Skip to main content

Combining Mathematical and Simulation Approaches to Understand the Dynamics of Computer Models

Part of the Understanding Complex Systems book series (UCS)

Keywords

  • Computer Simulation
  • Markov Chain
  • Computer Model
  • Formal Model
  • Mathematical Analysis

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-540-93813-2_11
  • Chapter length: 37 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   189.00
Price excludes VAT (USA)
  • ISBN: 978-3-540-93813-2
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Hardcover Book
USD   249.99
Price excludes VAT (USA)
Fig. 11.1
Fig. 11.2
Fig. 11.3
Fig. 11.4
Fig. 11.5
Fig. 11.6
Fig. 11.7
Fig. 11.8
Fig. 11.9
Fig. 11.10
Fig. 11.11
Fig. 11.12
Fig. 11.13

Notes

  1. 1.

    Note that simulations of stochastic models are actually using pseudo-random number generators, which are deterministic algorithms that require a seed as an input.

  2. 2.

    A formal model is a model expressed in a formal system (Cutland 1980). A formal system consists of a formal language and a deductive apparatus (a set of axioms and inference rules). Formal systems are used to derive new expressions by applying the inference rules to the axioms and/or previously derived expressions in the same system.

  3. 3.

    The mere fact that the model has been implemented and can be run in a computer is a proof that the model is formal (Suber 2002).

  4. 4.

    As a matter of fact, strictly speaking, inputs and outputs in a computer model are never numbers. We may interpret strings of bits as numbers, but we could equally well interpret the same strings of bits as e.g. letters. More importantly, a bit itself is already an abstraction, an interpretation we make of an electrical pulse that can be above or below a critical voltage threshold.

  5. 5.

    A sufficient condition for a programming language to be “sophisticated enough” is to allow for the implementation of the following three control structures:

    • Sequence (i.e. executing one subprogram, and then another subprogram),

    • Selection (i.e. executing one of two subprograms according to the value of a boolean variable, e.g. IF[boolean == true]-THEN[subprogram1]-ELSE[subprogram2]), and

    • Iteration (i.e. executing a subprogram until a boolean variable becomes false, e.g. WHILE[boolean == true]-DO[subprogram]).

    Any programming language that can combine subprograms in these three ways can implement any computable function; this statement is known as the “structured program theorem” (Böhm and Jacopini 1966; Harel 1980; Wikipedia 2007).

  6. 6.

    Note that statistics extracted from the model can be of any nature, as long as they are unambiguously defined. For example, they can refer to various time-steps, and only to certain agents (e.g. “average wealth of female agents in odd time-steps from 1 to 99”).

  7. 7.

    We use the term “mathematical analysis” in its broadest sense, i.e. we do not refer to any particular branch of mathematics, but to the general use of (any type of) mathematical technique to analyse a system.

  8. 8.

    Unless, of course, all possible particular instances are explored.

  9. 9.

    The frequency of the event “there are i walkers in a patch with a house” calculated over n simulation runs can be seen as the mean of a sample of n i.i.d. Bernouilli random variables where success denotes that the event occurred and failure denotes that it did not. Thus, the frequency f is the maximum likelihood (unbiased) estimator of the exact probability with which the event occurs. The standard error of the calculated frequency f is the standard deviation of the sample divided by the square root of the sample size. In this particular case, the formula reads:

    $$ \rm Std. error(\it f,\it n)=(\it f(\rm 1-\it f)/(\it n-\rm 1))^{1/2}$$

    Where f is the frequency of the event, n is the number of samples, and the standard deviation of the sample has been calculated dividing by (n – 1).

  10. 10.

    The term ‘Markov chain’ allows for countably infinite state spaces too (Karr 1990).

  11. 11.

    Formally, the occupancy of state i is defined as:

    $$ \pi_i^{*}=\mathop{\lim}\limits_{{n\to \infty }}\frac{{E({N_i}(n))}}{n+1 } $$

    where N i (n) denotes the number of times that the THMC visits state i over the time span {0, 1,…, n}.

