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Moment Semantics for Reversible Rule-Based Systems

  • Vincent DanosEmail author
  • Tobias Heindel
  • Ricardo Honorato-Zimmer
  • Sandro Stucki
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9138)

Abstract

We develop a notion of stochastic rewriting over marked graphs – i.e. directed multigraphs with degree constraints. The approach is based on double-pushout (DPO) graph rewriting. Marked graphs are expressive enough to internalize the ‘no-dangling-edge’ condition inherent in DPO rewriting. Our main result is that the linear span of marked graph occurrence-counting functions – or motif functions – form an algebra which is closed under the infinitesimal generator of (the Markov chain associated with) any such rewriting system. This gives a general procedure to derive the moment semantics of any such rewriting system, as a countable (and recursively enumerable) system of differential equations indexed by motif functions. The differential system describes the time evolution of moments (of any order) of these motif functions under the rewriting system. We illustrate the semantics using the example of preferential attachment networks; a well-studied complex system, which meshes well with our notion of marked graph rewriting. We show how in this case our procedure obtains a finite description of all moments of degree counts for a fixed degree.

Keywords

Stochastic processes Moment semantics Reversible Computing Graph rewriting Rule-based systems 

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References

  1. 1.
    The Preferential Attachment ODE Generator (2015). https://github.com/sstucki/pa-ode-gen/
  2. 2.
    Bapodra, M., Heckel, R.: From graph transformations to differential equations. ECEASST 30 (2010)Google Scholar
  3. 3.
    Barabási, A.L., Albert, R.: Emergence of scaling in random networks. Science 286(5439), 509–512 (1999)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Barabási, A.L., Albert, R., Jeong, H.: Mean-field theory for scale-free random networks. Physica A: Statistical Mechanics and its Applications 272(1), 173–187 (1999)CrossRefGoogle Scholar
  5. 5.
    Chaput, P., Danos, V., Panangaden, P., Plotkin, G.D.: Approximating Markov processes by averaging. Journal of the ACM 61(1), 5 (2014)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Corradini, A., Heindel, T., Hermann, F., König, B.: Sesqui-pushout rewriting. In: Corradini, A., Ehrig, H., Montanari, U., Ribeiro, L., Rozenberg, G. (eds.) ICGT 2006. LNCS, vol. 4178, pp. 30–45. Springer, Heidelberg (2006) CrossRefGoogle Scholar
  7. 7.
    Corradini, A., Montanari, U., Rossi, F., Ehrig, H., Heckel, R., Löwe, M.: Algebraic approaches to graph transformation. Part I: basic concepts and double pushout approach. In: Handbook of Graph Grammars and Computing by Graph Transformation, pp. 163–245 (1997)Google Scholar
  8. 8.
    Danos, V., Heindel, T., Honorato-Zimmer, R., Stucki, S.: Approximations for stochastic graph rewriting. In: Merz, S., Pang, J. (eds.) ICFEM 2014. LNCS, vol. 8829, pp. 1–10. Springer, Heidelberg (2014) Google Scholar
  9. 9.
    Danos, V., Honorato-Zimmer, R., Jaramillo-Riveri, S., Stucki, S.: Deriving rate equations for site graph rewriting systems. In: SASB (2013)Google Scholar
  10. 10.
    Danos, V., Honorato-Zimmer, R., Jaramillo-Riveri, S., Stucki, S.: Coarse-graining the dynamics of ideal branched polymers. In: Electronic Notes in Theoretical Computer Science, Workshop on Static Analysis and Systems Biology, SASB 2012, Deauville, pp. 47–64, April 2015Google Scholar
  11. 11.
    Dorogovtsev, S.N., Mendes, J.F.F.: Evolution of networks with aging of sites. Phys. Rev. E 62, 1842–1845 (2000)Google Scholar
  12. 12.
    Dorogovtsev, S.N., Mendes, J.F.F., Samukhin, A.N.: Structure of growing networks with preferential linking. Phys. Rev. Lett. 85, 4633–4636 (2000)CrossRefGoogle Scholar
  13. 13.
    Durrett, R., Gleeson, J.P., Lloyd, A.L., Mucha, P.J., Shi, F., Sivakoff, D., Socolar, J.E., Varghese, C.: Graph fission in an evolving voter model. Proceedings of the National Academy of Sciences 109(10), 3682–3687 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Ehrig, H., Ehrig, K., Habel, A., Pennemann, K.H.: Theory of constraints and application conditions: From graphs to high-level structures. Fundamenta Informaticae 74(1), 135–166 (2006)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Ehrig, H., Heckel, R., Korff, M., Löwe, M., Ribeiro, L., Wagner, A., Corradini, A.: Algebraic approaches to graph transformation. Part II: Single pushout approach and comparison with double pushout approach. In: Rozenberg, G. (ed.) Handbook of Graph Grammars and Computing by Graph Transformation, pp. 247–312. World Scientific, River Edge (1997) CrossRefGoogle Scholar
