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Temporal networks - Alain Barrat

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COMPLEX NETWORKS: THEORY, METHODS, AND APPLICATIONS (2ND EDITION)
May 16-20, 2016

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Temporal networks - Alain Barrat

  1. 1. Alain Barrat Centre de Physique Théorique, Marseille, France ISI Foundation, Turin, Italy http://www.cpt.univ-mrs.fr/~barrat http://www.sociopatterns.org alain.barrat@cpt.univ-mrs.fr Temporal networks
  2. 2. • From static to temporal – examples – representations, aggregation • Paths • Structure – statistics, burstiness, persistence, motifs • Models; Null models • Dynamical processes – toy models – epidemic spreading; interventions; incompleteness of data
  3. 3. Static networks set V of nodes joined by links (set E) very abstract representation very general convenient to represent many different systems
  4. 4. DATA: static representations Infrastructure networks Biological networks Communication networks Social networks Virtual networks ... • Empirical study and characterization: find generic characteristics (small-world, heterogeneities, hierarchies, communities...) , define statistical characterization tools • Modeling: understand formation mechanisms • Consequences of the empirically found properties on dynamical phenomena taking place on the networks (epidemic spreading, robustness and resilience, etc…) • Link structure and function - as first approach - whenever network does not change on relevant timescale
  5. 5. >Networks change over time
  6. 6. J. Stehlé et al. PLoS ONE 6(8):e23176 (2011) Example: contacts in a primary school, static view
  7. 7. Example: contacts in a primary school, dynamic view J. Stehlé et al. PLoS ONE 6(8):e23176 (2011)
  8. 8. Exemples of temporal networks • Social networks – friendships – collaborations • Communication networks – cell phone data – online social networks (Twitter, etc) • Transportation networks – air transportation – animal transportation • Biological networks
  9. 9. Back to square one: Fundamental issue= data gathering!!! Networks= (often) dynamical entities (communication, social networks, online networks, transport networks, etc…) • Which dynamics? • Characterization? • Modeling? • Consequences on dynamical phenomena? (e.g. epidemics, information propagation…) • Link temporal+topological structure and function Time-varying networks: often represented by aggregated views • Lack of data • Convenience Temporal networks
  10. 10. Definition: temporal network Temporal network: T=(V,S) • V=set of nodes • S=set of event sequences assigned to pairs of nodes 10 sij 2 S : Other representation: time-dependent adjacency matrix: a(i,j,t)= 1 <=> i and j connected at time t sij = {(ts,1 ij , te,1 ij ) · · · (ts,` ij , te,` ij )}
  11. 11. Representations of temporal networks Contact sequences Time ID1 ID2 2 2 4 2 1 5 3 2 4 3 1 6 4 2 3 5 2 4 5 1 4 8 4 6 ……. Contact intervals ID1 ID2 Time interval 2 3 [1,5] 2 1 [2,4] 4 6 [5,9] 1 3 [7,15] 5 3 [7,9] …….
  12. 12. Representations of temporal networks Timelines of nodes
  13. 13. Representations of temporal networks Timelines of links (1,2) (1,3) (2,4) (4,5)
  14. 14. Representations of temporal networks Annotated graph A B D C E (1,4,15) (1,2,5,7,9) (6,8,10,12,15) (2,8)
  15. 15. Aggregation of temporal networks Temporal network Static weighted network weight=sum/number of events Static network Contact matrix
  16. 16. contacts in a primary school
  17. 17. School cumulative f2f network (2 days, 2 min threshold) J. Stehlé et al. PLoS ONE 6(8):e23176 (2011) temporal information: encoded in edges’ weights (contact duration(s), number of events, intermittency, etc...) and node properties
  18. 18. contact matrices J. Stehle, et al. High-Resolution Measurements of Face-to-Face Contact Patterns in a Primary School PLoS ONE 6(8), e23176 (2011) Average values: little information
  19. 19. Other representations? x (x,y) ? Distribution of weights, of contact numbers and durations,… Contact matrices of distributions
  20. 20. Example: negative binomial fits of the cumulated contact duration distributions between individuals of different roles in the hospital A. Machens et al., BMC Inf. Dis. (2013)
  21. 21. Temporality matters: reachability issue Review Holme-Saramaki, Phys. Rep. (2012), arXiv:1108.1780
  22. 22. >Paths in temporal networks
  23. 23. Paths in static networks G=(V,E) Path of length n = ordered collection of • n+1 vertices i0,i1,…,in ∈ V • n edges (i0,i1), (i1,i2)…,(in-1,in) ∈ E i2 i0 i1 i5 i4 i3 Notions of shortest path, of connectedness
  24. 24. Time-respecting paths in temporal networks i2 i0 i1 i5 i4 i3 Path = {(i0,i1,t1),(i1,i2,t2),….,(in-1,in,tn) | t1 < t2 <…< tn)} t1 t2 t3 t4 t5 Length of path: n Duration of path: tn - t1 Notions of shortest path and of fastest path Sequence of events
  25. 25. Reachability Node i at time t Source set: set of nodes that can reach i at time t Reachable set: set of nodes that can be reached from i starting at time t Time “Light cone”
  26. 26. Time-respecting paths in temporal networks • Not always reciprocal: the existence of a path from i to j does not guarantee the existence of a path from j to i • Not always transitive: the existence of paths from i to j and from j to k does not guarantee the existence of a path from i (to j) to k • Time-dependence: • there can be a path from i to j starting at t but no path starting at t’ > t • length of shortest path can depend on starting time • duration of fastest path can depend on starting time • there can be a path starting from i at t0, reaching j at t1, and another path starting form i at t’0 > t0 reaching j at t’1 > t1 , with t’1 - t’0 < t1 - t0 (i.e., smaller duration but arriving later), and/or of shorter length (smaller number of hops)
  27. 27. Centrality measures in temporal networks https://www.cl.cam.ac.uk/~cm542/phds/johntang.pdf Temporal betweenness centrality Temporal closeness centrality Coverage centrality Takaguchi et al., European Physical Journal B, 89, 35 (2016) http://arxiv.org/abs/1506.07032
  28. 28. >Structure in temporal networks
  29. 29. Static networks • Degree distribution P(k) • Clustering coefficient (of nodes of degree k) • Average degree of nearest neighbours knn • Motifs • Communities • Distribution of link weights P(w) • Distribution of node strengths P(s) •….
