Control Strategies of Contagion Processes in Time-Varying Networks

Part of the Theoretical Biology book series (THBIO)


The vast majority of strategies aimed at controlling contagion processes on networks consider a timescale separation between the evolution of the system and the unfolding of the process. However, in the real world, many networks are highly dynamical and evolve, in time, concurrently to the contagion phenomena. Here, we review the most commonly used immunization strategies on networks. In the first part of the chapter, we focus on controlling strategies in the limit of timescale separation. In the second part instead, we introduce results and methods that relax this approximation. In doing so, we summarize the main findings considering both numerical and analytically approaches in real as well as synthetic time-varying networks.



The results presented in Sect. 8.3.2 are adapted from Ref. [84] and obtained in collaboration with S. Liu and A. Vespignani.


  1. 1.
    Barrat, A., Barthélemy, M., Vespignani, A.: Dynamical Processes on Complex Networks. Cambridge University Press, Cambridge (2008)CrossRefzbMATHGoogle Scholar
  2. 2.
    Newman, M.E.J.: Networks. An Introduction. Oxford University Press, Oxford (2010)CrossRefzbMATHGoogle Scholar
  3. 3.
    Cohen, R., Havlin, S.: Complex Networks: Structure, Robustness and Function. Cambridge University Press, Cambridge (2010)CrossRefzbMATHGoogle Scholar
  4. 4.
    Butts, C.T.: Revisting the foundations of network analysis. Science 325, 414–416 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Gonçalves, B., Perra, N.: Social Phenomena: From Data Analysis to Models. Springer, Cham/New York (2015)CrossRefGoogle Scholar
  6. 6.
    Vespignani, A.: Predicting the behavior of techno-social systems. Science 325, 425–428 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Fortunato, S.: Community detection in graphs. Phys. Rep. 486, 75–174 (2010)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Wang, Z., Bauch, C.T., Bhattacharyya, S., d’Onofrio, A., Manfredi, P., Perc, M., Perra, N., Salathé, M., Zhao, D.: Statistical physics of vaccination. Phys. Rep. 664, 1–113 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kitsak, M., Gallos, L.K., Havlin, S., Makse, H.A.: Identification of infuential spreaders in complex networks. Nat. Phys. 6, 888 (2010)CrossRefGoogle Scholar
  10. 10.
    Pastor-Satorras, R., Vespignani, A.: Immunization of complex networks. Phys. Rev. E 63, 036104 (2002)CrossRefGoogle Scholar
  11. 11.
    Cohen, R., Havlin, S., and ben-Avraham, D.: Efficient immunization strategies for computer networks and populations. Phys. Rev. Lett. 91, 247901 (2003)Google Scholar
  12. 12.
    Morris, M.: Telling tails explain the discrepancy in sexual partner reports. Nature 365, 437 (1993)CrossRefGoogle Scholar
  13. 13.
    Morris, M., Goodreau, S., Moody, J.: Sexual networks, concurrency, and STD/HIV, Ch. 7. In: Holmes, K.K. et al. (eds.) Sexually Transmitted Diseases. McGraw-Hill, New York, USA (2007)Google Scholar
  14. 14.
    Clauset, A., Eagle, N.: Persistence and periodicity in a dynamic proximity network. In: DIMACS Workshop on Computational Methods for Dynamic Interaction Networks, pp. 1–5 (2007)Google Scholar
  15. 15.
    Vespignani, A.: Modeling dynamical processes in complex socio-technical systems. Nat. Phys. 8, 32–30 (2012)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Rocha, L.E.C., Liljeros, F., Holme, P.: Simulated epidemics in an empirical spatiotemporal network of 50,185 sexual contacts. PLoS Comput. Biol. 7(3), e1001109, 03 (2011)Google Scholar
  17. 17.
    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)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Stehlé, J., Voirin, N., Barrat, A., Cattuto, C., Colizza, V., Isella, L., Régis, C., Pinton, J.-F., Khanafer, N., Van den Broeck, W., Vanhems, P.: Simulation of an SEIR infectious disease model on the dynamic contact network of conference attendees. BMC Med. 9(87) (2011)Google Scholar
  19. 19.
    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 (2011)CrossRefGoogle Scholar
  20. 20.
    Miritello, G., Moro, E., Lara, R.: Dynamical strength of social ties in information spreading. Phys. Rev. E 83, 045102 (2011)CrossRefGoogle Scholar
  21. 21.
    Kivelä, M., Pan, R., Kaski, K., Kertész, J., Saramäki, J., Karsai, M.: Multiscale analysis of spreading in a large communication network. J. Stat. Mech. 2012(3), P03005 (2012)CrossRefGoogle Scholar
  22. 22.