  12. 12.

    Given that the system has entered the absorbing class C v .

  13. 13.

    This finding does not refute some of the most important conclusions obtained by the authors of the original model.

  14. 14.

    This is so because many assumptions we make in our models are, to some extent, for the sake of simplicity. As a matter of fact, in most cases the whole purpose of modelling is to build an abstraction of the world which is simpler than the world itself, so we can make inferences about the model that we cannot make directly from the real world (Edmonds 2001; Galán et al. 2009; Izquierdo et al. 2008a).

  15. 15.

    This comment was added by the editors as the authors are too modest to so describe their own work.

References

  • Arthur WB (1989) Competing technologies, increasing returns, and lock-in by historical events. Econ J 99(394):116–131

    CrossRef  Google Scholar 

  • Arthur WB, Holland JH, LeBaron B, Palmer R, Tayler P (1997) Asset pricing under endogenous expectations in an artificial stock market. In: Arthur WB, Durlauf S, Lane D (eds) The economy as an evolving complex system II. Addison-Wesley Longman, Reading, pp 15–44

    Google Scholar 

  • Axelrod RM (1986) An evolutionary approach to norms. Am Polit Sci Rev 80(4):1095–1111

    CrossRef  Google Scholar 

  • Axelrod RM (1997a) Advancing the art of simulation in the social sciences. In: Conte R, Hegselmann R, Terna P (eds) Simulating social phenomena (Lecture notes in economics and mathematical systems, 456). Springer, Berlin, pp 21–40

    Google Scholar 

  • Axelrod RM (1997b) The dissemination of culture: a model with local convergence and global polarization. J Confl Resolut 41(2):203–226

    CrossRef  Google Scholar 

  • Axelrod RM, Bennett DS (1993) A landscape theory of aggregation. Br J Polit Sci 23(2):211–233

    CrossRef  Google Scholar 

  • Axtell RL (2000) Why agents? On the varied motivations for agent computing in the social sciences. In: Macal VM, Sallach D (eds) Proceedings of the workshop on agent simulation: applications, models, and tools, Argonne National Laboratory, Argonne, pp 3–24

    Google Scholar 

  • Axtell RL, Epstein JM (1994) Agent based modeling: understanding our creations. The Bulletin of the Santa Fe Institute, Winter 1994, pp 28–32

    Google Scholar 

  • Balzer W, Brendel KR, Hofmann S (2001) Bad arguments in the comparison of game theory and simulation in social studies. J Artif Soc Soc Simul 4(2), http://jasss.soc.surrey.ac.uk/4/2/1.html

  • Benveniste A, Métivier M, Priouret P (1990) Adaptive algorithms and stochastic approximations. Springer, Berlin

    CrossRef  MATH  Google Scholar 

  • Böhm C, Jacopini G (1966) Flow diagrams, turing machines and languages with only two formation rules. Commun ACM 9(5):366–371

    CrossRef  MATH  Google Scholar 

  • Bush RR, Mosteller F (1955) Stochastic models for learning. Wiley, New York

    MATH  Google Scholar 

  • Castellano C, Marsili M, Vespignani A (2000) Nonequilibrium phase transition in a model for social influence. Phys Rev Lett 85(16):3536–3539

    CrossRef  Google Scholar 

  • Cutland N (1980) Computability: an introduction to recursive function theory. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • Edmonds B (2001) The use of models: making MABS actually work. In: Moss S, Davidsson P (eds) Multi-agent-based simulation (Lecture notes in artificial intelligence, 1979). Springer, Berlin, pp 15–32

    Google Scholar 

  • Edmonds B (2005) Simulation and complexity: how they can relate. In: Feldmann V, Mühlfeld K (eds) Virtual worlds of precision: computer-based simulations in the sciences and social sciences. Lit-Verlag, Münster, pp 5–32

