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Spreading processes on temporal networks

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My talk at the workshop Critical and collective effects in graphs and networks, April 28, 2016.
http://discrete-mathematics.org/?page_id=451

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Spreading processes on temporal networks

  1. 1. Spreading phenomena on temporal networks Petter Holme
  2. 2. Temporal networks
  3. 3. network
  4. 4. time
  5. 5. Temporal networks How can we measure them? sociopatterns.org
  6. 6. Temporal networks Contact sequences i 2 6 2 10 7 3 5 2 7 10 j 4 8 8 11 2 5 3 10 3 2 t 10 10 15 20 22 25 30 30 31 34
  7. 7. Temporal networks Timelines of nodes 0 5 10 15 20 1 2 3 4 5 6 t
  8. 8. Temporal networks Annotated graphs E D C B A 11,20 1,4,8 3,8,10,17 11,15 16
  9. 9. Temporal network epidemiology
  10. 10. Temporal nwk. epidemiology Susceptible meets Infectious Infectious With some probability or rate Susceptible or Recovered With some rate or after some time Step 1: Compartmental models
  11. 11. time Temporal nwk. epidemiology Step 2: Contact patterns
  12. 12. Time matters
  13. 13. Time matters E D C B A 11,20 1,4,8 3,8,10,17 11,15 16
  14. 14. Rocha, Liljeros, Holme, 2010. PNAS 107: 5706-5711. Escort/sex-buyer contacts: 16,730 individuals 50,632 contacts 2,232 days Time matters Rocha, Liljeros, Holme, 2011. PLoS Comp Biol 7: e1001109. 0 0.2 0.4 0.6 0 200 400 600 Time (days) 800 Fractionofinfectious 0 0.2 0.4 0.6 0 200 400 600 Time (days) Empirical Randomized 800 Fractionofinfectious 1555 i j t 5 7 1021 20 9 1119 4 30 1539 i j t 5 7 1555 20 9 1021 4 30 1119 4 20 15394 20
  15. 15. Time matters
  16. 16. Rocha, Liljeros, Holme Karsai, & al. Time matters 0 0.2 0.4 0.6 0 200 400 600 Time (days) Empirical Randomized 800 Fractionofinfectious 1 0.8 0.6 0.4 0.2 0 0 100 200 300 Time (days) Fractionofinfectious
  17. 17. 1 Understanding Complex Systems Petter Holme Jari Saramäki Editors Temporal Networks Physics Reports 519 (2012) 97–125 Contents lists available at SciVerse ScienceDirect Physics Reports journal homepage: www.elsevier.com/locate/physrep Temporal networks Petter Holmea,b,c,⇤ , Jari Saramäkid a IceLab, Department of Physics, Umeå University, 901 87 Umeå, Sweden b Department of Energy Science, Sungkyunkwan University, Suwon 440–746, Republic of Korea c Department of Sociology, Stockholm University, 106 91 Stockholm, Sweden d Department of Biomedical Engineering and Computational Science, School of Science, Aalto University, 00076 Aalto, Espoo, Finland a r t i c l e i n f o Article history: Accepted 1 March 2012 Available online 6 March 2012 editor: D.K. Campbell a b s t r a c t A great variety of systems in nature, society and technology – from the web of sexual contacts to the Internet, from the nervous system to power grids – can be modeled as graphs of vertices coupled by edges. The network structure, describing how the graph is wired, helps us understand, predict and optimize the behavior of dynamical systems. In many cases, however, the edges are not continuously active. As an example, in networks of communication via e-mail, text messages, or phone calls, edges represent sequences of instantaneous or practically instantaneous contacts. In some cases, edges are active for non-negligible periods of time: e.g., the proximity patterns of inpatients at hospitals can be represented by a graph where an edge between two individuals is on throughout the time they are at the same ward. Like network topology, the temporal structure of edge activations can affect dynamics of systems interacting through the network, from disease contagion on the network of patients to information diffusion over an e-mail network. In this review, we present the emergent field of temporal networks, and discuss methods for analyzing topological and temporal structure and models for elucidating their relation to the behavior of dynamical systems. In the light of traditional network theory, one can see this framework as moving the information of when things happen from the dynamical system on the network, to the network itself. Since fundamental properties, such as the transitivity of edges, do not necessarily hold in temporal networks, many of these methods need to be quite different from those for static networks. The study of temporal networks is very interdisciplinary in nature. Reflecting this, even the object of study has many names— temporal graphs, evolving graphs, time-varying graphs, time-aggregated graphs, time- stamped graphs, dynamic networks, dynamic graphs, dynamical graphs, and so on. This review covers different fields where temporal graphs are considered, but does not attempt to unify related terminology—rather, we want to make papers readable across disciplines. © 2012 Elsevier B.V. All rights reserved. Contents 1. Introduction.......................................................................................................................................................................................... 98 2. Types of temporal networks................................................................................................................................................................ 100 2.1. Person-to-person communication.......................................................................................................................................... 100 2.2. One-to many information dissemination............................................................................................................................... 101 2.3. Physical proximity ................................................................................................................................................................... 101 2.4. Cell biology............................................................................................................................................................................... 101 ⇤ Corresponding author at: IceLab, Department of Physics, Umeå University, 901 87 Umeå, Sweden. E-mail address: petter.holme@physics.umu.se (P. Holme). 0370-1573/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.physrep.2012.03.001
