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Continuous-Time Random Walks and Temporal Networks

  • Renaud LambiotteEmail author
Chapter
Part of the Computational Social Sciences book series (CSS)

Abstract

Real-world networks often exhibit complex temporal patterns that affect their dynamics and function. In this chapter, we focus on the mathematical modelling of diffusion on temporal networks, and on its connection with continuous-time random walks. In that case, it is important to distinguish active walkers, whose motion triggers the activity of the network, from passive walkers, whose motion is restricted by the activity of the network. One can then develop renewal processes for the dynamics of the walker and for the dynamics of the network respectively, and identify how the shape of the temporal distribution affects spreading. As we show, the system exhibits non-Markovian features when the renewal process departs from a Poisson process, and different mechanisms tend to slow down the exploration of the network when the temporal distribution presents a fat tail. We further highlight how some of these ideas could be generalised, for instance to the case of more general spreading processes.

Keywords

Temporal networks Random walks Diffusion Renewal process Mixing 

Notes

Acknowledgements

I would like to thank my many collaborators without whom none of this work would have been done and, in particular, Naoki Masuda for co-writing [14] that was a great inspiration for this chapter.

References

  1. 1.
    Balescu, R.: Statistical Dynamics: Matter Out of Equilibrium. Imperial College, London (1997)CrossRefGoogle Scholar
  2. 2.
    Klafter, J., Sokolov, I.M.: First Steps in Random Walks: From Tools to Applications. Oxford University Press, New York (2011)CrossRefGoogle Scholar
  3. 3.
    Lovász, L., et al.: Random walks on graphs: a survey. Combinatorics, Paul Erdos is Eighty 2(1), 1–46 (1993)Google Scholar
  4. 4.
    Masuda, N., Porter, M.A., Lambiotte, R. Random walks and diffusion on networks. Phys. Rep. 716, 1–58 (2017)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    Blondel, V.D., Hendrickx, J.M., Olshevsky, A., Tsitsiklis, J.N.: Convergence in multiagent coordination, consensus, and flocking. In: Proceedings of the 44th IEEE Conference on Decision and Control, pp. 2996–3000. IEEE, Piscataway (2005)Google Scholar
  6. 6.
    Brin, S., Page, L.: Anatomy of a large-scale hypertextual web search engine. In: Proceedings of the Seventh International World Wide Web Conference, pp. 107–117 (1998)Google Scholar
  7. 7.
    Fouss, F., Saerens, M., Shimbo, M.: Algorithms and Models for Network Data and Link Analysis. Cambridge University Press, Cambridge (2016)CrossRefGoogle Scholar
  8. 8.
    Rosvall, M., Bergstrom, C.T.: Maps of random walks on complex networks reveal community structure. Proc. Natl. Acad. Sci. USA 105, 1118–1123 (2008)ADSCrossRefGoogle Scholar
  9. 9.
    Delvenne, J.C., Yaliraki, S.N., Barahona, M.: Stability of graph communities across time scales. Proc. Natl. Acad. Sci. USA 107, 12755–12760 (2010)ADSCrossRefGoogle Scholar
  10. 10.
    Lambiotte, R., Delvenne, J.C., Barahona, M.: Random walks, Markov processes and the multiscale modular organization of complex networks. IEEE Trans. Netw. Sci. Eng. 1, 76–90 (2014)Google Scholar
  11. 11.
    Newman, M.: Networks: An Introduction. Oxford University Press, Oxford (2010)CrossRefGoogle Scholar
  12. 12.
    Holme, P., Saramäki, J.: Temporal networks. Phys. Rep. 519(3), 97–125 (2012)ADSCrossRefGoogle Scholar
  13. 13.
    Holme, P.: Modern temporal network theory: a colloquium. Eur. Phys. J. B 88(9), 1–30 (2015)CrossRefGoogle Scholar
  14. 14.
    Masuda, N., Lambiotte, R.: A Guide to Temporal Networks. World Scientific, London (1996)zbMATHGoogle Scholar
  15. 15.
    Barabasi, A.-L.: The origin of bursts and heavy tails in human dynamics. Nature 435(7039), 207 (2005)ADSCrossRefGoogle Scholar
  16. 16.
    Malmgren, R.D., Stouffer, D.B., Motter, A.E., Amaral, L.A.N.: A poissonian explanation for heavy tails in e-mail communication. Proc. Natl. Acad. Sci. 105(47), 18153–18158 (2008)ADSCrossRefGoogle Scholar
  17. 17.