    Fujiwara, N., Kurths, J., Díaz-Guilera, A.: Synchronization in networks of mobile oscillators. Phys. Rev. E 83(2), 025101 (2011)CrossRefGoogle Scholar
  23. 23.
    Parshani, R., Dickison, M., Cohen, R., Stanley, H. E., Havlin, S.: Dynamic networks and directed percolation. EPL (Europhys. Lett.) 90(3), 38004 (2010)Google Scholar
  24. 24.
    Bajardi, P., Barrat, A., Natale, F., Savini, L., Colizza, V.: Dynamical patterns of cattle trade movements. PLoS One 6(5), e19869, 05 (2011)Google Scholar
  25. 25.
    Panisson, A., Barrat, A., Cattuto, C., Van den Broeck, W., Ruffo, G., Schifanella, R.: On the dynamics of human proximity for data diffusion in ad-hoc networks. Ad Hoc Netw. 10, 1532–1543 (2011)CrossRefGoogle Scholar
  26. 26.
    Baronchelli, A., Díaz-Guilera, A.: Consensus in networks of mobile communicating agents. Phys. Rev. E 85, 016113 (2012)CrossRefGoogle Scholar
  27. 27.
    Starnini, M., Baronchelli, A., Barrat, A., Pastor-Satorras, R.: Random walks on temporal networks. Phys. Rev. E 85, 056115 (2012)CrossRefGoogle Scholar
  28. 28.
    Pfitzner, R., Scholtes, I., Garas, A., Tessone, C.J., Schweitzer, F.: Betweenness preference: quantifying correlations in the topological dynamics of temporal networks. Phys. Rev. Lett. 110, 19 (2013)CrossRefGoogle Scholar
  29. 29.
    Karsai, M., Perra, N., Vespignani, A.: Time varying networks and the weakness of strong ties. Sci. Rep. 4, 4001 (2014)CrossRefGoogle Scholar
  30. 30.
    Hoffmann, T., Porter, M.A., Lambiotte, R.: Generalized master equations for non-poisson dynamics on networks. Phys. Rev. E 86, 046102 (2012)CrossRefGoogle Scholar
  31. 31.
    Toroczkai, Z., Guclu, H.: Proximity networks and epidemics. Phys. A 378(1), 68–75 (2007)CrossRefGoogle Scholar
  32. 32.
    Perra, N., Gonçalves, B., Pastor-Satorras, R., Vespignani, A.: Time scales and dynamical processes in activity driven networks. Sci. Rep. 2, 469 (2012)CrossRefGoogle Scholar
  33. 33.
    Ribeiro, B., Perra, N., Baronchelli, A.: Quantifying the effect of temporal resolution on time-varying networks. Sci. Rep. 3, 3006 (2013)CrossRefGoogle Scholar
  34. 34.
    Perra, N., Baronchelli, A., Mocanu, D., Gonçalves, B., Pastor-Satorras, R., Vespignani, A.: Random walks and search in time varying networks. Phys. Rev. Lett. 109, 238701 (2012)CrossRefGoogle Scholar
  35. 35.
    Liu, S., Baronchelli, A., Perra, N.: Contagion dynamics in time-varying metapopulations networks. Phys. Rev. E 87(032805) (2013)Google Scholar
  36. 36.
    Starnini, M., Pastor-Satorras, R.: Topological properties of a time-integrated activity-driven network. Phys. Rev. E 87, 062807 (2013)CrossRefGoogle Scholar
  37. 37.
    Takaguchi, T., Sato, N., Yano, K., Masuda, N.: Importance of individual events in temporal networkss. New J. Phys. 14, 093003 (2012)CrossRefGoogle Scholar
  38. 38.
    Takaguchi, T., Masuda, N., Holme, P.: Bursty communication patterns facilitate spreading in a threshold-based epidemic dynamics. PLoS One 8(7), e68629 (2013)CrossRefGoogle Scholar
  39. 39.
    Holme, P., Liljeros, F.: Birth and death of links control disease spreading in empirical contact networks. Sci. Rep. 4, 4999 (2014)CrossRefGoogle Scholar
  40. 40.
    Holme, P., Masuda, N.: The basic reproduction number as a predictor for epidemic outbreaks in temporal networks. PLoS One 10(3), e0120567 (2015)CrossRefGoogle Scholar
  41. 41.
    Holme, P., Saramäki, J.: Temporal networks. Phys. Rep. 519, 97 (2012)CrossRefGoogle Scholar
  42. 42.