  18. 18. Holme, Scientific Reports, 2015. Time matters 0.2 0.3 0.1 0 P(Ω) Temporal network 0.001 0.01 0.1 1 0.001 0.01 0.1 1 λ δ/T 0.2 0.6 0.4 0 0.5 0 1 P(Ω) P(Ω) Static network Fully mixed 0.001 0.01 0.1 1 0.001 0.01 0.1 1 λ δ/T 0.001 0.01 0.1 1 0.001 0.01 0.1 1 λ δ/T
  19. 19. Temporal structures
  20. 20. History Network 1. Laszlo Barabási discovers a power-law. 2. It makes a difference for spreading dynamics. 3. It helps us to understand real epidemics. Time 1. Laszlo Barabási discovers a power-law. 2. It makes a difference for spreading dynamics. 3. It helps us to understand real epidemics.
  21. 21. Fat-tailed interevent time distributions Slowing down of spreading. 10-12 10 -10 10 -8 10-6 10 -4 10 -2 10 0 10 2 10 4 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 Poisson Power-law Time Incidence/N Min, Goh, Vazquez, 2011. PRE 83, 036102. But both the cell phone and the prostitution data are bursty. So why are they different w.r.t. spreading? Interevent times
  22. 22. Europhys. Lett., 64 (3), pp. 427–433 (2003) EUROPHYSICS LETTERS 1 November 2003 Network dynamics of ongoing social relationships P. Holme(∗ ) Department of Physics, Ume˚a University - 901 87 Ume˚a, Sweden (received 21 July 2003; accepted in final form 22 August 2003) PACS. 89.65.-s – Social and economic systems. PACS. 89.75.Hc – Networks and genealogical trees. PACS. 89.75.-k – Complex systems. Abstract. – Many recent large-scale studies of interaction networks have focused on networks of accumulated contacts. In this letter we explore social networks of ongoing relationships with an emphasis on dynamical aspects. We find a distribution of response times (times between consecutive contacts of different direction between two actors) that has a power law shape over a large range. We also argue that the distribution of relationship duration (the time between the first and last contacts between actors) is exponentially decaying. Methods to reanalyze the data to compensate for the finite sampling time are proposed. We find that the degree distribution for networks of ongoing contacts fits better to a power law than the degree distribution of the network of accumulated contacts do. We see that the clustering and assortative mixing coefficients are of the same order for networks of ongoing and accumulated contacts, and that the structural fluctuations of the former are rather large. Introduction. – The recent development in database technology has allowed researchers to extract very large data sets of human interaction sequences. These large data sets are suitable to the methods and modeling techniques of statistical physics, and thus, the last years have witnessed the appearance of an interdisciplinary field between physics and sociology [1–3]. More specifically, these studies have focused on network structure —in what ways the networks
  23. 23. Limited communication capacity unveils strategies for human interaction Giovanna Miritello1,2 , Rube´n Lara2 , Manuel Cebrian3,4 & Esteban Moro1,5 1 Departamento de Matema´ticas & GISC, Universidad Carlos III de Madrid, 28911 Legane´s, Spain, 2 Telefo´nica Research, 28050 Madrid, Spain, 3 NICTA, Melbourne, Victoria 3010, Australia, 4 Department of Computer Science & Engineering, University of California at San Diego, La Jolla, CA 92093, USA, 5 Instituto de Ingenierı´a del Conocimiento, Universidad Auto´noma de Madrid, 28049 Madrid, Spain. Connectivity is the key process that characterizes the structural and functional properties of social networks. However, the bursty activity of dyadic interactions may hinder the discrimination of inactive ties from large interevent times in active ones. We develop a principled method to detect tie de-activation and apply it to a large longitudinal, cross-sectional communication dataset (<19 months, <20 million people). Contrary to the perception of ever-growing connectivity, we observe that individuals exhibit a finite communication capacity, which limits the number of ties they can maintain active in time. On average men display higher capacity than women, and this capacity decreases for both genders over their lifespan. Separating communication capacity from activity reveals a diverse range of tie activation strategies, from stable to exploratory. This allows us to draw novel relationships between individual strategies for human interaction and the evolution of social networks at global scale. any different forces govern the evolution of social relationships making them far from random. In recent years, the understanding of what mechanisms control the dynamics of activating or deactivating social SUBJECT AREAS: SCIENTIFIC DATA COMPLEX NETWORKS APPLIED MATHEMATICS STATISTICAL PHYSICS Received 15 January 2013 Accepted 2 May 2013 Published 6 June 2013 Correspondence and requests for materials should be addressed to E.M. (emoro@math.
