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Rejection-Based Simulation of Stochastic Spreading Processes on Complex Networks

Part of the Lecture Notes in Computer Science book series (LNBI,volume 11705)

Abstract

Stochastic processes can model many emerging phenomena on networks, like the spread of computer viruses, rumors, or infectious diseases. Understanding the dynamics of such stochastic spreading processes is therefore of fundamental interest. In this work we consider the wide-spread compartment model where each node is in one of several states (or compartments). Nodes change their state randomly after an exponentially distributed waiting time and according to a given set of rules. For networks of realistic size, even the generation of only a single stochastic trajectory of a spreading process is computationally very expensive.

Here, we propose a novel simulation approach, which combines the advantages of event-based simulation and rejection sampling. Our method outperforms state-of-the-art methods in terms of absolute runtime and scales significantly better while being statistically equivalent.

Keywords

  • Spreading process
  • SIR
  • Epidemic modeling
  • Monte-Carlo simulation
  • Gillespie Algorithm

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Notes

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    github.com/gerritgr/Rejection-Based-Epidemic-Simulation.

References

  1. Barabási, A.-L.: Network Science. Cambridge University Press, Cambridge (2016)

    MATH  Google Scholar 

  2. Barrat, A., Barthelemy, M., Vespignani, A.: Dynamical Processes on Complex Networks. Cambridge University Press, Cambridge (2008)

    CrossRef  Google Scholar 

  3. Porter, M., Gleeson, J.: Dynamical Systems on Networks: A Tutorial, vol. 4. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-319-26641-1

    CrossRef  MATH  Google Scholar 

  4. Goutsias, J., Jenkinson, G.: Markovian dynamics on complex reaction networks. Phys. Rep. 529(2), 199–264 (2013)

    MathSciNet  CrossRef  Google Scholar 

  5. Pastor-Satorras, R., Castellano, C., Van Mieghem, P., Vespignani, A.: Epidemic processes in complex networks. Rev. Mod. Phys. 87(3), 925 (2015)

    MathSciNet  CrossRef  Google Scholar 

  6. Kiss, I.Z., Miller, J.C., Simon, P.L.: Mathematics of Epidemics on Networks: From Exact to Approximate Models. IAM, vol. 46. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-50806-1

    CrossRef  MATH  Google Scholar 

  7. Simon, P.L., Taylor, M., Kiss, I.Z.: Exact epidemic models on graphs using graph-automorphism driven lumping. J. Math. Biol. 62(4), 479–508 (2011)

    MathSciNet  CrossRef  Google Scholar 

  8. Van Mieghem, P., Omic, J., Kooij, R.: Virus spread in networks. IEEE/ACM Trans. Netw. (TON) 17(1), 1–14 (2009)

    CrossRef  Google Scholar 

  9. Sahneh, F.D., Scoglio, C., Van Mieghem, P.: Generalized epidemic mean-field model for spreading processes over multilayer complex networks. IEEE/ACM Trans. Netw. (TON) 21(5), 1609–1620 (2013)

    CrossRef  Google Scholar 

  10. Gleeson, J.P.: High-accuracy approximation of binary-state dynamics on networks. Phys. Rev. Lett. 107(6), 068701 (2011)

    CrossRef  Google Scholar 

  11. Gleeson, J.P., Melnik, S., Ward, J.A., Porter, M.A., Mucha, P.J.: Accuracy of mean-field theory for dynamics on real-world networks. Phys. Rev. E 85(2), 026106 (2012)

    CrossRef  Google Scholar 

  12. Gleeson, J.P.: Binary-state dynamics on complex networks: pair approximation and beyond. Phys. Rev. X 3(2), 021004 (2013)

    Google Scholar 

  13. Devriendt, K., Van Mieghem, P.: Unified mean-field framework for susceptible-infected-susceptible epidemics on networks, based on graph partitioning and the isoperimetric inequality. Phys. Rev. E 96(5), 052314 (2017)

    CrossRef  Google Scholar 

  14. Bortolussi, L., Hillston, J., Latella, D., Massink, M.: Continuous approximation of collective system behaviour: a tutorial. Perform. Eval. 70(5), 317–349 (2013)

    CrossRef  Google Scholar 

  15. Prakash, B.A., Vreeken, J., Faloutsos, C.: Spotting culprits in epidemics: how many and which ones? In: 2012 IEEE 12th International Conference on Data Mining (ICDM), pp. 11–20. IEEE (2012)