    Holme, P.: Modern temporal network theory: a colloquium. Eur. Phys. J. B 88(9), 1–30 (2015)CrossRefGoogle Scholar
  43. 43.
    Masuda, N., Lambiotte, R.: A Guide to Temporal Networks. World Scientific, New Jersey (2016)CrossRefzbMATHGoogle Scholar
  44. 44.
    Kermack, W.O., McKendrick, A.G.: A contribution to the mathematical theory of epidemics. Proc. R. Soc. A 115, 700 (1927)CrossRefzbMATHGoogle Scholar
  45. 45.
    Keeling, M.J., Rohani, P.: Modeling Infectious Disease in Humans and Animals. Princeton University Press, Princeton (2008)zbMATHGoogle Scholar
  46. 46.
    Pastor-Satorras, R., Vespignani, A.: Epidemic spreading in scale-free networks. Phys. Rev. Lett. 86, 3200 (2001)CrossRefGoogle Scholar
  47. 47.
    Wang, Y., Chakrabarti, D., Wang, G., Faloutsos, C.: Epidemic spreading in real networks: an eigenvalue viewpoint. In: Proceedings of 22nd International Symposium on Reliable Distributed Systems, pp. 25–34 (2003)Google Scholar
  48. 48.
    Castellano, C., Pastor-Satorras, R.: Thresholds for epidemic spreading in networks. Phys. Rev. Lett. 105, 218701 (2010)CrossRefGoogle Scholar
  49. 49.
    Durrett, R.: Some features of the spread of epidemics and information on a random graph. Proc. Natl. Acad. Sci. 107, 4491–4498 (2010)CrossRefGoogle Scholar
  50. 50.
    Goltsev, A.V., Dorogovtsev, S.N., Oliveira, J.G., Mendes, J.F.F.: Localization and spreading of diseases in complex networks. Phys. Rev. Lett. 109(12), 128702 (2012)CrossRefGoogle Scholar
  51. 51.
    Boguñá, M., Castellano, C., Pastor-Satorras, R.: Nature of the epidemic threshold for the susceptible-infected-susceptible dynamics in networks. Phys. Rev. Lett. 111(6), 068701 (2013)CrossRefGoogle Scholar
  52. 52.
    Castellano, C., Pastor-Satorras, R.: Thresholds for epidemic spreading in networks. Phys. Rev. Lett. 105(21), 218701 (2010)CrossRefGoogle Scholar
  53. 53.
    Lee, H.K., Shim, P.S., Noh, J.D.: Epidemic threshold of the susceptible-infected-susceptible model on complex networks. Phys. Rev. E 87(6), 062812 (2013)CrossRefGoogle Scholar
  54. 54.
    Albert, R., Jeong, H., Barabási, A.L.: Error and attack tolerance of complex networks. Nature 406, 378 (2000)CrossRefGoogle Scholar
  55. 55.
    Pastor-Satorras, R., Vespignani, A.: Epidemics and Immunization in Scale-Free Networks. In: Bornholdt, S., Schuster, H.G. (eds.) Handbook of Graphs and Networks: From the Genome to the Internet, pp. 111–130. Wiley-VCH Verlag GmbH Co/KGaA, Weinheim (2005)Google Scholar
  56. 56.
    Perra, N., Fortunato, S.: Spectral centrality measures in complex networks. Phys. Rev. E 78(3), 036107 (2008)MathSciNetCrossRefGoogle Scholar
  57. 57.
    Hébert-Dufresne, L., Allard, A., Young, J.-G., Dubé, L.J.: Global efficiency of local immunization on complex networks. Sci. Rep. 3, 2171 (2013)CrossRefGoogle Scholar
  58. 58.
    Thedchanamoorthy, G., Piraveenan, M., Uddin, S., Senanayake, U.: Influence of vaccination strategies and topology on the herd immunity of complex networks. Soc. Netw. Anal. Min. 4(1), 213 (2014)CrossRefGoogle Scholar
  59. 59.
    Chen, Y., Paul, G., Havlin, S., Liljeros, F., Eugene Stanley, H.: Finding a better immunization strategy. Phys. Rev. Lett. 101, 058701 (2008)CrossRefGoogle Scholar
  60. 60.
    Holme, P., Kim, B.J., Yoon, C.N., Han, S.K.: Attack vulnerability of complex networks. Phys. Rev. E 65(5), 056109+ (2002)Google Scholar
  61. 61.
    Schneider, C.M., Mihaljev, T., Herrmann, H.J.: Inverse targeting – an effective immunization strategy. EPL (Europhys. Lett.) 98(4), 46002+ (2012)Google Scholar
  62. 62.
