–Vlasov equationmetastabilityexit from a domaincyclessimulated annealing

MSC classification

Primary: 60F10: Large deviations 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 47A75: Eigenvalue problems 68M20: Performance evaluation; queueing; scheduling

Information

Type
Original Article
Information
Advances in Applied Probability , Volume 55 , Issue 1 , March 2023 , pp. 85 - 125
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Aghajani, R., Li, X. and Ramanan, K. (2017). The PDE method for the analysis of randomized load balancing networks. Proc. ACM Measurement Anal. Comput. Systems 1, article no. 38, 28 pp.CrossRefGoogle Scholar
Aghajani, R. and Ramanan, K. (2019). The hydrodynamic limit of a randomized load balancing network. Ann. Appl. Prob. 29, 2114–2174.CrossRefGoogle Scholar
Akhil, P. T., Altman, E. and Sundaresan, R. (2019). A mean-field approach for controlling singularly perturbed multi-population SIS epidemics. Preprint. Available at https://arxiv.org/abs/1902.05713.Google Scholar
Anantharam, V. (1991). A mean field limit for a lattice caricature of dynamic routing in circuit switched networks. Ann. Appl. Prob. 1, 481–503.CrossRefGoogle Scholar
Bakry, D., Gentil, I. and Ledoux, M. (2014). Analysis and Geometry of Markov Diffusion Operators. Springer, Cham.CrossRefGoogle Scholar
Benaïm, M. and Boudec, J.-Y. L. (2008). A class of mean field interaction models for computer and communication systems. Performance Evaluation 65, 823–838.CrossRefGoogle Scholar
Bhattacharya, A. and Kumar, A. (2017). Analytical modeling of IEEE 802.11-type CSMA/CA networks with short term unfairness. IEEE/ACM Trans. Networking 25, 3455–3472.CrossRefGoogle Scholar
Bianchi, G. (1998). IEEE 802.11—saturated throughput analysis. IEEE Commun. Lett. 12, 318–320.CrossRefGoogle Scholar
Bodineau, T. and Giacomin, G. (2004). From dynamic to static large deviations in boundary driven exclusion particle systems. Stoch. Process. Appl. 110, 67–81.CrossRefGoogle Scholar
Boorstyn, R., Kershenbaum, A., Maglaris, B. S. and Sahin, V. (1987). Throughput analysis in multihop CSMA packet radio networks. IEEE Trans. Commun. 35, 267–274.CrossRefGoogle Scholar
Bordenave, C., McDonald, D. and Proutière, A. (2010). A particle system in interaction with a rapidly varying environment: mean field limits and applications. Networks Heterogeneous Media 5, 31–62.CrossRefGoogle Scholar
Borkar, V. S. and Sundaresan, R. (2012). Asymptotics of the invariant measure in mean field models with jumps. Stoch. Systems 2, 322–380.CrossRefGoogle Scholar
Comets, F. (1987). Nucleation for a long range magnetic model. Ann. Inst. H. Poincaré Prob. Statist. 23, 135–178.Google Scholar
Dawson, D. A. and Gärtner, J. (1987). Large deviations from the McKean-Vlasov limit for weakly interacting diffusions. Stochastics 20, 247–308.CrossRefGoogle Scholar
Del Moral, P. and Miclo, L. (1999). On the convergence and applications of generalized simulated annealing. SIAM J. Control Optimization 37, 1222–1250.CrossRefGoogle Scholar
Del Moral, P. and Zajic, T. (2003). A note on the Laplace–Varadhan integral lemma. Bernoulli 9, 49–65.CrossRefGoogle Scholar
Dembo, A. and Zeitouni, O. (2010). Large Deviations Techniques and Applications, 2nd edn. Springer, Berlin, Heidelberg.CrossRefGoogle Scholar
Djehiche, B. and Kaj, I. (1995). The rate function for some measure-valued jump processes. Ann. Prob. 23, 1414–1438.CrossRefGoogle Scholar
Ethier, S. N. and Kurtz, T. G. (2005). Markov Processes: Characterization and Convergence. John Wiley, New York.Google Scholar
Freidlin, M. I. and Wentzell, A. D. (2012). Random Perturbations of Dynamical Systems, 3rd edn. Springer, Berlin, Heidelberg.CrossRefGoogle Scholar
Gan, T. and Cameron, M. (2017). A graph-algorithmic approach for the study of metastability in Markov chains. J. Nonlinear Sci. 27, 927–972.CrossRefGoogle Scholar
Gärtner, J. (1988). On the McKean-Vlasov limit for interacting diffusions. Math. Nachr. 137, 197–248.CrossRefGoogle Scholar
Gibbens, R. J., Hunt, P. J. and Kelly, F. P. (1990). Bistability in communication networks. In Disorder in Physical Systems, Oxford University Press, pp. 113–127.Google Scholar
Graham, C. (2000). Chaoticity on path space for a queueing network with selection of the shortest queue among several. J. Appl. Prob. 37, 198–211.CrossRefGoogle Scholar
