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From temporal to static networks, and back

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Infectious diseases are a major burden to global health. Understanding their mechanisms and being able to predict and intervene epidemic outbreaks is an important challenge for researchers and decision makers alike. It should not be too hard either―if we include human contact patterns, the mechanisms of contagion and the typical features of the disease, we could model most infectious-disease related phenomena. Of these three components, the network epidemiology of the last decade has shown that our limited understanding of human contact patterns is probably the most important focus are for advancing infectious disease epidemiology. We will discuss what is known about human contact patterns and how to include this knowledge in epidemic modeling. First, we discuss recent work on what the epidemiologically most important temporal structures of human contacts are. We use about 80 empirical temporal network datasets, several arguably important for disease spreading, and scan the entire parameter space of disease-spreading models. By comparing to null-models, we identify important, simple temporal patterns that affect disease spreading stronger than the bursty interevent time distributions. Furthermore, we investigate how to eliminate the temporal information to make an as relevant static network as possible. After all, static network epidemiology has more methods and results than temporal network epidemiology and it for some purposes it is necessary. We find that an “exponential threshold” representation almost always the best performance, but time-sliced network (with a carefully chosen window, usually considerably different than the sampling time of the data) works almost as good. In contrast, networks of concurrent contacts do not seem to carry so important information.

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From temporal to static networks, and back

  1. 1. TNETS edition
  2. 2. Susceptible Infectious Infectious Compartmental models
  3. 3. Infectious Recovered Susceptible
  4. 4. Contact patterns
  5. 5. ID1 ID2 Time 1 2 34 3 4 55 1 5 56 4 2 70 5 4 77 6 1 102 5 6 110 5 7 122 6 7 130 2 5 198 3 4 205 4 2 210 2 7 230
  6. 6. Sociopatterns gallery P H Y S I C A L P R O X I M I T Y Prostitution Sociopatterns conference Hospital system N = 16,730, L = 50,632, T = 6.0y N = 113, L = 20,818, T = 59h N = 159(8), L = 6,027(350), T = 7.3(1)h N = 293,878, L = 64,625,283, T = 3,570dReality mining N = 63, L = 26,260, T = 8.6h
  7. 7. ELECTRONIC COMMUNICATION N = 57,189, L = 444,162, T = 112.0d Bornholdt’s e-mail Eckmann’s e-mail N = 3,188, L = 115,684, T = 81.6d Filmtipset forum N = 7,084, L = 1,412,401, T = 8.61y Filmtipset messages Pussokram dating N = 28,972, L = 529,890, T = 512.0d QX dating N = 80,683, L = 4,337,203, T = 63.7d N = 35,624, L = 472,496, T = 8.27y Facebook wall posts N = 293,878, L = 876,993, T = 1591d
  8. 8. TEMPORAL TO STATIC
  9. 9. tstart tstop 0 5 10 15 20 1 2 3 4 5 6 t 1 2 3 4 5 6 time-slice networks
  10. 10. tstart tstop 0 5 10 15 20 1 2 3 4 5 6 t 1 2 3 4 5 6 ongoing networks
  11. 11. 0 5 10 15 20 1 2 3 4 5 6 t 1 2 3 4 5 6 1 2 3 4 5 6 Ω = 2.5 τ = 10 exponential-threshold networks