    Karrer, B., Newman, M.E.J.: Stochastic blockmodels and community structure in networks. Phys. Rev. E 83(1), 016107 (2011)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    Ispolatov, I., Krapivsky, P.L., Yuryev, A.: Duplication-divergence model of protein interaction network. Phys. Rev. E 71(6),061911 (2005)ADSCrossRefGoogle Scholar
  19. 19.
    Lambiotte, R., Krapivsky, P.L., Bhat, U., Redner, S.: Structural transitions in densifying networks. Phys. Rev. Lett. 117(21), 218301 (2016)ADSCrossRefGoogle Scholar
  20. 20.
    Hawkes, A.G.: Point spectra of some mutually exciting point processes. J. R. Stat. Soc. B 33, 438–443 (1971)ADSMathSciNetzbMATHGoogle Scholar
  21. 21.
    Masuda, N., Takaguchi, T., Sato, N., Yano, K.: Self-exciting point process modeling of conversation event sequences. In: Temporal Networks, pp. 245–264. Springer, Berlin (2013)Google Scholar
  22. 22.
    Kobayashi, R., Lambiotte, R.: Tideh: time-dependent hawkes process for predicting retweet dynamics. In: Tenth International AAAI Conference on Web and Social Media (2016)Google Scholar
  23. 23.
    Brockmann, D., Hufnagel, L., Geisel, T.: The scaling laws of human travel. Nature 439(7075), 462 (2006)ADSCrossRefGoogle Scholar
  24. 24.
    Perraudin, N., Vandergheynst, P.: Stationary signal processing on graphs. IEEE Trans. Signal Process. 65(13), 3462–3477 (2017)ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    Chung, F.R.K., Graham, F.C.: Spectral Graph Theory. Number 92. American Mathematical Society, Providence (1997)Google Scholar
  26. 26.
    De Nigris, S., Hastir, A., Lambiotte, R.: Burstiness and fractional diffusion on complex networks. Eur. Phys. J. B 89(5), 114 (2016)ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    Hoffmann, T., Porter, M.A., Lambiotte, R.: Generalized master equations for non-Poisson dynamics on networks. Phys. Rev. E 86, 046102 (2012)ADSCrossRefGoogle Scholar
  28. 28.
    Speidel, L., Lambiotte, R., Aihara, K., Masuda, N.: Steady state and mean recurrence time for random walks on stochastic temporal networks. Phys. Rev. E 91, 012806 (2015)ADSMathSciNetCrossRefGoogle Scholar
  29. 29.
    Allen, A.O.: Probability, Statistics, and Queueing Theory: With Computer Science Applications, 2nd edn. Academic Press, Boston (1990)zbMATHGoogle Scholar
  30. 30.
    Saramäki, J., Holme, P.: Exploring temporal networks with greedy walks. Eur. Phys. J. B 88(12), 334 (2015)ADSCrossRefGoogle Scholar
  31. 31.
    Gueuning, M., Lambiotte, R., Delvenne, J.-C.: Backtracking and mixing rate of diffusion on uncorrelated temporal networks. Entropy 19(10), 542 (2017)ADSCrossRefGoogle Scholar
  32. 32.
    Karsai, M., Kivelä, M., Pan, R.K., Kaski, K., Kertész, J., Barabási, A.-L., Saramäki, J.: Small but slow world: how network topology and burstiness slow down spreading. Phys. Rev. E 83(2), 025102 (2011)ADSCrossRefGoogle Scholar
  33. 33.
    Moinet, A., Starnini, M., Pastor-Satorras, R.: Random walks in non-poissoinan activity driven temporal networks. arXiv preprint arXiv:1904.10749 (2019)Google Scholar
  34. 34.
    Moinet, A., Starnini, M., Pastor-Satorras, R.: Burstiness and aging in social temporal networks. Phys. Rev. Lett. 114(10), 108701 (2015)ADSCrossRefGoogle Scholar
  35. 35.
    Scholtes, I., Wider, N., Pfitzner, R., Garas, A., Tessone, C.J., Schweitzer, F.: Causality-driven slow-down and speed-up of diffusion in non-markovian temporal networks. Nat. Commun. 5, 5024 (2014)ADSCrossRefGoogle Scholar
  36. 36.
    Lambiotte, R., Rosvall, M., Scholtes, I.: From networks to optimal higher-order models of complex systems. Nat. Phys. 1 (2019)Google Scholar
  37. 37.
    Petit, J., Gueuning, M., Carletti, T., Lauwens, B., Lambiotte, R.: Random walk on temporal networks with lasting edges. Phys. Rev. E 98(5), 052307 (2018)ADSCrossRefGoogle Scholar
  38. 38.
    Zhao, Q., Erdogdu, M.A., He, H.Y., Rajaraman, A., Leskovec, J.: Seismic: a self-exciting point process model for predicting tweet popularity. In: Proceedings of the 21th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 1513–1522. ACM, New York (2015)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Mathematical InstituteUniversity of OxfordOxfordUK

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