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Random Walks on Stochastic Temporal Networks

  • Till Hoffmann
  • Mason A. Porter
  • Renaud Lambiotte
Chapter
Part of the Understanding Complex Systems book series (UCS)

Abstract

In the study of dynamical processes on networks, there has been intense focus on network structure—i.e., the arrangement of edges and their associated weights—but the effects of the temporal patterns of edges remains poorly understood. In this chapter, we develop a mathematical framework for random walks on temporal networks using an approach that provides a compromise between abstract but unrealistic models and data-driven but non-mathematical approaches. To do this, we introduce a stochastic model for temporal networks in which we summarize the temporal and structural organization of a system using a matrix of waiting-time distributions. We show that random walks on stochastic temporal networks can be described exactly by an integro-differential master equation and derive an analytical expression for its asymptotic steady state. We also discuss how our work might be useful to help build centrality measures for temporal networks.

Keywords

Static Network Probability Distribution Function Probability Mass Function Temporal Network Dominant Eigenvector 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

This chapter is based on [20], which contains additional calculations and numerical simulations. RL would like to acknowledge support from FNRS (MIS-2012-F.4527.12) and Belspo (PAI Dysco). MAP acknowledges a research award (#220020177) from the James S. McDonnell Foundation and a grant from the EPSRC (EP/J001759/1). We thank T. Carletti, J.-C. Delvenne, P. J. Mucha, M. Rosvall, and J. Saramäki for fruitful discussions, and we thank P. Holme for helpful comments in his review of this manuscript.