  24. 24. time time (2,3) (2,4) (2,5) (3,4) (3,5) (4,5) (5,6) (1,2) (1,2) (2,3) (2,4) (2,5) (3,4) (3,5) (4,5) (5,6) Ongoing link picture
  25. 25. time (2,3) (2,4) (2,5) (3,4) (3,5) (4,5) (5,6) (1,2) (1,2) (2,3) (2,4) (2,5) (3,4) (3,5) (4,5) (5,6) time Link turnover picture
  26. 26. T0 T0 Beginning interval neutralized T0 T0 Interevent times neutralized End interval neutralized T0 T0 Reference models
  27. 27. SIR on prostitution data 0 0.1 0.2 0.3 0.1 0.2 0.90.8 10.70.60.50.40.3 0.1 1 0.01 0.001 per-contact transmission probability durationofinfection Ω
  28. 28. SIR on prostitution data 0 0.1 0.2 0.3 0.1 0.2 0.90.8 10.70.60.50.40.3 0.1 1 0.01 0.001 per-contact transmission probability durationofinfection Ω Interevent times neutralized
  29. 29. SIR on prostitution data 0 0.1 0.2 0.3 0.1 0.2 0.90.8 10.70.60.50.40.3 0.1 1 0.01 0.001 per-contact transmission probability durationofinfection Ω Beginning times neutralized
  30. 30. SIR on prostitution data 0 0.1 0.2 0.3 0.1 0.2 0.90.8 10.70.60.50.40.3 0.1 1 0.01 0.001 per-contact transmission probability durationofinfection Ω End times neutralized
  31. 31. SIR, average deviations 0 0.02 0.04 0.06 E-mail 1 0.1 0 0.05 Film 0 0.05 0.1 Dating 1 0.05 0.1 0.15 0.2 0 Forum 0 0.02 0.04 0.06 E-mail 2 0 0.02 0.06 0.08 0.04 Facebook 0 0.01 0.02 0.03 0.04 Prostitution 0 0.1 0.2 0.3 Hospital 0 0.04 0.06 0.08 0.02 Gallery 0 0.02 0.04 0.06 Conference 0.05 0.1 0 Dating 2 endtimes beginningtimes intereventtimes Holme, Liljeros, 2014. Scientific Reports 4: 4999.
  32. 32. Temporal structures 2 …a no-brain (low-brain?) approach but what about yet other structures?
  33. 33. School 2 ConferenceProstitution Hospital Reality Gallery 1 School 1 Gallery 2 –3 –2 –1 0lg λ –3 –2 –1 0 lgδ –3 –2 –1 0lg λ –3 –2 –1 0 lgδ –3 –2 –1 0lg λ –3 –2 –1 0 lgδ –3 –2 –1 0lg λ –3 –2 –1 0 lgδ –3 –2 –1 0lg λ –3 –2 –1 0 lgδ –3 –2 –1 0lg λ –3 –2 –1 0 lgδ –3 –2 –1 0lg λ –3 –2 –1 0 lgδ –3 –2 –1 0lg λ –3 –2 –1 0lgδ 0.75 10.50 τ / T 0.25
  34. 34. Outbreak duration School 2 ConferenceProstitution Hospital Reality Gallery 1 School 1 Gallery 2 –3 –2 –1 0lg λ –3 –2 –1 0 lgδ –3 –2 –1 0lg λ –3 –2 –1 0 lgδ –3 –2 –1 0lg λ –3 –2 –1 0 lgδ –3 –2 –1 0lg λ –3 –2 –1 0 lgδ –3 –2 –1 0lg λ –3 –2 –1 0 lgδ –3 –2 –1 0lg λ –3 –2 –1 0 lgδ –3 –2 –1 0lg λ –3 –2 –1 0 lgδ –3 –2 –1 0lg λ –3 –2 –1 0lgδ 0.75 10.50 τ / T 0.25
  35. 35. vs static nwks School 2 ConferenceProstitution Hospital Reality Gallery 1 School 1 Gallery 2 –3 –2 –1 0lg λ –3 –2 –1 0 lgδ –3 –2 –1 0lg λ –3 –2 –1 0 lgδ –3 –2 –1 0lg λ –3 –2 –1 0 lgδ –3 –2 –1 0lg λ –3 –2 –1 0 lgδ –3 –2 –1 0lg λ –3 –2 –1 0 lgδ –3 –2 –1 0lg λ –3 –2 –1 0 lgδ –3 –2 –1 0lg λ –3 –2 –1 0 lgδ –3 –2 –1 0lg λ –3 –2 –1 0lgδ 0.5 10–0.5 lg σ
  36. 36. –3 –2 –1 0lg λ –3 –2 –1 0 lgδ –3 –2 –1 0lg λ –3 –2 –1 0 lgδ –3 –2 –1 0lg λ –3 –2 –1 0 lgδ –3 –2 –1 0lg λ –3 –2 –1 0 lgδ –3 –2 –1 0lg λ –3 –2 –1 0 lgδ –3 –2 –1 0lg λ –3 –2 –1 0 lgδ –3 –2 –1 0lg λ –3 –2 –1 0 lgδ –3 –2 –1 0lg λ –3 –2 –1 0lgδ 0.5 10–0.5 lg σ School 2School 1 ConferenceProstitution Hospital Reality Gallery 1 Gallery 2 vs fully-connected nwks
  37. 37. Ridiculograms (network)
  38. 38. Ridiculograms (time) 0 T/2 Tt 0 T/2 Tt 0 T/2 Tt 0 T/2 Tt 0 T/2 Tt 0 T/2 Tt 0 T/2 Tt 0 T/2 Tt School 2School 1 ConferenceProstitution Hospital Reality Gallery 1 Gallery 2