Hwang, C.-R. and Sheu, S.-J. (1990). Large-time behavior of perturbed diffusion Markov processes with applications to the second eigenvalue problem for Fokker–Planck operators and simulated annealing. Acta Appl. Math. 19, 253–295.CrossRefGoogle Scholar
Kifer, Y. (1990). A discrete-time version of the Wentzell–Friedlin theory. Ann. Prob. 18, 1676–1692.CrossRefGoogle Scholar
Kraaij, R. (2016). Large deviations for finite state Markov jump processes with mean-field interaction via the comparison principle for an associated Hamilton–Jacobi equation. J. Statist. Phys. 164, 321–345.CrossRefGoogle Scholar
Kumar, A., Altman, E., Miorandi, D. and Goyal, M. (2005). New insights from a fixed point analysis of single cell IEEE 802.11 WLANs. In Proc. 24th Annual Joint Conference of the IEEE Computer and Communications Societies, Vol. 3, Institute of Electrical and Electronics Engineers, Piscataway, NJ, pp. 1550–1561.Google Scholar
Léonard, C. (1990). Some epidemic systems are long range interacting particle systems. In Stochastic Processes in Epidemic Theory, Springer, Berlin, Heidelberg, pp. 170–183.CrossRefGoogle Scholar
Léonard, C. (1995). Large deviations for long range interacting particle systems with jumps. Ann. Inst. H. Poincaré Prob. Statist. 31, 289–323.Google Scholar
Li, J. et al. (2018). Mean field games in nudge systems for societal networks. ACM Trans. Modeling Performance Evaluation Comput. Systems 3, article no. 15, 31 pp.CrossRefGoogle Scholar
Manjrekar, M., Ramaswamy, V. and Shakkottai, S. (2014). A mean field game approach to scheduling in cellular systems. In IEEE INFOCOM 2014—IEEE Conference on Computer Communications, Institute of Electrical and Electronics Engineers, Piscataway, NJ, pp. 1554–1562.CrossRefGoogle Scholar
McKean, H. P. (1967). Propagation of chaos for a class of non-linear parabolic equations. In Stochastic Differential Equations (Lecture Series in Differential Equations, Session 7), Air Force Office of Scientific Research, Arlington, VA, pp. 41–57.Google Scholar
Miclo, L. (1995). Comportement de spectres d’opérateurs de Schrödinger à basse température. Bull. Sci. Math. 119, 529–553.Google Scholar
Miclo, L. (1995). Remarques sur l’ergodicité des algorithmes de recuit simulé sur un graphe. Stoch. Process. Appl. 58, 329–360.CrossRefGoogle Scholar
Mitzenmacher, M. (2001). The power of two choices in randomized load balancing. IEEE Trans. Parallel Distributed Systems 12, 1094–1104.CrossRefGoogle Scholar
Mukhopadhyay, A., Karthik, A. and Mazumdar, R. R. (2016). Randomized assignment of jobs to servers in heterogeneous clusters of shared servers for low delay. Stoch. Systems 6, 90–131.CrossRefGoogle Scholar
Olesker-Taylor, S. (2022). Metastability in loss networks with dynamic alternative routing. To appear in Ann. Appl. Prob. CrossRefGoogle Scholar
Panageas, I., Srivastava, P. and Vishnoi, N. K. (2016). Evolutionary dynamics in finite populations mix rapidly. In Proc. Twenty-Seventh Annual ACM–SIAM Symposium on Discrete Algorithms, Society for Industrial and Applied Mathematics, Philadelphia, pp. 480–497.CrossRefGoogle Scholar
Panageas, I. and Vishnoi, N. K. (2016). Mixing time of Markov chains, dynamical systems and evolution. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016), Schloss Dagstuhl—Leibniz-Zentrum fÜr Informatik, Dagstuhl, pp. 63:1–63:14.Google Scholar
Ramaiyan, V., Kumar, A. and Altman, E. (2008). Fixed point analysis of single cell IEEE 802.11e WLANs: uniqueness and multistability. IEEE/ACM Trans. Networking 16, 1080–1093.CrossRefGoogle Scholar
Reiffers-Masson, A. and Sundaresan, R. (2019). Reputation-based information design for inducing prosocial behavior. Preprint. Available at https://arxiv.org/abs/1905.00585.Google Scholar
Salins, M. (2019). Equivalences and counterexamples between several definitions of the uniform large deviations principle. Prob. Surveys 16, 99–142.CrossRefGoogle Scholar
Sznitman, A.-S. (1991). Topics in propagation of chaos. In Ecole d’Eté de Probabilités de Saint-Flour XIX—1989, Springer, Berlin, Heidelberg, pp. 165–251.CrossRefGoogle Scholar
Thai, M.-N. (2015). Birth and death process in mean field type interaction. Preprint. Available at https://arxiv.org/abs/1510.03238.Google Scholar
Trouvé, A. (1996). Cycle decompositions and simulated annealing. SIAM J. Control Optimization 34, 966–986.CrossRefGoogle Scholar
Wentzel, A. D. (1975). On the asymptotic behaviour of the first eigenvalue of a second-order differential operator with small parameter by the higher derivatives. Teor. Veroyat. Primen. 20, 610–613.Google Scholar