  12. 12. GOOD REPRESENTATION: RANKING OF IMPORTANT VERTICES CONSERVED FOR ALL PARAMETER VALUES: MEASURE AVG OUTBREAK SIZE WHEN SPREADING STARTS AT i FOR ALL PARAMETER VALUES: MEASURE DEGREE OF i FOR ALL PARAMETER VALUES: MEASURE CORENESS OF i degree 4 coreness 0 coreness 2 coreness 3 coreness 4 static importance optimal params. dynamic importance Spearman rank correlation coefficent = Quality of representation
  13. 13. E-mail 1 E-mail 2 Dating Gallery Conference Prostitution Results, Degree Time-slice Ongoing Exponential-threshold Accumulated
  14. 14. E-mail 1 E-mail 2 Dating Gallery Conference Prostitution Time-slice Ongoing Exponential-threshold Accumulated Results, Coreness
  15. 15. Time sliceTime sliceTime slice OngoingOngoingOngoing Expo. thresholdExpo. thresholdExpo. threshold Acc. ρmax tstart tstop ρmax tstart tstop ρmax τ ΩΩ ρ E-mail 1 0.73 0 0.42 0.50 0.25 0.25 0.77 0.40 0.30 0.46 E-mail 2 0.91 0 0.25 0.91 0.20 0.20 0.93 1.0 0.26 0.88 Dating 0.82 0 0.65 0.42 0.25 0.25 0.86 0.10 0.16 0.71 Gallery 0.77 0 0.72 0.53 0.39 0.39 0.87 0.70 0.71 0.76 Conference 0.79 0 0.10 0.74 0.10 0.11 0.77 0.04 0.02 0.53 Prostitution 0.71 0 0.77 0.30 0.60 0.60 0.72 0.04 0.20 0.49 Performance & parameter values Degree
  16. 16. Parameter dependence of performance 0 0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 00 0.2 0.4 0.6 0.8 1 ρ tstart/ T tstop/T Time slice
  17. 17. Parameter dependence of performance 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 0.2 0.25 0.3 ρ tstart/ T tstop/T Concurrency
  18. 18. Parameter dependence of performance 0 0.5 1 1.5 2 2.5 3 3.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.5 1 1.5 2 ρ Ω τ/T Exponential threshold
  19. 19. STEP 1 Assign stubs to vertices from a random number distribution. 1 2 3 4 5 6
  20. 20. STEP 2 Connect random pairs of stubs to form a simple graph. 1 2 3 4 5 6
  21. 21. STEP 3 Create active intervals for each edge. (1,2) (2,3) (2,4) (2,5) (3,4) (3,5) (4,5) (5,6) time
  22. 22. STEP 4 Create a time series of contacts from some interevent-time distribution. time
  23. 23. STEP 5 Split the time series into segments proportional to the intervals and impose the contacts of the segments to the intervals. (1,2) (2,3) (2,4) (2,5) (3,4) (3,5) (4,5) (5,6) time
  24. 24. STEP 6 Forget the active intervals. (1,2) (2,3) (2,4) (2,5) (3,4) (3,5) (4,5) (5,6) time
  25. 25. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.1 1µ ρ max 0.50.05 Exponential threshold Time-slice Accumulated Ongoing
  26. 26. break
  27. 27. STATIC TO TEMPORAL
  28. 28. (1,2) (2,3) (2,4) (2,5) (3,4) (3,5) (4,5) (5,6)
  29. 29. (1,2) (2,3) (2,4) (2,5) (3,4) (3,5) (4,5) (5,6) time time (1,2) (2,3) (2,4) (2,5) (3,4) (3,5) (4,5) (5,6) ONGOINGLINKPICTURE
  30. 30. 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 LINKTURNOVERPICTURE
  31. 31. time Beginning time Interevent times End time 0 T t1 t2 t3 t4 t5 t6 t7 t8 tB tE DEFI NITI ONS
  32. 32. Compensate for the size bias on intervals because of finite T0 t’ t sampling time (t’ would only be recorded if it starts within [0,T–t’])
  33. 33. Compensate for the size bias on intervals because of finite T0 t’ t sampling time (t’ would only be recorded if it starts within [0,T–t’]) Compensate for the chance an interevent time t is active 0 t at the start of the sampling is proportional to t
  34. 34. Compensate for the size bias on intervals because of finite T0 t’ t sampling time (t’ would only be recorded if it starts within [0,T–t’]) Compensate for the chance an interevent time t is active 0 t at the start of the sampling is proportional to t ti T–tii: ti≥t ∑ / ti T–tii ∑ Sum up and normalize