    Google Scholar 

  • Edwards M, Huet S, Goreaud F, Deffuant G (2003) Comparing an individual-based model of behaviour diffusion with its mean field aggregate approximation. J Artif Soc Soc Simul 6(4), http://jasss.soc.surrey.ac.uk/6/4/9.html

  • Ehrentreich N (2002) The Santa Fe Artificial Stock Market re-examined: suggested corrections (Betriebswirtschaftliche Diskussionsbeiträge Nr. 45/02). Wirtschaftswissenschaftliche Fakultät, Martin-Luther-Universität, Halle/Saale, http://econwpa.wustl.edu:80/eps/comp/papers/0209/0209001.pdf

  • Ehrentreich N (2006) Technical trading in the Santa Fe Institute Artificial Stock Market revisited. J Econ Behav Organ 61(4):599–616

    CrossRef  Google Scholar 

  • Epstein JM (2006) Remarks on the foundations of agent-based generative social science. In: Judd KL, Tesfatsion L (eds) Handbook of computational economics, vol. 2: agent-based computational economics. North-Holland, Amsterdam, pp 1585–1604

    Google Scholar 

  • Epstein JM, Axtell RL (1996) Growing artificial societies: social science from the bottom up. Brookings Institution Press/MIT Press, Cambridge

    Google Scholar 

  • Evans A, Heppenstall A, Birkin M (2013) Understanding simulation results. Chapter 9 in this volume

    Google Scholar 

  • Flache A, Hegselmann R (1999) Rationality vs. learning in the evolution of solidarity networks: a theoretical comparison. Comput Math Organ Theory 5(2):97–127

    CrossRef  MATH  Google Scholar 

  • Flache A, Hegselmann R (2001) Do irregular grids make a difference? Relaxing the spatial regularity assumption in cellular models of social dynamics. J Artif Soc Soc Simul 4(4), http://jasss.soc.surrey.ac.uk/4/4/6.html

  • Flache A, Macy MW (2002) Stochastic collusion and the power law of learning. J Confl Resolut 46(5):629–653

    CrossRef  Google Scholar 

  • Galán JM, Izquierdo LR (2005) Appearances can be deceiving: lessons learned re-implementing Axelrod’s ‘Evolutionary approach to norms’. J Artif Soc Soc Simul 8(3), http://jasss.soc.surrey.ac.uk/8/3/2.html

  • Galán JM et al (2009) Errors and artefacts in agent-based modelling. J Artif Soc Soc Simul 12(1), http://jasss.soc.surrey.ac.uk/12/1/1.html

  • Genz A, Kwong KS (2000) Numerical evaluation of singular multivariate normal distributions. J Stat Comput Sim 68(1):1–21

    MathSciNet  CrossRef  MATH  Google Scholar 

  • Gilbert N (1999) Simulation: a new way of doing social science. Am Behav Sci 42(10):1485–1487

    Google Scholar 

  • Gilbert N (2007) Agent-based models, vol 153, Quantitative applications in the social sciences. Sage, London

    Google Scholar 

  • Gilbert N, Terna P (2000) How to build and use agent-based models in social science. Mind Soc 1(1):57–72

    CrossRef  Google Scholar 

  • Gilbert N, Troitzsch KG (1999) Simulation for the social scientist. Open University Press, Buckingham

    Google Scholar 

  • Gotts NM, Polhill JG, Law ANR (2003a) Agent-based simulation in the study of social dilemmas. Artif Intell Rev 19(1):3–92

    CrossRef  Google Scholar 

  • Gotts NM, Polhill JG, Adam WJ (2003b) Simulation and analysis in agent-based modelling of land use change. In: Online proceedings of the first conference of the European social simulation association, Groningen, The Netherlands, 18–21 Sept 2003, http://www.uni-koblenz.de/~essa/ESSA2003/proceedings.htm

