Skip to main content

Rejection-Based Simulation of Non-Markovian Agents on Complex Networks

  • Conference paper
  • First Online:
Complex Networks and Their Applications VIII (COMPLEX NETWORKS 2019)

Part of the book series: Studies in Computational Intelligence ((SCI,volume 881))

Included in the following conference series:

Abstract

Stochastic models in which agents interact with their neighborhood according to a network topology are a powerful modeling framework to study the emergence of complex dynamic patterns in real-world systems. Stochastic simulations are often the preferred—sometimes the only feasible—way to investigate such systems. Previous research focused primarily on Markovian models where the random time until an interaction happens follows an exponential distribution.

In this work, we study a general framework to model systems where each agent is in one of several states. Agents can change their state at random, influenced by their complete neighborhood, while the time to the next event can follow an arbitrary probability distribution. Classically, these simulations are hindered by high computational costs of updating the rates of interconnected agents and sampling the random residence times from arbitrary distributions.

We propose a rejection-based, event-driven simulation algorithm to overcome these limitations. Our method over-approximates the instantaneous rates corresponding to inter-event times while rejection events counter-balance these over-approximations. We demonstrate the effectiveness of our approach on models of epidemic and information spreading.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 1.

    github.com/gerritgr/non-markovian-simulation.

References

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

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  4. Kiss, I.Z., Miller, J.C., Simon, P.L.: Mathematics of Epidemics on Networks. Forthcoming in Springer TAM Series, Cham (2016)

    Google Scholar 

  5. Porter, M., Gleeson, J.: Dynamical Systems on Networks: A Tutorial, vol. 4. Springer, Cham (2016)

    Book  Google Scholar 

  6. Rodrigues, H.S.: Application of SIR epidemiological model: new trends. arXiv preprint. arXiv:1611.02565 (2016)

  7. Kitsak, M., Gallos, L.K., Havlin, S., Liljeros, F., Muchnik, L., Stanley, H.E., Makse, H.A.: Identification of influential spreaders in complex networks. Nat. Phys. 6(11), 888 (2010)

    Article  Google Scholar 

  8. Zhao, L., Wang, J., Chen, Y., Wang, Q., Cheng, J., Cui, H.: SIHR rumor spreading model in social networks. Phys. A Stat. Mech. Appl. 391(7), 2444–2453 (2012)

    Article  Google Scholar 

  9. Goltsev, A.V., De Abreu, F.V., Dorogovtsev, S.N., Mendes, J.F.F.: Stochastic cellular automata model of neural networks. Phys. Rev. E 81(6), 061921 (2010)

    Article  MathSciNet  Google Scholar 

  10. Meier, J., Zhou, X., Hillebrand, A., Tewarie, P., Stam, C.J., Van Mieghem, P.: The epidemic spreading model and the direction of information flow in brain networks. NeuroImage 152, 639–646 (2017)

    Article  Google Scholar 

  11. Gan, C., Yang, X., Liu, W., Zhu, Q., Zhang, X.: Propagation of computer virus under human intervention: a dynamical model. Discret. Dyn. Nat. Soc. 2012 (2012)

    Article  Google Scholar 

  12. May, R.M., Arinaminpathy, N.: Systemic risk: the dynamics of model banking systems. J. R. Soc. Interface 7(46), 823–838 (2009)

    Article  Google Scholar 

  13. Peckham, R.: Contagion: epidemiological models and financial crises. J. Public Health 36(1), 13–17 (2013)

    Article  Google Scholar 

  14. Masuda, N., Rocha, L.E.C.: A Gillespie algorithm for non-Markovian stochastic processes. SIAM Rev. 60(1), 95–115 (2018)

    Article  MathSciNet  Google Scholar 

  15. Lloyd, A.L.: Realistic distributions of infectious periods in epidemic models: changing patterns of persistence and dynamics. Theor. Popul. Biol. 60(1), 59–71 (2001)

    Article  Google Scholar 

  16. Yang, G.L.: Empirical study of a non-Markovian epidemic model. Math. Biosci. 14(1–2), 65–84 (1972)

    Article  Google Scholar 

  17. Blythe, S.P., Anderson, R.M.: Variable infectiousness in HFV transmission models. Math. Med. Biol. J. IMA 5(3), 181–200 (1988)

    Article  MathSciNet  Google Scholar 

  18. Hollingsworth, T.D., Anderson, R.M., Fraser, C.: HIV-1 transmission, by stage of infection. J. Infect. Dis. 198(5), 687–693 (2008)

    Article  Google Scholar 

  19. Feng, Z., Thieme, H.R.: Endemic models for the spread of infectious diseases with arbitrarily distributed disease stages I: general theory. SIAM J. Appl. Math. 61(3), 803–833 (2000)