  16. 16.
    Ehrig, H., Pfender, M., Schneider, H.J.: Graph-grammars: an algebraic approach. In: 14th Annual IEEE Symposium on Switching and Automata Theory, pp. 167–180 (1973)Google Scholar
  17. 17.
    Ethier, S.N., Kurtz, T.G.: Markov Processes: Characterization and Convergence. Wiley (1986)Google Scholar
  18. 18.
    Evans, M.R., Ferrari, P.A., Mallick, K.: Matrix representation of the stationary measure for the multispecies TASEP. Journal of Statistical Physics 135(2), 217–239 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Fages, F., Soliman, S.: Formal cell biology in Biocham. In: Bernardo, M., Degano, P., Zavattaro, G. (eds.) SFM 2008. LNCS, vol. 5016, pp. 54–80. Springer, Heidelberg (2008) CrossRefGoogle Scholar
  20. 20.
    Feret, J., Danos, V., Harmer, R., Krivine, J., Fontana, W.: Internal coarse-graining of molecular systems. PNAS 106(16), 6453–6458 (2009)CrossRefGoogle Scholar
  21. 21.
    Gleeson, J.P.: High-accuracy approximation of binary-state dynamics on networks. Physical Review Letters 107(6), 068701 (2011)CrossRefGoogle Scholar
  22. 22.
    Harmer, R., Danos, V., Feret, J., Krivine, J., Fontana, W.: Intrinsic information carriers in combinatorial dynamical systems. Chaos 20(3) (2010)Google Scholar
  23. 23.
    Hayman, J., Heindel, T.: Pattern graphs and rule-based models: the semantics of Kappa. In: Pfenning, F. (ed.) FOSSACS 2013 (ETAPS 2013). LNCS, vol. 7794, pp. 1–16. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  24. 24.
    Heckel, R.: DPO transformation with open maps. In: Ehrig, H., Engels, G., Kreowski, H.-J., Rozenberg, G. (eds.) ICGT 2012. LNCS, vol. 7562, pp. 203–217. Springer, Heidelberg (2012) CrossRefGoogle Scholar
  25. 25.
    Heckel, R., Lajios, G., Menge, S.: Stochastic graph transformation systems. Fundam. Inform. 74(1), 63–84 (2006)zbMATHMathSciNetGoogle Scholar
  26. 26.
    Heckel, R., Wagner, A.: Ensuring consistency of conditional graph grammars - a constructive approach. Electronic Notes in Theoretical Computer Science 2(0), 118–126 (1995)CrossRefGoogle Scholar
  27. 27.
    van Kampen, N.: Stochastic processes in physics and chemistry, 3rd edition, North-Holland (2007)Google Scholar
  28. 28.
    Lack, S., Sobociński, P.: Adhesive categories. In: Walukiewicz, I. (ed.) FOSSACS 2004. LNCS, vol. 2987, pp. 273–288. Springer, Heidelberg (2004) CrossRefGoogle Scholar
  29. 29.
    Lack, S., Sobocinski, P.: Adhesive and quasiadhesive categories. Theoretical Informatics and Applications 39(2), 522–546 (2005)MathSciNetGoogle Scholar
  30. 30.
    Lopez, C.F., Muhlich, J.L., Bachman, J.A., Sorger, P.K.: Programming biological models in Python using PySB. Molecular Systems Biology 9(1) (2013)Google Scholar
  31. 31.
    Löwe, M.: Algebraic Approach to Single-Pushout Graph Transformation. Theoretical Computer Science 109(1&2), 181–224 (1993)zbMATHMathSciNetCrossRefGoogle Scholar
  32. 32.
    Lynch, J.F.: A logical characterization of individual-based models. In: 23rd Annual IEEE Symposium on Logic in Computer Science, LICS 2008, pp. 379–390. IEEE (2008)Google Scholar
  33. 33.
    Norris, J.R.: Markov chains. Cambridge series in statistical and probabilistic mathematics. Cambridge University Press (1998)Google Scholar
  34. 34.
    Shkarin, S.A.: Some results on solvability of ordinary linear differential equations in locally convex spaces. Mathematics of the USSR-Sbornik 71(1), 29 (1992)zbMATHMathSciNetCrossRefGoogle Scholar
  35. 35.
    Stukalin, E.B., Phillips III, H., Kolomeisky, A.B.: Coupling of two motor proteins: a new motor can move faster. Physical Review Letters 94(23), 238101 (2005)CrossRefGoogle Scholar
  36. 36.
    Thomas, P., Matuschek, H., Grima, R.: Intrinsic noise analyzer: a software package for the exploration of stochastic biochemical kinetics using the system size expansion. PloS ONE 7(6), e38518 (2012)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Vincent Danos
    • 1
    Email author
  • Tobias Heindel
    • 2
  • Ricardo Honorato-Zimmer
    • 2
  • Sandro Stucki
    • 3
  1. 1.Département d’InformatiqueÉcole Normale SupérieureParisFrance
  2. 2.School of InformaticsUniversity of EdinburghEdinburghUK
  3. 3.Programming Methods LaboratoryEPFLLausanneSwitzerland

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