    Google Scholar 

  16. Farajtabar, M., Gomez-Rodriguez, M., Du, N., Zamani, M., Zha, H., Song, L.: Back to the past: source identification in diffusion networks from partially observed cascades. In: Artificial Intelligence and Statistics (2015)

    Google Scholar 

  17. Schneider, C.M., Mihaljev, T., Havlin, S., Herrmann, H.J.: Suppressing epidemics with a limited amount of immunization units. Phys. Rev. E 84(6), 061911 (2011)

    CrossRef  Google Scholar 

  18. Cohen, R., Havlin, S., Ben-Avraham, D.: Efficient immunization strategies for computer networks and populations. Phys. Rev. Lett. 91(24), 247901 (2003)

    CrossRef  Google Scholar 

  19. Buono, C., Braunstein, L.A.: Immunization strategy for epidemic spreading on multilayer networks. EPL (Europhys. Lett.) 109(2), 26001 (2015)

    CrossRef  Google Scholar 

  20. Wu, Q., Fu, X., Jin, Z., Small, M.: Influence of dynamic immunization on epidemic spreading in networks. Phys. A 419, 566–574 (2015)

    CrossRef  Google Scholar 

  21. Cota, W., Ferreira, S.C.: Optimized Gillespie algorithms for the simulation of Markovian epidemic processes on large and heterogeneous networks. Comput. Phys. Commun. 219, 303–312 (2017)

    CrossRef  Google Scholar 

  22. St-Onge, G., Young, J.-G., Hébert-Dufresne, L., Dubé, L.J.: Efficient sampling of spreading processes on complex networks using a composition and rejection algorithm. arXiv preprint arXiv:1808.05859 (2018)

  23. Sahneh, F.D., Vajdi, A., Shakeri, H., Fan, F., Scoglio, C.: GEMFsim: a stochastic simulator for the generalized epidemic modeling framework. J. Comput. Sci. 22, 36–44 (2017)

    CrossRef  Google Scholar 

  24. Hayward, R., McDiarmid, C.: Average case analysis of heap building by repeated insertion. J. Algorithms 12(1), 126–153 (1991)

    MathSciNet  CrossRef  Google Scholar 

  25. Porter, T., Simon, I.: Random insertion into a priority queue structure. IEEE Trans. Softw. Eng. SE–1(3), 292–298 (1975)

    MathSciNet  CrossRef  Google Scholar 

  26. Masuda, N., Konno, N.: Multi-state epidemic processes on complex networks. J. Theor. Biol. 243(1), 64–75 (2006)

    MathSciNet  CrossRef  Google Scholar 

  27. Vestergaard, C.L., Génois, M.: Temporal gillespie algorithm: fast simulation of contagion processes on time-varying networks. PLoS Comput. Biol. 11(10), e1004579 (2015)

    CrossRef  Google Scholar 

  28. Masuda, N., Holme, P.: Temporal Network Epidemiology. Springer, Heidelberg (2017). https://doi.org/10.1007/978-981-10-5287-3

    CrossRef  MATH  Google Scholar 

  29. Holme, P., Saramäki, J.: Temporal networks. Phys. Rep. 519(3), 97–125 (2012)

    CrossRef  Google Scholar 

  30. Holme, P.: Modern temporal network theory: a colloquium. Eur. Phys. J. B 88(9), 234 (2015)

    CrossRef  Google Scholar 

  31. Fosdick, B.K., Larremore, D.B., Nishimura, J., Ugander, J.: Configuring random graph models with fixed degree sequences. SIAM Rev. 60(2), 315–355 (2018)

    MathSciNet  CrossRef  Google Scholar 

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Acknowledgments

This research has been partially funded by the German Research Council (DFG) as part of the Collaborative Research Center “Methods and Tools for Understanding and Controlling Privacy”. We thank Michael Backenköhler for his comments on the manuscript.

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Correspondence to Gerrit Großmann .

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Großmann, G., Wolf, V. (2019). Rejection-Based Simulation of Stochastic Spreading Processes on Complex Networks. In: Češka, M., Paoletti, N. (eds) Hybrid Systems Biology. HSB 2019. Lecture Notes in Computer Science(), vol 11705. Springer, Cham. https://doi.org/10.1007/978-3-030-28042-0_5

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  • DOI: https://doi.org/10.1007/978-3-030-28042-0_5

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