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Complexity of Inferring Local Transition Functions of Discrete Dynamical Systems

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Implementation and Application of Automata (CIAA 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9223))

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Abstract

We consider the problem of inferring the local transition functions of discrete dynamical systems from observed behavior. Our focus is on synchronous systems whose local transition functions are threshold functions. We assume that the topology of the system is known and that the goal is to infer a threshold value for each node so that the system produces the observed behavior. We show that some of these inference problems are efficiently solvable while others are NP-complete, even when the underlying graph of the dynamical system is a simple path. We also identify a fixed parameter tractable problem in this context.

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Notes

  1. 1.

    We will use the term “stable configuration” instead of “fixed point” throughout this paper since we will be using the word “fixed” in the context of fixed parameter tractability.

References

  1. Abrahao, B., Chierichetti, F., Kleinberg, R., Panconesi, A.: Trace complexity of network inference. In: Proceedings of the 19th ACM SIGKDD, pp. 491–499. ACM (2013)

    Google Scholar 

  2. Adiga, A., Kuhlman, C.J., Marathe, M.V., Ravi, S.S., Rosenkrantz, D.J., Stearns, R.E.: Complexity of inferring local transition functions of discrete dynamical systems. Technical report NDSSL-TR-15-048, NDSSL, Virginia Bioinformatics Institute, Virginia Tech, Blacksburg, VA, June 2015

    Google Scholar 

  3. Barrett, C.L., Hunt III, H.B., Marathe, M.V., Ravi, S.S., Rosenkrantz, D.J., Stearns, R.E.: Complexity of reachability problems for finite discrete dynamical systems. J. Comput. Syst. Sci. 72(8), 1317–1345 (2006)

    Article  MathSciNet  Google Scholar 

  4. Barrett, C.L., Hunt III, H.B., Marathe, M.V., Ravi, S.S., Rosenkrantz, D.J., Stearns, R.E.: Modeling and analyzing social network dynamics using stochastic discrete graphical dynamical systems. Theo. Comput. Sci. 412(30), 3932–3946 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Barrett, C.L., Hunt III, H.B., Marathe, M.V., Ravi, S.S., Rosenkrantz, D.J., Stearns, R.E., Thakur, M.: Predecessor existence problems for finite discrete dynamical systems. Theo. Comput. Sci. 386(1–2), 3–37 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Barrett, C.L., Hunt III, H.B., Marathe, M.V., Ravi, S.S., Rosenkrantz, D.J., Stearns, R.E., Tosic, P.T.: Gardens of Eden and fixed points in sequential dynamical systems. In: DM-CCG, pp. 95–110 (2001)

    Google Scholar 

  7. Cormen, T., Leiserson, C., Rivest, R., Stein, C.: Introduction to Algorithms. MIT Press and McGraw-Hill, Cambridge (2009)

    MATH  Google Scholar 

  8. Durand, B.: A random NP-complete problem for inversion of 2D cellular automata. Theor. Comput. Sci. 148(1), 19–32 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  9. Easley, D., Kleinberg, J.: Networks, crowds and markets: reasoning about a highly connected world. Cambridge University Press, New York (2010)

    Book  Google Scholar 

  10. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-completeness. W. H. Freeman and Co., San Francisco (1979)

    MATH  Google Scholar 

  11. Goles, E., Martínez, S.: Neural and automata networks. Kluwer, Dordrecht (1990)

    Book  MATH  Google Scholar 

  12. Gomez Rodriguez, M., Leskovec, J., Krause, A.: Inferring networks of diffusion and influence. In: Proceedings of the 16th ACM SIGKDD, pp. 1019–1028. ACM (2010)

    Google Scholar 

  13. Gonzalez-Bailon, S., Borge-Holthoefer, J., Rivero, A., Moreno, Y.: The Dynamics of Protest Recruitment Through an Online Network. In: Nature Scientific Reports, pp. 1–7 (2011), doi:10.1038/srep00197

  14. Graham, R., Knuth, D., Patashnik, O.: Concrete Mathematics. Addison-Wesley, Reading (1994)

    MATH  Google Scholar 

  15. Granovetter, M.: Threshold models of collective behavior. Am. J. Sociol. 83(6), 1420–1443 (1978)

    Article  Google Scholar 

  16. Green, F.: NP-complete problems in cellular automata. Complex Syst. 1(3), 453–474 (1987)

    MATH  Google Scholar 

  17. Håstad, J.: Clique is hard to approximate within \(n^{1-\epsilon }\). Acta Mathematica 182, 105–142 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kearns, M.J., Vazirani, U.V.: An Introduction to Computational Learning Theory. MIT Press, Cambridge (1994)