  • Gotts NM, Polhill JG, Law ANR (2003c) Aspiration levels in a land-use simulation. Cybern Syst 34(8):663–683

    CrossRef  MATH  Google Scholar 

  • Gotts NM, Polhill JG, Law ANR, Izquierdo LR (2003d) Dynamics of imitation in a land use simulation. In: Dautenhahn K, Nehaniv C (eds) Proceedings of the second international symposium on imitation in animals and artefacts, University of Wales, Aberystwyth, 7–11 April 2003, pp 39–46

    Google Scholar 

  • Grinstead CM, Snell JL (1997) Chapter 11: Markov chains. In: Grinstead CM, Snell JL (eds) Introduction to probability (Second revised edition). American Mathematical Society, Providence, pp 405–470, http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/book.html

  • Häggström O (2002) Finite Markov chains and algorithmic applications. Cambridge University Press, Cambridge

    CrossRef  MATH  Google Scholar 

  • Harel D (1980) On folk theorems. Commun ACM 23(7):379–389

    CrossRef  MATH  Google Scholar 

  • Hauert C, Doebeli M (2004) Spatial structure often inhibits the evolution of cooperation in the snowdrift game. Nature 428(6983):643–646

    CrossRef  Google Scholar 

  • Hegselmann R, Flache A (2000) Rational and adaptive playing. Anal Krit 22(1):75–97

    Google Scholar 

  • Holland JH, Miller JH (1991) Artificial adaptive agents in economic theory. Am Econ Rev 81(2): 365–370

    Google Scholar 

  • Huet S, Edwards M, Deffuant G (2007) Taking into account the variations of neighbourhood sizes in the mean-field approximation of the threshold model on a random network. J Artif Soc Soc Simul 10(1), http://jasss.soc.surrey.ac.uk/10/1/10.html

  • Imhof LA, Fudenberg D, Nowak MA (2005) Evolutionary cycles of cooperation and defection. Proc Natl Acad Sci USA 102(31):10797–10800

    CrossRef  Google Scholar 

  • Izquierdo SS, Izquierdo LR (2006) On the structural robustness of evolutionary models of cooperation. In: Corchado E, Yin H, Botti VJ, Fyfe C (eds) Intelligent data engineering and automated learning – IDEAL 2006 (Lecture notes in computer science, 4224). Springer, Berlin/Heidelberg, pp 172–182

    Google Scholar 

  • Izquierdo SS, Izquierdo LR (2007) The impact on market efficiency of quality uncertainty without asymmetric information. J Bus Res 60(8):858–867

    CrossRef  Google Scholar 

  • Izquierdo LR, Polhill JG (2006) Is your model susceptible to floating point errors? J Artif Soc Soc Simul 9(4), http://jasss.soc.surrey.ac.uk/9/4/4.html

  • Izquierdo LR, Izquierdo SS, Gotts NM, Polhill JG (2007) Transient and asymptotic dynamics of reinforcement learning in games. Game Econ Behav 61(2):259–276

    MathSciNet  CrossRef  MATH  Google Scholar 

  • Izquierdo LR, Galán JM, Santos JI, Olmo R (2008a) Modelado de sistemas complejos mediante simulación basada en agentes y mediante dinámica de sistemas. Empiria 16:85–112

    Google Scholar 

  • Izquierdo SS, Izquierdo LR, Gotts NM (2008b) Reinforcement learning dynamics in social dilemmas. J Artif Soc Soc Simul 11(2), http://jasss.soc.surrey.ac.uk/11/2/1.html

  • Izquierdo LR, Izquierdo SS, Galán JM, Santos JI (2009) Techniques to understand computer simulations: Markov chain analysis. J Artif Soc Soc Simul 12(1), http://jasss.soc.surrey.ac.uk/12/1/6.html