  35. 35. 0 0.2 0.4 0.6 0.8 1 0 500 1000 1500 2000 time t (days) predicted from interevent times end times beginning times PROSTITUTION P(t)B
  36. 36. Dating 2 1 0 0.5 0.75 0.25 Dating 1 Forum 1 0 0.5 0.75 0.25 1 0 0.5 0.75 0.25 Prostitution Hospital 1 0 0.5 0.75 0.25 1 0 0.5 0.75 0.25 E-mail 2 Facebook 1 0 0.5 0.75 0.25 1 0 0.5 0.75 0.25 E-mail 1 Film 1 0 0.5 0.75 0.25 1 0 0.5 0.75 0.25 Conference Gallery 1 0 0.5 0.75 0.25 1 0 0.5 0.75 0.25 End Times Beginning Times Predictable edges w.r.t. beginning / end times
  37. 37. (1,2) (1,3) (1,4) (2,3) reference networks
  38. 38. (1,2) (1,3) (1,4) (2,3) (1,2) (1,3) (1,4) (2,3) reference network: identical interevent times
  39. 39. (1,2) (1,3) (1,4) (2,3) (1,2) (1,3) (1,4) (2,3) reference network: identical beginning times
  40. 40. (1,2) (1,3) (1,4) (2,3) (1,2) (1,3) (1,4) (2,3) reference network: identical end times
  41. 41. 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 durationofinfectivestage fractionofinfectives O r i g i n a l d a t a S I R
  42. 42. 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 durationofinfectivestage fractionofinfectives I n t e r e v e n t t i m e s S I R
  43. 43. 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 durationofinfectivestage fractionofinfectives B e g i n n i n g t i m e s S I R
  44. 44. 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 durationofinfectivestage fractionofinfectives E n d t i m e s S I R
  45. 45. 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
  46. 46. 0.1 0.2 0.90.8 10.70.60.50.40.3 0.1 1 0.01 0.001 per-contact transmission probability durationofinfectivestage O r i g i n a l d a t a 0 0.2 0.3 0.4 averagenumberofinfections S I S 0.1
  47. 47. 0.1 0.2 0.90.8 10.70.60.50.40.3 0.1 1 0.01 0.001 per-contact transmission probability durationofinfectivestage I n t e r e v e n t t i m e s 0 0.2 0.3 0.4 averagenumberofinfections S I S 0.1
  48. 48. 0.1 0.2 0.90.8 10.70.60.50.40.3 0.1 1 0.01 0.001 per-contact transmission probability durationofinfectivestage B e g i n n i n g t i m e s 0 0.2 0.3 0.4 averagenumberofinfections S I S 0.1
  49. 49. 0 0.2 0.3 0.4 0.1 0.2 0.90.8 10.70.60.50.40.3 0.1 1 0.01 0.001 per-contact transmission probability durationofinfectivestage averagenumberofinfections E n d t i m e s S I S 0.1
  50. 50. 0 0.01 0.02 0 0.0005 0.001 0.0015 0 0.05 0.1 0 0.01 0.02 0.03 0.04 0 0.0005 0.001 0.0005 0.001 0 0 0.05 0.1 0 0.001 0.0015 0.0005 0.05 0.1 0.15 0.2 0 0 0.02 0.04 0.06 0.08 0 0.01 0.03 0.04 0.02 E-mail 1 Film Dating 1 Forum E-mail 2 Facebook Prostitution Hospital Gallery Conference Dating 2 endtimes beginningtimes intereventtimes
  51. 51. Science by: Illustrations by: Petter Holme Fredrik Liljeros Mi Jin Lee P Holme, 2013, PLoS Comp. Biol. 9:e1003142. P Holme, F Liljeros, 2013, arxiv:1307.6436.
  52. 52. 0 0.2 0.4 0.6 0.8 E-mail 1 0 0.2 0.6 0.4 Film 0 0.1 0.2 0.3 0.4 E-mail 2 0 0.2 0.4 Facebook 0 0.4 0.6 0.2 Dating 1 0.8 0.2 0.4 0.6 0 ForumDating 2 0.2 0.8 0 0.4 0.6 –0.4 –0.2 0.2 0.8 0 0.4 0.6 Conference Hospital 0 0.2 0.4 Prostitution 0 0.1 0.2 Gallery 0 0.2 0.8 0.4 0.6endtimes beginningtimes intereventtimes
  53. 53. 0 0.1 0.2 0.3 0.4 E-mail 1 0 0.2 0.4 0.6 E-mail 2 0 0.4 0.6 0.2 Dating 1 0 0.005 0.01 Facebook 0.8 0.2 0.4 0 ForumDating 2 –0.005 0 0.005 endtimes –0.5 0 0.5 Conference Hospital –0.001 0 0.001 Prostitution 0 0.1 0.2 beginningtimes intereventtimes Gallery 0 0.2 0.8 0.4 0.6 0 0.2 0.5 0.4 Film 0.3 0.1

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