  39. 39. Network structures avg. fraction of nodes present when 50% of contact happened avg. fraction of links present when 50% of contact happened avg. fraction of nodes present at 50% of the sampling time avg. fraction of links present at 50% of the sampling time frac. of nodes present 1st and last 10% of the contacts frac. of links present 1st and last 10% of the contacts frac. of nodes present 1st and last 10% of the sampling time frac. of links present 1st and last 10% of the sampling time Time evolution degree distribution, mean degree distribution, s.d. degree distribution, coefficient of variation degree distribution, skew Degree distribution link duration, mean link duration, s.d. link duration, coefficient of variation link duration, skew link interevent time, mean link interevent time, s.d. link interevent time, coefficient of variation link interevent time, skew Link activity Node activity node duration, mean node duration, s.d. node duration, coefficient of variation node duration, skew node interevent time, mean node interevent time, s.d. node interevent time, coefficient of variation node interevent time, skew Other network structure number of nodes clustering coefficient assortativity
  40. 40. Network structures 0 10.5 average life time of nodes 0.10 0.2 0.3 average life time of links fraction of nodes present after half of the contacts 0.5 10.6 0.7 0.8 0.9 x = 0.392 x = 0.418 x = 0.487 Prostitution Conference Hospital Gallery 2Gallery 1School 1 School 2 Reality
  41. 41. Vaccination
  42. 42. Assume we can vaccinate a fraction f, then how can we choose the people to vaccinate? Using only local info? Vaccination
  43. 43. Chose a person at random. Neighborhood vaccination Cohen, Havlin, ben Avraham, 2002.
  44. 44. Ask her to name a friend. Neighborhood vaccination Cohen, Havlin, ben Avraham, 2002.
  45. 45. Vaccinate the friend. Neighborhood vaccination Cohen, Havlin, ben Avraham, 2002.
  46. 46. Lee, Rocha, Liljeros, Holme, 2012. PLoS ONE 7:e36439. Vaccination
  47. 47. Lee, Rocha, Liljeros, Holme, 2012. PLoS ONE 7:e36439. Vaccination
  48. 48. Lee, Rocha, Liljeros, Holme, 2012. PLoS ONE 7:e36439. Vaccination
  49. 49. Lee, Rocha, Liljeros, Holme, 2012. PLoS ONE 7:e36439. Vaccination
  50. 50. Vaccination
  51. 51. Other temporal networks results
  52. 52. Spreading by threshold dynamics Takaguchi, Masuda, Holme, 2013. Bursty communication patterns facilitate spreading in a threshold-based epidemic dynamics. PLoS ONE 8:e68629. Karimi, Holme, 2013. Threshold model of cascades in empirical temporal networks. Physica A 392:3476– 3483. Random walks Holme, Saramäki, 2015. Exploring temporal networks with greedy walks. Eur J Phys B 88:334. Review papers Masuda, Holme, 2013. Predicting and controlling infectious disease epidemics using temporal networks. F1000Prime Rep. 5:6. Holme, 2015. Modern temporal network theory: A colloquium. Eur J Phys B 88:234. Holme, 2014. Analyzing temporal networks in social media, Proc. IEEE 102:1922–1933.
  53. 53. A B
  54. 54. A B C
  55. 55. Thank you! Спасибо! Collaborators: Jari Saramäki Naoki Masuda Luis Rocha Sungmin Lee Fredrik Liljeros Fariba Karimi Illustrations by: Mi Jin Lee

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