  • Janssen J, Manca R (2006) Applied semi-Markov processes. Springer, New York

    MATH  Google Scholar 

  • Karr AF (1990) Markov processes. In: Heyman DP, Sobel MJ (eds) Stochastic models, vol 2, Handbooks in operations research and management science. Elsevier, Amsterdam, pp 95–123

    CrossRef  Google Scholar 

  • Klemm K, Eguíluz VM, Toral R, San Miguel M (2003a) Nonequilibrium transitions in complex networks: a model of social interaction. Phys Rev E 67(2):026120

    CrossRef  Google Scholar 

  • Klemm K, Eguíluz VM, Toral R, San Miguel M (2003b) Role of dimensionality in Axelrod’s model for the dissemination of culture. Phys A 327(1–2):1–5

    MathSciNet  MATH  Google Scholar 

  • Klemm K, Eguíluz VM, Toral R, San Miguel M (2003c) Global culture: a noise-induced transition in finite systems. Phys Rev E 67(4):045101

    CrossRef  Google Scholar 

  • Klemm K, Eguíluz VM, Toral R, San Miguel M (2005) Globalization, polarization and cultural drift. J Econ Dyn Control 29(1–2):321–334

    CrossRef  MATH  Google Scholar 

  • Kulkarni VG (1995) Modeling and analysis of stochastic systems. Chapman & Hall/CRC, Boca Raton

    MATH  Google Scholar 

  • Kulkarni VG (1999) Modeling, analysis, design, and control of stochastic systems. Springer, New York

    MATH  Google Scholar 

  • Kushner HJ, Yin GG (1997) Stochastic approximation algorithms and applications. Springer, New York

    MATH  Google Scholar 

  • Lebaron B, Arthur WB, Palmer R (1999) Time series properties of an artificial stock market. J Econ Dyn Control 23(9–10):1487–1516

    CrossRef  MATH  Google Scholar 

  • Leombruni R, Richiardi M (2005) Why are economists sceptical about agent-based simulations? Phys A 355(1):103–109

    MathSciNet  CrossRef  Google Scholar 

  • Lieberman E, Havlin S, Nowak MA (2009) Evolutionary dynamics on graphs. Nature 433(7023): 312–316

    CrossRef  Google Scholar 

  • Mabrouk N, Deffuant G, Lobry C (2007) Confronting macro, meso and micro scale modelling of bacteria dynamics. In: M2M 2007: Third international model-to-model workshop, Marseille, France, 15–16 Mar 2007, http://m2m2007.macaulay.ac.uk/M2M2007-Mabrouk.pdf

  • Macy MW, Flache A (2002) Learning dynamics in social dilemmas. Proc Natl Acad Sci USA 99(3):7229–7236

    CrossRef  Google Scholar 

  • Miller JH, Page SE (2004) The standing ovation problem. Complexity 9(5):8–16

    CrossRef  Google Scholar 

  • Nowak MA, May RM (1992) Evolutionary games and spatial chaos. Nature 359(6398):826–829

    CrossRef  Google Scholar 

  • Nowak MA, Sigmund K (1992) Tit for tat in heterogeneous populations. Nature 355(6357): 250–253

    CrossRef  Google Scholar 

  • Nowak MA, Sigmund K (1993) A strategy of win-stay, lose-shift that outperforms tit for tat in the Prisoner’s Dilemma game. Nature 364(6432):56–58

    CrossRef  Google Scholar 

  • Ostrom T (1988) Computer simulation: the third symbol system. J Exp Soc Psychol 24(5):381–392

    CrossRef  Google Scholar 

  • Polhill JG, Izquierdo LR, Gotts NM (2006) What every agent based modeller should know about floating point arithmetic. Environ Model Software 21(3):283–309

    CrossRef  Google Scholar 

  • Richiardi M, Leombruni R, Saam NJ, Sonnessa M (2006) A common protocol for agent-based social simulation. J Artif Soc Soc Simul 9(1), http://jasss.soc.surrey.ac.uk/9/1/15.html