  30. 30. Temporal networks • Properties of networks aggregated on different time windows: P(k), P(w), P(s), etc… • Evolution of averaged properties when time window length increases (e.g., <k>(t), <s>(t)) example: face-to-face contacts in a primary school J. Stehlé et al. PLoS ONE 6(8):e23176 (2011)
  31. 31. Temporal networks Temporal statistical properties • distribution of contact numbers P(n), • distribution of contact durations P(τ), • distribution of inter-contact times P(Δt) for nodes and links Stationarity of distributions (Non-)stationarity of activity Burstiness
  32. 32. Example: US airport network Stationarity of distributions
  33. 33. However: dynamically evolving Total traffic Average degree Example: US airport network
  34. 34. Example: US airport network 10 -4 10 0 smin ) 1 month 1/2 year 1 year 2 years 10 yearsτ , Δt 10 -4 10 -2 10 0 P(τ),P(Δt) P(τ) P(Δt) Slope -2 10 0 a f A = duration of a link Δt = interval between active periods of a link ⌧
  35. 35. Example: face-to-face contacts Contact duration Stationarity of distributions 9am 11am 1pm 3pm 5pm 6pm4pm2pm10am 12pm time of the day 10 20 30 40 #persons 0 1 2 3 4 〈k〉
  36. 36. Burstiness 36 Poisson process Bursty behavior A.-L. Barabási, Nature (2006)
  37. 37. Example: US airport network 10 -4 10 0 smin ) 1 month 1/2 year 1 year 2 years 10 yearsτ , Δt 10 -4 10 -2 10 0 P(τ),P(Δt) P(τ) P(Δt) Slope -2 10 0 a f A = duration of a link Δt = interval between active periods of a link ⌧
  38. 38. Example: face-to-face contacts Inter-contact duration
  39. 39. Measure of burstiness B = ⌧ m⌧ ⌧ + m⌧ where m𝜏 is the mean and σ𝜏 the std deviation of the inter-event time distribution Poisson: B = 0 (exponential distribution) Periodic: B = -1 (Delta distribution) Broad distribution: B = 1 (if σ𝜏 diverges) Burstiness = clustering of events in time Goh, Barabási, EPL 2008 See also: Karsai et al., Sci Rep 2, 397 (2012)
  40. 40. Consequence of burstiness Bursty Poisson randomly chosen time Bursty timeline implies larger waiting time with higher probability => typically slower diffusion (if no correlations)
  41. 41. Structure: Persistence of patterns Compare successive snapshots or time-aggregated networks: • correlation between (weighted) adjacency matrices • correlation between contact matrices • local similarity of neighbourhoods • detection of timescales http://arxiv.org/abs/1604.00758 i = P j wij,(1)wij,(2) qP j w2 ij,(1) P i,j w2 ij,(2)
  42. 42. Persistence of patterns Example: contacts in a primary school, neighbourhood similarities between different days, same gender vs different gender 1A 1B 2A 2B 3A 3B 4A 4B 5A 5B classes 0 0.25 0.5 0.75 1 cosinesimilarity σsg σog
  43. 43. Persistence of patterns Example: contacts in a high school, neighbourhood similarities between different days NB: need to compare with null model(s)
  44. 44. What are null models? • ensemble of instances of randomly built systems • that preserve some properties of the studied systems Aim: • understand which properties of the studied system are simply random, and which ones denote an underlying mechanism or organizational principle • compare measures with the known values of a random case
  45. 45. Static null models • Fixing size (N, E): random (Erdös-Renyi) graph • Fixing degree sequence: reshuffling/rewiring methods Original network Rewired network i i j n mm j n rewiring step • preserves the degree of each node • destroys topological correlations
  46. 46. Example: contacts in a high school, neighbourhood similarities between different days, vs different null models (b): respecting class structure
  47. 47. Null models for temporal networks Many properties and correlations many possible null models see later
  48. 48. >Structure: Temporal motifs
  49. 49. Same aggregated network, different temporal sequences detection of temporal patterns? L. Kovanen et al., PNAS (2013) L. Kovanen et al., J. Stat. Mech. (2011) P11005
  50. 50. L. Kovanen et al., J. Stat. Mech. (2011) P11005 Δt-adjacent events A B C t=1 t=4 events Δt-adjacents for Δt=4 events Δt-adjacent: • at least one node in common • at most Δt between end of 1st event and start of 2nd event