References

  1. 1.
    Allesina, S., Pascual, M.: Googling food webs: can an eigenvector measure species’ importance for coextinctions? PLoS Comput. Biol. 5, e1000494 (2009)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Balescu, R.: Statistical Dynamics. Imperial College Press, London (1997)zbMATHGoogle Scholar
  3. 3.
    Barabási, A.-L.: The origin of bursts and heavy tails in human dynamics. Nature 435, 207 (2005)ADSCrossRefGoogle Scholar
  4. 4.
    Beguerisse Díaz, M., Porter, M.A., Onnela, J.-P.: Competition for popularity in catalog networks. Chaos 20, 043101 (2010)ADSCrossRefGoogle Scholar
  5. 5.
    Bergstrom, C., West, J., Wiseman, M.: The eigenfactor metrics. J. Neurosci. 28, 11433 (2008)CrossRefGoogle Scholar
  6. 6.
    Boccaletti, S., Latora, V., Moreno, Y., Chavez, M., Hwang, D.-U.: Complex networks: Structure and dynamics. Phys. Rep. 424, 175–308 (2006)MathSciNetADSCrossRefGoogle Scholar
  7. 7.
    Brin, S., Page, L.: The anatomy of a large-scale hypertextual Web search engine. In: Proceedings of the 7th International Conference on World Wide Web (WWW), pp. 107–117, Elsevier, Amsterdam, The Netherlands (1998)Google Scholar
  8. 8.
    Caley, P., Becker, N.G., Philp, D.J.: The waiting time for inter-country spread of pandemic influenza. PLoS ONE 2, e143 (2007)ADSCrossRefGoogle Scholar
  9. 9.
    Callaghan, T., Mucha, P.J., Porter, M.A.: Random walker ranking for NCAA Division IA football. Am. Math. Mon. 114, 761–777 (2007)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Chung, F.: Spectral Graph Theory. CBMS Regional Conference Series in Mathematics, No. 92. American Mathematical Society, Providence (1996)Google Scholar
  11. 11.
    Delvenne, J.-C., Yaliraki, S., Barahona, M.: Stability of graph communities across time scales. Proc. Natl. Acad. Sci. USA 107, 12755 (2010)ADSCrossRefGoogle Scholar
  12. 12.
    Eckmann, J.-P., Moses, E., Sergi, D.: Entropy of dialogues creates coherent structures in e-mail traffic. Proc. Natl. Acad. Sci. USA 101, 14333 (2004)MathSciNetADSzbMATHCrossRefGoogle Scholar
  13. 13.
    Evans, T.S.: Complex networks. Contemp. Phys. 45, 455 (2004)ADSCrossRefGoogle Scholar
  14. 14.
    Fernández-Gracia, J., Eguíluz, V., San Miguel, M.: Update rules and interevent time distributions: slow ordering versus no ordering in the voter model. Phys. Rev. E 84, 015103 (2011)ADSCrossRefGoogle Scholar
  15. 15.
    Ferreira, A.: On models and algorithms for dynamic communication networks: the case for evolving graphs. In: Proceedings of 4e Rencontres Francophones sur les Aspects Algorithmiques des Télécommunications (ALGOTEL2002), pp. 155–161, INRIA Press, Mèze, France (2002)Google Scholar
  16. 16.
    Ghosh, R., Lerman, K., Surachawala, T., Voevodski, K., Teng, S.-T.: Non-conservative diffusion and its application to social network analysis. arXiv:1102.4639 (2011)Google Scholar
  17. 17.
    Grindrod, P., Parsons, M.C., Higham, D.J., Estrada, E.: Communicability across evolving networks. Phys. Rev. E 83, 046120 (2011)ADSCrossRefGoogle Scholar
  18. 18.
    Hethcote, H.W., Tudor, D.W.: Integral equation models for endemic infectious diseases. J. Math. Biol. 9, 37 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Hoel, P., Port, S., Stone, C.: Introduction to Probability Theory. Houghton Mifflin, Boston, MA (1971)zbMATHGoogle Scholar
  20. 20.
    Hoffmann, T., Porter, M.A., Lambiotte, R.: Generalized master equations for non-Poisson dynamics on networks. Phys. Rev. E 86, 046102 (2012)ADSCrossRefGoogle Scholar
  21. 21.
    Holme, P., Saramäki, J.: Temporal networks. Phys. Rep. 519, 97 (2012)CrossRefGoogle Scholar
  22. 22.
    Iribarren, J.L., Moro, E.: Impact of human activity patterns on the dynamics of information diffusion. Phys. Rev. Lett. 103, 038702 (2009)ADSCrossRefGoogle Scholar
  23. 23.
    Iribarren, J.L., Moro, E.: Branching dynamics of viral information spreading. Phys. Rev. E 84, 046116 (2011)ADSCrossRefGoogle Scholar
  24. 24.
    Isella, L., Stehlé, J., Barrat, A., Cattuto, C., Pinton, J.-F., Van den Broeck, W.: What’s in a crowd? Analysis of face-to-face behavioral networks. J. Theor. Biol. 271, 166 (2011)CrossRefGoogle Scholar
  25. 25.
    Jeh, G., Widom, J.: SimRank: a measure of structural-context similarity. In: KDD’02: Proceedings of the eighth ACM SIGKDD international conference on Knowledge discovery and data mining, pp. 538–543, ACM, New York, NY (2002)Google Scholar
  26. 26.
    Karrer, B., Newman, M.E.J.: A message passing approach for general epidemic models. Phys. Rev. E 82, 016101 (2010)MathSciNetADSCrossRefGoogle Scholar
  27. 27.
    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, 025102(R) (2011)Google Scholar
  28. 28.
    Karsai, M., Kaski, K., Barabási, A.-L., Kertész, J.: Universal features of correlated bursty behaviour. Sci. Rep. 2, 397 (2012)ADSCrossRefGoogle Scholar
  29. 29.
    Kempe, D., Kleinberg, J., Kumar, A.: Connectivity and inference problems for temporal networks. J. Comp. Sys. Sci. 64, 820 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Kenkre, V.M., Andersen, J.D., Dunlap, D.H., Duke, C.B.: Unified theory of the mobilities of photo-injected electrons in naphthalene. Phys. Rev. Lett. 62, 1165 (1989)ADSCrossRefGoogle Scholar