  • Santos FC, Pacheco JM, Lenaerts T (2006) Evolutionary dynamics of social dilemmas in structured heterogeneous populations. Proc Natl Acad Sci USA 103(9):3490–3494

    CrossRef  Google Scholar 

  • Suber P (2002) Formal systems and machines: an isomorphism (Electronic hand-out for the course “Logical Systems”, Earlham College, Richmond, IN) http://www.earlham.edu/~peters/courses/logsys/machines.htm

  • Takadama K, Suematsu YL, Sugimoto N, Nawa NE, Shimohara K (2003) Cross-element validation in multiagent-based simulation: switching learning mechanisms in agents. J Artif Soc Soc Simul 6(4), http://jasss.soc.surrey.ac.uk/6/4/6.html

  • Takahashi N (2000) The emergence of generalized exchange. Am J Sociol 10(4):1105–1134

    CrossRef  Google Scholar 

  • Traulsen A, Nowak MA, Pacheco JM (2006) Stochastic dynamics of invasion and fixation. Phys Rev E 74(1):011909

    CrossRef  Google Scholar 

  • Vilà X (2008) A model-to-model analysis of Bertrand competition. J Artif Soc Soc Simul 11(2), http://jasss.soc.surrey.ac.uk/11/2/11.html

  • Wikipedia (2007) Structured program theorem, http://en.wikipedia.org/w/index.php?title=Structured_program_theorem&oldid=112885072

  • Wilensky U (1999) NetLogo, http://ccl.northwestern.edu/netlogo

Download references

Acknowledgements

The authors have benefited from the financial support of the Spanish Ministry of Education and Science (projects DPI2004-06590, DPI2005-05676 and TIN2008-06464-C03-02), the Spanish Ministry for Science and Innovation (CSD2010-00034), and the JCyL (projects VA006B09, BU034A08 and GREX251-2009). We are also very grateful to Nick Gotts, Bruce Edmonds, Gary Polhill, and Cesáreo Hernández for many extremely useful discussions.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Luis R. Izquierdo .

Editor information

Editors and Affiliations

Further Reading

Further Reading

Firstly we suggest three things to read to learn more about Markov Chain models. Grinstead and Snell (1997) provides an excellent introduction to the theory of finite Markov Chains, with many examples and exercises. Häggström (2002) gives a clear and concise introduction to Probability theory and Markov Chain theory, and then illustrates the usefulness of these theories by studying a range of stochastic algorithms with important applications in optimisation and other problems in computing. One of the algorithms covered is the Markov chain Monte Carlo method. Finally, Kulkarni (1995) provides a rigorous analysis of many types of useful stochastic processes, e.g. discrete and continuous time Markov Chains, renewal processes, regenerative processes, and Markov regenerative processes.

The reader may find three other papers helpful. Izquierdo et al. (2009) analyses the dynamics of ten well-known models in the social simulation literature using the theory of Markov Chains, and is thus a good illustration of the approach in practice within the context of social simulation.Footnote 15 Epstein (2006) is a more general discussion, treating a variety of foundational and epistemological issues surrounding generative explanation in the social sciences, and discussing the role of agent-based computational models in generative social science. Finally, Leombruni and Richiardi (2005) usefully discusses several issues surrounding the interpretation of simulation dynamics and the generalisation of the simulation results. For a different approach to analysing the dynamics of a simulation model we refer the interested reader to Chap. 9 in this volume (Evans et al. 2013).

Rights and permissions

Reprints and Permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Izquierdo, L.R., Izquierdo, S.S., Galán, J.M., Santos, J.I. (2013). Combining Mathematical and Simulation Approaches to Understand the Dynamics of Computer Models. In: Edmonds, B., Meyer, R. (eds) Simulating Social Complexity. Understanding Complex Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-93813-2_11

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-93813-2_11

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-93812-5

  • Online ISBN: 978-3-540-93813-2

  • eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)