    Feld, S.L.: Why your friends have more friends than you do. Am. J. Sociol. 96(6), 1464–1477 (1991)CrossRefGoogle Scholar
  63. 63.
    Hodas, N.O., Kooti, F., Lerman, K.: Friendship paradox redux: your friends are more interesting than you. In: Proceedings of 7th International Conference on Weblogs and Social Media, Apr (2013)Google Scholar
  64. 64.
    Lattanzi, S., Singer, Y.: The power of random neighbors in social networks. In: Proceedings of the Eighth ACM International Conference on Web Search and Data Mining, WSDM ’15. ACM, New York, pp. 77–86 (2015)Google Scholar
  65. 65.
    Madar, N., Kalisky, T., Cohen, R., ben Avraham, D., Havlin, S.: Immunization and epidemic dynamics in complex networks. Eur. Phys. J. B 38(2), 269–276 (2004)Google Scholar
  66. 66.
    Dezső, Z., Barabási, A.: Halting viruses in scale-free networks. Phys. Rev. E 65, 055103 (2002)CrossRefGoogle Scholar
  67. 67.
    Gallos, L.K., Liljeros, F., Argyrakis, P., Bunde, A., Havlin, S.: Improving immunization strategies. Phys. Rev. E 75, 045104(R) (2007)Google Scholar
  68. 68.
    Tanaka, G., Urabe, C., Aihara, K.: Random and targeted interventions for epidemic control in metapopulation models. Sci. Rep. 4, 5522 EP–07 (2014)Google Scholar
  69. 69.
    Gong, K., Tang, M., Hui, P.M., Zhang, H.F., Younghae, D., Lai, Y.-C.: An efficient immunization strategy for community networks. PLoS One 8(12), 1–11 (2013)Google Scholar
  70. 70.
    Salathé, M., Jones, J.H.: Dynamics and control of diseases in networks with community structure. PLoS Comput. Biol. 6(4), e1000736 (2010)MathSciNetCrossRefGoogle Scholar
  71. 71.
    Eames, K.T.D.: Modelling disease spread through random and regular contacts in clustered populations. Theor. Popul. Biol. 73(1), 104–111 (2008)CrossRefzbMATHGoogle Scholar
  72. 72.
    Smieszek, T., Fiebig, L., Scholz, R.W.: Models of epidemics: when contact repetition and clustering should be included. Theor. Biol. Med. Model. 6(1), 1 (2009)CrossRefGoogle Scholar
  73. 73.
    Miller, J.C.: Spread of infectious disease through clustered populations. J. R. Soc. Interface 6, 1121–1134 (2009)CrossRefGoogle Scholar
  74. 74.
    Britton, T., Deijfen, M., Lagerås, A.N., Lindholm, M., et al.: Epidemics on random graphs with tunable clustering. J. Appl. Probab. 45(3), 743–756 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  75. 75.
    Cardillo, A., Reyes-Suárez, C., Naranjo, F., Gómez-Gardeñes, J.: Evolutionary vaccination dilemma in complex networks. Phys. Rev. E 88, 032803 (2013)CrossRefGoogle Scholar
  76. 76.
    Campbell E., Salathé, M.: Complex social contagion makes networks more vulnerable to disease outbreaks. Sci. Rep. 3, 1905 EP–05 (2013)Google Scholar
  77. 77.
    Prakash, B.A., Tong, H., Valler, N., Faloutsos, M., Faloutsos, C.: Virus propagation on time-varying networks: theory and immunization algorithms. In: Joint European Conference on Machine Learning and Knowledge Discovery in Databases. Springer, pp. 99–114 (2010)Google Scholar
  78. 78.
    Valdano, E., Ferreri, L., Poletto, C., Colizza, V.: Analytical computation of the epidemic threshold on temporal networks. Phys. Rev. X 5(2), 021005 (2015)Google Scholar
  79. 79.
    Tomasello, M.V., Perra, N., Tessone, C.J., Karsai, M., Schweitzer, F.: The role of endogenous and exogenous mechanisms in the formation of R&D networks. Sci. Rep. 4, 5679 (2014)Google Scholar
  80. 80.
    Ubaldi, E., Perra, N., Karsai, M., Vezzani, A., Burioni, R., Vespignani, A.: Asymptotic theory of time-varying social networks with heterogeneous activity and tie allocation. Sci. Rep. 6, 35724 (2016)CrossRefGoogle Scholar
  81. 81.