  31. 31.
    Kivelä, M., Pan, R.K., Kaski, K., Kertész, J., Saramäki, J., Karsai, M.: Multiscale analysis of spreading in a large communication network. J. Stat. Mech. P03005 (2012)Google Scholar
  32. 32.
    Klafter, J., Sokolov, I.M.: Anomalous diffusion spreads its wings. Phys. World 18, 29 (2005)Google Scholar
  33. 33.
    Kleinberg, J.: Bursty and hierarchical structure in streams. Data Min. Knowl. Disc. 7, 373 (2003)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Kumar, R., Novak, J., Raghavan, P., Tomkins, A.: On the bursty evolution of blogspace. In: Proceedings of the 12th International Conference on World Wide Web (WWW), pp. 568–576, ACM, New York, NY (2003)Google Scholar
  35. 35.
    Lambiotte, R., Rosvall, M.: Ranking and clustering of nodes in networks with smart teleportation. Phys. Rev. E 85, 056107 (2012)ADSCrossRefGoogle Scholar
  36. 36.
    Lambiotte, R., Ausloos, M., Thelwall, M.: Word statistics in blogs and RSS feeds: Towards empirical universal evidence. J. Informetrics 1, 277 (2007)CrossRefGoogle Scholar
  37. 37.
    Lambiotte, R., Sinatra, R., Delvenne, J.-C., Evans, T.S., Barahona, M., Latora, V.: Flow graphs: Interweaving dynamics and structure. Phys. Rev. E 84, 017102 (2011)ADSCrossRefGoogle Scholar
  38. 38.
    Langville, A., Meyer, C.: Google’s PageRank and Beyond: The Science of Search Engine Rankings. Princeton University Press, Princeton (2006)Google Scholar
  39. 39.
    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. USA 105, 18153 (2008)ADSCrossRefGoogle Scholar
  40. 40.
    Miritello, G., Moro, E., Lara, R.: Dynamical strength of social ties in information spreading. Phys. Rev. E 83, 045102(R) (2011)Google Scholar
  41. 41.
    Montroll, E.W., Weiss, G.H.: Random walks on lattices. J. Math. Phys. 6, 167 (1965)MathSciNetADSCrossRefGoogle Scholar
  42. 42.
    Mucha, P.J., Richardson, T., Macon, K., Porter, M.A., Onnela, J.-P.: Community structure in time-dependent, multiscale, and multiplex networks. Science 328, 876 (2010)MathSciNetADSzbMATHCrossRefGoogle Scholar
  43. 43.
    Newman, M.E.J.: Networks: An Introduction. Oxford University Press, London (2010)zbMATHGoogle Scholar
  44. 44.
    Oliveira, J.G., Barabási, A.-L.: Darwin and Einstein correspondence patterns. Nature 437, 1251 (2005)ADSCrossRefGoogle Scholar
  45. 45.
    Pan, R.K., Saramäki, J.: Path lengths, correlations, and centrality in temporal networks. Phys. Rev. E 84, 016105 (2011)ADSCrossRefGoogle Scholar
  46. 46.
    Radicchi, F.: Who is the best player ever? A complex network analysis of the history of professional tennis. PloS ONE 6, e17249 (2011)ADSCrossRefGoogle Scholar
  47. 47.
    Rocha, L.E.C., Liljeros, F., Holme, P.: Information dynamics shape the sexual networks of Internet-mediated prostitution. Proc. Natl. Acad. Sci. USA 107, 5706 (2010)ADSzbMATHCrossRefGoogle Scholar
  48. 48.
    Rosvall, M., Bergstrom, C.: Maps of information flow reveal community structure in complex networks. Proc. Natl. Acad. Sci. USA 105, 1118 (2008)ADSCrossRefGoogle Scholar
  49. 49.
    Sabatelli, L., Keating, S., Dudley, J., Richmond, P.: Waiting time distributions in financial markets. Eur. Phys. J. B 27, 273 (2002)MathSciNetADSGoogle Scholar
  50. 50.
    Scher, H., Lax, M.: Stochastic transport in a disordered solid. I. Theor. Phys. Rev. B 7, 4491 (1973)MathSciNetADSCrossRefGoogle Scholar
  51. 51.
    Starnini, M., Baronchelli, A., Barrat, A., Pastor-Satorras, R.: Random walks on temporal networks. Phys. Rev. E 85, 056115 (2012)ADSCrossRefGoogle Scholar
  52. 52.
    Stewart, W.J.: Probability, Markov Chains, Queues, and Simulation: The Mathematical Basis of Performance Modeling. Princeton University Press, Princeton (2009)Google Scholar
  53. 53.
    Takaguchi, T., Masuda, N.: Voter model with non-Poissonian interevent intervals. Phys. Rev. E 84, 036115 (2011)ADSCrossRefGoogle Scholar
  54. 54.
    Tang, J., Musolesi, M., Mascolo, C., Latora, V.: Characterising Temporal Distance and Reachability in Mobile and Online Social Networks. In: Proceedings of the 2nd ACM SIGCOMM Workshop on Online Social Networks (WOSN’09), pp. 118–124, ACM, New York, NY (2009)Google Scholar
  55. 55.
    Tang, J., Scellato, S., Musolesi, M., Mascolo, C., Latora, V.: Small-world behavior in time-varying graphs. Phys. Rev. E 81, 055101(R) (2010)Google Scholar
  56. 56.
    Vazquez, A., Balazs, R., Andras, L., Barabási, A.-L.: Impact of non-Poisson activity patterns on spreading processes. Phys. Rev. Lett. 98, 158702 (2007)ADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Till Hoffmann
    • 1
  • Mason A. Porter
    • 2
  • Renaud Lambiotte
    • 3
  1. 1.Department of PhysicsUniversity of OxfordOxfordUK
  2. 2.Oxford Centre for Industrial and Applied Mathematics, Mathematical Institute and CABDyN Complexity CentreUniversity of OxfordOxfordUK
  3. 3.Department of MathematicsUniversity of NamurNamurBelgium

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