    Google Scholar 

  19. Kleinberg, J.: Cascading behavior in networks: Algorithmic and economic issues. In: Nissan, N., Roughgarden, T., Tardos, E., Vazirani, V. (eds.) Algorithmic Graph Theory. ch. 24. Cambridge University Press, New York (2008)

    Google Scholar 

  20. Kosub, S., Homan, C.M.: Dichotomy results for fixed point counting in Boolean dynamical systems. In: Proceedings of the ICTCS, pp. 163–174 (2007)

    Google Scholar 

  21. Macy, M., Willer, R.: From factors to actors: Computational sociology and agent-based modeling. Ann. Rev. Sociol. 28, 143–166 (2002)

    Article  Google Scholar 

  22. Mortveit, H.S., Reidys, C.M.: An Introduction to Sequential Dynamical Systems. Springer, Heidelberg (2007)

    Google Scholar 

  23. Murphy, K.P.: Passively learning finite automata. Technical report, Computer Science Department, UC Davis (1995)

    Google Scholar 

  24. Neidermeier, R.: Invitation to Fixed Parameter Algorithms. Oxford University Press, New York (2006)

    Book  Google Scholar 

  25. Romero, D.M., Meeder, B., Kleinberg, J.: Differences in the mechanics of information diffusion across topics: idioms, political hashtags, and complex contagion on twitter. In: Proceedings of the 20th WWW, pp. 695–704. ACM (2011)

    Google Scholar 

  26. Shah, D., Zaman, T.: Rumors in a network: Who’s the culprit? IEEE Trans. Inf. Theory 57(8), 5163–5181 (2011)

    Article  MathSciNet  Google Scholar 

  27. Soundarajan, S., Hopcroft, J.E.: Recovering social networks from contagion information. In: Kratochvíl, J., Li, A., Fiala, J., Kolman, P. (eds.) TAMC 2010. LNCS, vol. 6108, pp. 419–430. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  28. Sutner, K.: Computational classification of cellular automata. Int. J. Gen. Syst. 41(6), 595–607 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  29. Trucano, T.G., Swiler, L.P., Igusa, T., Oberkampf, W.L., Pilch, M.: Calibration, validation and sensitivity analysis: What is what. Reliab. Eng. Syst. Saf. 91, 1331–1357 (2006)

    Article  Google Scholar 

  30. Ugander, J., Backstrom, L., Marlow, C., Kleinberg, J.: Structural diversity in social contagion. PNAS 109(9), 5962–5966 (2012)

    Article  Google Scholar 

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Acknowledgments

We thank the reviewers for carefully reading the manuscript and providing valuable suggestions. This work has been partially supported by DTRA Grant HDTRA1-11-1-0016 and DTRA CNIMS Contract HDTRA1-11-D-0016-0010, NSF NetSE Grant CNS-1011769, NSF SDCI Grant OCI-1032677 and NIH MIDAS Grant 5U01GM070694-11.

Author information

Authors and Affiliations

  1. Virginia Bioinformatics Institute, Virginia Tech, Blacksburg, VA, USA

    Abhijin Adiga, Chris J. Kuhlman & Madhav V. Marathe

  2. Computer Science Department, University at Albany – SUNY, Albany, NY, USA

    S. S. Ravi, Daniel J. Rosenkrantz & Richard E. Stearns

Authors
  1. Abhijin Adiga
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  2. Chris J. Kuhlman
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  3. Madhav V. Marathe
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  4. S. S. Ravi
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  5. Daniel J. Rosenkrantz
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  6. Richard E. Stearns
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Corresponding author

Correspondence to Abhijin Adiga .

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Editors and Affiliations

  1. Umeå University, Umeå, Sweden

    Frank Drewes

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Adiga, A., Kuhlman, C.J., Marathe, M.V., Ravi, S.S., Rosenkrantz, D.J., Stearns, R.E. (2015). Complexity of Inferring Local Transition Functions of Discrete Dynamical Systems. In: Drewes, F. (eds) Implementation and Application of Automata. CIAA 2015. Lecture Notes in Computer Science(), vol 9223. Springer, Cham. https://doi.org/10.1007/978-3-319-22360-5_3

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  • DOI: https://doi.org/10.1007/978-3-319-22360-5_3

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-22359-9

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