Analytical Approaches to Agent-Based Models

Abstract

The aim of this article is to present an approach to the analysis of simple systems composed of a large number of units in interaction. Suppose to have a large number of agents belonging to a finite number of different groups: as the agents randomly interact with each other, they move from a group to another as a result of the interaction. The object of interest is the stochastic process describing the number of agents in each group. As this is generally intractable, it has been proposed in the literature to approximate it in several ways. We review these approximations and we illustrate them with reference to a version of the epidemic model. The tools presented in the paper should be considered as a complement rather than as a substitute of the classical analysis of ABMs through simulation.

Keywords

Individual-based models Markov processes Differential equations Diffusion approximation Central limit theorem 

References

  1. Allain, M.-F. (1976). Approximation par un processus de diffusion des oscillations, autour d’une valeur moyenne, d’une suite de processus de Markov de saut pur. Comptes rendus de l’Académie des sciences Paris Séries A-B, 282(16), Aiii, A891–A894.Google Scholar
  2. Allain, M.-F. (1976). Étude de la vitesse de convergence d’une suite de processus de Markov de saut pur. Comptes rendus de l’Académie des sciences Paris Séries A-B, 282(17), Aiii, A1015–A1018.Google Scholar
  3. Alm, S. E. (1978). On the rate of convergence in diffusion approximation of jump Markov processes. Report 8, Uppsala University, Department of Mathematics.Google Scholar
  4. Axelrod, R. (1986). An evolutionary approach to norms. American Political Science Review, 80(4), 1095–1111.CrossRefGoogle Scholar
  5. Barbour, A. D. (1972). The principle of the diffusion of arbitrary constants. Journal of Applied Probability, 9, 519–541.CrossRefGoogle Scholar
  6. Barbour, A. D. (1974). On a functional central limit theorem for Markov population processes. Advances in Applied Probability, 6, 21–39.CrossRefGoogle Scholar
  7. Black, A. J., & McKane, A. J. (2012). Stochastic formulation of ecological models and their applications. Trends in Ecology & Evolution, 27(6), 337–345.CrossRefGoogle Scholar
  8. Bortolussi, L., Hillston, J., Latella, D., & M. Massink (2013). Continuous approximation of collective system behaviour: A tutorial. Performance Evaluation, 70, 317–349.CrossRefGoogle Scholar
  9. Centola, D. (2010). The spread of behavior in an online social network experiment. Science, 329(5996), 1194–1197.CrossRefGoogle Scholar
  10. Challenger, J. D., Fanelli, D., & McKane, A. J. (2014). The theory of individual based discrete-time processes. The Journal of Statistical Physics, 156(1), 131–155.CrossRefGoogle Scholar
  11. Collet, F., Dai Pra, P., & Sartori, E. (2010). A simple mean field model for social interactions: Dynamics, fluctuations, criticality. The Journal of Statistical Physics, 139(5), 820–858.CrossRefGoogle Scholar
  12. Ethier, S. N., & Kurtz, T. G. (1986). Markov processes. Characterization and convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. New York: John Wiley & Sons, Inc.Google Scholar
  13. Feller, W. (1951). Diffusion processes in genetics. In J. Neyman (Ed.), Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability (pp. 227–246). Berkeley: University of California Press.Google Scholar
  14. Galán, J. M., & Izquierdo, L. R. (2005). Appearances can be deceiving: Lessons learned re-implementing Axelrod’s ‘evolutionary approach to norms’. Journal of Artificial Societies and Social Simulation, 8(3), 2.Google Scholar
  15. Goutsias, J., & Jenkinson G. (2013). Markovian dynamics on complex reaction networks. Physics Reports, 529(2), 199–264.CrossRefGoogle Scholar
  16. Hirshman, B. R., Charles, J. St., & Carley, K. M. (2011). Leaving us in tiers: can homophily be used to generate tiering effects? Computational & Mathematical Organization Theory, 17(4), 318–343.CrossRefGoogle Scholar
  17. Karlin, S., & Taylor, H. M. (1975). A first course in stochastic processes, 2nd ed. New York/London: Academic Press.Google Scholar