    Article  MathSciNet  Google Scholar 

  20. Barabasi, A.-L.: The origin of bursts and heavy tails in human dynamics. Nature 435(7039), 207 (2005)

    Article  Google Scholar 

  21. Vázquez, A., Oliveira, J.G., Dezsö, Z., Goh, K.-I., Kondor, I., Barabási, A.-L.: Modeling bursts and heavy tails in human dynamics. Phys. Rev. E 73(3), 036127 (2006)

    Article  Google Scholar 

  22. Softky, W.R., Koch, C.: The highly irregular firing of cortical cells is inconsistent with temporal integration of random EPSPs. J. Neurosci. 13(1), 334–350 (1993)

    Article  Google Scholar 

  23. Boguná, M., Lafuerza, L.F., Toral, R., Serrano, M.A.: Simulating non-Markovian stochastic processes. Phys. Rev. E 90(4), 042108 (2014)

    Article  Google Scholar 

  24. 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)

    Article  Google Scholar 

  25. 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)

  26. Großmann, G., Wolf, V.: Rejection-based simulation of stochastic spreading processes on complex networks. In: International Workshop on Hybrid Systems Biology, pp. 63–79. Springer, Cham (2019)

    Chapter  Google Scholar 

  27. Cox, D.R.: Renewal Theory (1962)

    Google Scholar 

  28. Pasupathy, R.: Generating Homogeneous Poisson Processes. Wiley Encyclopedia of Operations Research and Management Science, Hoboken (2010)

    Google Scholar 

  29. Kiss, I.Z., Röst, G., Vizi, Z.: Generalization of pairwise models to non-Markovian epidemics on networks. Phys. Rev. Lett. 115(7), 078701 (2015)

    Article  Google Scholar 

  30. Pellis, L., House, T., Keeling, M.J.: Exact and approximate moment closures for non-Markovian network epidemics. J. Theor. Biol. 382, 160–177 (2015)

    Article  MathSciNet  Google Scholar 

  31. Jo, H.-H., Perotti, J.I., Kaski, K., Kertész, J.: Analytically solvable model of spreading dynamics with non-Poissonian processes. Phys. Rev. X 4(1), 011041 (2014)

    Google Scholar 

  32. Sherborne, N., Miller, J.C., Blyuss, K.B., Kiss, I.Z.: Mean-field models for non-Markovian epidemics on networks: from edge-based compartmental to pairwise models. arXiv preprint. arXiv:1611.04030 2016

  33. Starnini, M., Gleeson, J.P., Boguñá, M.: Equivalence between non-Markovian and Markovian dynamics in epidemic spreading processes. Phys. Rev. Lett. 118(12), 128301 (2017)

    Article  Google Scholar 

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

    Article  Google Scholar 

  35. Großmann, G., Bortolussi, L., Wolf, V.: Rejection-based simulation of non-Markovian agents on complex networks. arxiv.org/abs/1910.03964 (2019)

  36. Ogata, Y.: On Lewis’ simulation method for point processes. IEEE Trans. Inf. Theor. 27(1), 23–31 (1981)

    Article  Google Scholar 

  37. Dassios, A., Zhao, H., et al.: Exact simulation of Hawkes process with exponentially decaying intensity. Electron. Commun. 18 (2013)

    Google Scholar 

  38. 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)

    Article  MathSciNet  Google Scholar 

  39. Röst, G., Vizi, Z., Kiss, I.Z.: Impact of non-Markovian recovery on network epidemics. In: BIOMAT 2015: International Symposium on Mathematical and Computational Biology, pp. 40–53. World Scientific (2016)

    Google Scholar 

  40. Van Mieghem, P., Van de Bovenkamp, R.: Non-Markovian infection spread dramatically alters the susceptible-infected-susceptible epidemic threshold in networks. Phys. Rev. Lett. 110(10), 108701 (2013)

    Article  Google Scholar 

  41. Jo, H.-H., Lee, B.-H., Hiraoka, T., Jung, W.-S.: Copula-based algorithm for generating bursty time series. arXiv preprint. arXiv:1904.08795 (2019)

Download references

Acknowledgements

We thank Guillaume St-Onge for helpful comments on non-Markovian dynamics. This research was been partially funded by the German Research Council (DFG) as part of the Collaborative Research Center “Methods and Tools for Understanding and Controlling Privacy”.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Gerrit Großmann .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Großmann, G., Bortolussi, L., Wolf, V. (2020). Rejection-Based Simulation of Non-Markovian Agents on Complex Networks. In: Cherifi, H., Gaito, S., Mendes, J., Moro, E., Rocha, L. (eds) Complex Networks and Their Applications VIII. COMPLEX NETWORKS 2019. Studies in Computational Intelligence, vol 881. Springer, Cham. https://doi.org/10.1007/978-3-030-36687-2_29

Download citation

Publish with us

Policies and ethics