    Ubaldi, E., Vezzani, A., Karsai, M., Perra, N., Burioni, R.: Burstiness and tie activation strategies in time-varying social networks. Sci. Rep. 7, 46225 (2017)CrossRefGoogle Scholar
  82. 82.
    Alessandretti, L., Sun, K., Baronchelli, A., Perra, N.: Random walks on activity-driven networks with attractiveness. Phys. Rev. E 95, 052318 (2017)CrossRefGoogle Scholar
  83. 83.
    Starnini, M., Pastor-Satorras, R.: Topological properties of a time-integrated activity-driven network. Phys. Rev. E 87(6), 062807 (2013)CrossRefGoogle Scholar
  84. 84.
    Liu, S., Perra, N., Karsai, M., Vespignani, A.: Controlling contagion processes in activity driven networks. Phys. Rev. Lett. 112(11), 118702 (2014)CrossRefGoogle Scholar
  85. 85.
    Pentland, A., Eagle, N., Lazer, D.: Inferring social network structure using mobile phone data. Proc. Natl. Acad. Sci. (PNAS) 106(36), 15274–15278 (2009)Google Scholar
  86. 86.
    Lee, S., Rocha, L.E.C., Liljeros, F., Holme, P.: Exploiting temporal network structures of human interaction to effectively immunize populations. PLoS One 7, e36439 (2012)CrossRefGoogle Scholar
  87. 87.
    Rocha, L., Liljeros, F., Holme, P.: Information dynamics shape the sexual networks of internet-mediated prostitution. Proc. Natl. Acad. Sci. 107(13), 5706–5711 (2010)CrossRefzbMATHGoogle Scholar
  88. 88.
    Liljeros, F., Giesecke, J., Holme, P.: The contact network of inpatients in a regional healthcare system. a longitudinal case study. Math. Popul. Stud. 14(4), 269–284 (2007)Google Scholar
  89. 89.
    Eckmann, J.-P., Moses, E., Sergi, D.: Entropy of dialogues creates coherent structures in e-mail traffic. Proc. Natl. Acad. Sci. U. S. A. 101(40), 14333–14337 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  90. 90.
    Holme, P., Edling, CR., Liljeros, F.: Structure and time evolution of an internet dating community. Soc. Netw. 26(2), 155–174 (2004)CrossRefGoogle Scholar
  91. 91.
    Starnini, M., Machens, A., Cattuto, C., Barrat, A., Pastor-Satorras, R.: Immunization strategies for epidemic processes in time-varying contact networks. J. Theor. Biol. 337, 89–100 (2013)MathSciNetCrossRefGoogle Scholar
  92. 92.
  93. 93.
    Tang, J., Mascolo, C., Musolesi, M., Latora, V.: Exploiting temporal complex network metrics in mobile malware containment. In: 2011 IEEE International Symposium on a World of Wireless, Mobile and Multimedia Networks (WoWMoM), IEEE, pp. 1–9 (2011)Google Scholar
  94. 94.
    Rachuri, K.K., Musolesi, M., Mascolo, C., Rentfrow, P.J., Longworth, C., Aucinas, A.: Emotionsense: a mobile phones based adaptive platform for experimental social psychology research. In: Proceedings of the 12th ACM International Conference on Ubiquitous Computing. ACM, pp. 281–290 (2010)Google Scholar
  95. 95.
    Scott, J., Gass, R., Crowcroft, J., Hui, P., Diot, C., Chaintreau, A.: CRAWDAD dataset Cambridge/haggle (v. 2009-05-29). Downloaded from, May 2009
  96. 96.
    Funk, S., Salathé, M., Jansen, V.A.A.: Modelling the influence of human behaviour on the spread of infectious diseases: a review. J. R. Soc. Interface 7(50), 1247–1256 (2010)CrossRefGoogle Scholar
  97. 97.
    Wang, Z., Andrews, M.A., Wu, Z.X., Wang, L., Bauch, C.T.: Coupled disease–behavior dynamics on complex networks: a review. Phys. Life Rev. 15, 1–29 (2015)CrossRefGoogle Scholar
  98. 98.
    Weng, L., Ratkiewicz, J., Perra, N., Gonçalves, B., Castillo, C., Bonchi, F., Schifanella, R., Menczer, F., Flammini, A.: The role of information diffusion in the evolution of social networks. In: Proceedings of the 19th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. ACM, pp. 356–364 (2013)Google Scholar

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© Springer Nature Singapore Pte Ltd. 2017

Authors and Affiliations

  1. 1.Univ de Lyon, ENS de Lyon, INRIA, CNRSLyonFrance
  2. 2.Centre for Business Network AnalysisUniversity of GreenwichLondonUK

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