  18. Kurtz, T. G. (1970). Solutions of ordinary differential equations as limits of pure jump Markov processes. Journal of Applied Probability, 7, 49–58.CrossRefGoogle Scholar
  19. Kurtz, T. G. (1971). Limit theorems for sequences of jump Markov processes approximating ordinary differential processes. Journal of Applied Probability, 8, 344–356.CrossRefGoogle Scholar
  20. Kurtz, T. G. (1972). The relationship between stochastic and deterministic models for chemical reactions. The Journal of Chemical Physics, 57(7), 2976–2978.CrossRefGoogle Scholar
  21. Kurtz, T. G. (1976). Limit theorems and diffusion approximations for density dependent Markov chains. Mathematical Programming Studies, 5, 67–78. Stochastic systems: Modeling, identification and optimization, I (Proc. Sympos., Univ. Kentucky, Lexington).Google Scholar
  22. Kurtz, T. G. (1977/1978). Strong approximation theorems for density dependent Markov chains. Stochastic Processes and their Applications, 6(3), 223–240.Google Scholar
  23. Kurtz, T. G. (1980). Relationships between stochastic and deterministic population models. In Biological growth and spread (Proc. Conf., Heidelberg, 1979). Lecture Notes in Biomathematics (Vol. 38, pp. 449–467). Berlin/New York: Springer.Google Scholar
  24. Kurtz, T. G. (1981). Approximation of population processes. CBMS-NSF Regional Conference Series in Applied Mathematics (Vol. 36). Philadelphia: Society for Industrial and Applied Mathematics (SIAM).Google Scholar
  25. Kurtz, T. G. (1983). Gaussian approximations for Markov chains and counting processes. In Proceedings of the 44th session of the International Statistical Institute, Vol. 1 (Madrid, 1983) (Vol. 50, pp. 361–376). With a discussion in Vol. 3, pp. 237–248.Google Scholar
  26. Lotka, A. J. (1925). Elements of Physical Biology. Baltimore: Williams & Wilkins Company.Google Scholar
  27. Matis, J. H., & Kiffe, T. R. (2000). Stochastic Population Models: A Compartmental Perspective. Lecture Notes in Statistics. New York: Springer.Google Scholar
  28. Norman, M. F. (1968). Slow learning. British Journal of Mathematical and Statistical Psychology, 21, 141–159.CrossRefGoogle Scholar
  29. Norman, M. F. (1972). Markov processes and learning models. Mathematics in Science and Engineering (Vol. 84). New York/London: Academic Press.Google Scholar
  30. Norman, M. F. (1974a). A central limit theorem for Markov processes that move by small steps. The Annals of Probability, 2, 1065–1074.CrossRefGoogle Scholar
  31. Norman, M. F. (1974b). Markovian learning processes. SIAM Review, 16, 143–162.CrossRefGoogle Scholar
  32. Plikynas, D., & Masteika, S. (2014). Agent-based nonlocal social systems: Neurodynamic oscillations approach. In G. Meiselwitz (Ed.), Social Computing and Social Media. Lecture Notes in Computer Science (Vol. 8531, pp. 253–264). New York: Springer International Publishing.Google Scholar
  33. Pollett, P. K. (2001). Diffusion approximations for ecological models. In F. Ghassemi, P. Whetton, R. Little & M. Littleboy (Eds.), MODSIM 2001 International Congress on Modelling and Simulation (pp. 843–848). Townsville: Modelling and Simulation Society of Australia and New Zealand Inc.Google Scholar
  34. Railsback, S. F., & Grimm, V. (2011). Agent-based and individual-based modeling: A practical introduction. Princeton: Princeton University Press.Google Scholar
  35. Sigmund, K. (2007). Kolmogorov and population dynamics. In Kolmogorov’s heritage in mathematics (pp. 177–186). Berlin: Springer.CrossRefGoogle Scholar
  36. Volterra, V. (1931). Leçons sur la théorie mathématique de la lutte pour la vie. Cahiers scientifiques, Fascicule VII. Paris: Gauthier-Villars.Google Scholar
  37. Volterra, V. (1962). Opere matematiche. Memorie e Note. Volume quinto: 1926–1940. Accademia nazionale dei Lincei.Google Scholar
  38. Wang, X., Tao, H., Xie, Z., & Yi, D. (2013). Mining social networks using wave propagation. Computational & Mathematical Organization Theory, 19(4), 569–579.CrossRefGoogle Scholar
  39. Zhang, Y., & Wu, Y. (2012). How behaviors spread in dynamic social networks. Computational & Mathematical Organization Theory, 18(4), 419–444.CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Dipartimento di EconomiaUniversità degli Studi dell’InsubriaVareseItaly

Personalised recommendations