  51. 51. L. Kovanen et al., J. Stat. Mech. (2011) P11005 Δt-connected events A B C t=1 t=4 =events connected by a chain of Δt-adjacent events D t=8 A B C t=2 t=3 D t=6 Examples (Δt=5): t=10
  52. 52. Temporal subgraph = set of Δt-connected events L. Kovanen et al., J. Stat. Mech. (2011) P11005 A B D C E t=1 t=5 t=20 t=24 t=26 t=9 F t=7 A B C E t=1 t=5 t=9 F t=7 A B D E t=20 t=24 t=26 Maximal temporal subgraphs (Δt=5):
  53. 53. Temporal motifs = Equivalence classes of valid temporal subgraphs Valid: no events are skipped at each node Equivalence classes: forget identities of nodes and exact timing J. Saramäki
  54. 54. Temporal motifs reveal homophily, gender-specific patterns, and group talk in call sequences L. Kovanen et al., PNAS (2013) which motifs are most frequent? metadata on nodes, compare with null models, …
  55. 55. >Models of temporal networks
  56. 56. > activity-driven model
  57. 57. Activity-driven network Perra et al., Sci. Rep. (2012) Model: N nodes, each with an “activity” a, taken from a distribution F(a) At each time step: • node i active with probability a(i) • each active node generate m links to other randomly chosen nodes • iterate with no memory Aggregate degree distribution ~ F No memory, no correlations… “Toy” model allowing for analytical computations
  58. 58. Activity-driven network with memory Model: N nodes, each with an “activity” a, taken from a distribution F(a) At each time step: • each node i becomes active with probability a(i) • each active node generate m links to other nodes; if it has already been in contact with n distinct nodes, the new link is • towards a new node with proba p(n)=c/(c+n) • towards an already contacted node with proba 1-p(n) (reinforcement mechanism seen in data) Karsai et al., Sci. Rep. (2014)
  59. 59. Memory-less With Memory Sun et al., EPJB (2015)
  60. 60. >a model of interacting agents
  61. 61. Time  t Time  t+1 Time  t+2 Time  t+3 Transition probabilities depending on: -present state -time of last state change Interacting agents
  62. 62. At each timestep: choose an agent i at random: • i isolated: with proba b0 f( t,ti ), agent i changes its state, and chooses an agent j with probability Π( t,tj ) • i in a group: with probability b1f( t,ti ), agent i changes its state : – with probability λ, agent i leaves the group – with probability 1-λ, it introduces an isolated agent n chosen with probability Π( t,tn ) to the group Parameters: b0, b1, λ A simple model of interacting agents J. Stehlé et al., Physical Review E 81, 035101 (2010) K. Zhao et al., Physical Review E 83, 056109 (2011)
  63. 63. Analytical and numerical results where N=1000,  b0=0.7  ,  b1=0.7,  λ=0.8,  Tmax=105 N Duration  in  a  given  state  p J. Stehlé et al., Physical Review E 81, 035101 (2010) K. Zhao et al., Physical Review E 83, 056109 (2011)
  64. 64. Model SocioPatterns data (face-to-face contacts) Distributions of times spent with p neighbours J. Stehlé et al., Physical Review E 81, 035101 (2010) K. Zhao et al., Physical Review E 83, 056109 (2011)
  65. 65. >another model of interacting agents
  66. 66. N agents; ti= last time i changed state tij = last time (i,j) changed state at each time step dt: (i) Each active link (i, j) is inactivated with proba dt z f(t − tij) (ii) Each agent i initiates a contact with another agent j with probability dt b fa(t − ti), j chosen among agents that are not in contact with i with probability Πa(t−tj)Π(t− tij) memory kernels f, fa, Πa, Π: • constant: Poissonian dynamics • empirics: decrease as 1/x a) b) c) C. Vestergaard et al., Phys. Rev. E 90, 042805 (2014)
  67. 67. Memory mechanisms can be incorporated separately Need all to reproduce empirical data C. Vestergaard et al., Phys. Rev. E 90, 042805 (2014) FullmodelEmpirical
  68. 68. >still another model of interacting agents
  69. 69. Agents with different “attractiveness” performing random walks M. Starnini, A. Baronchelli, R. Pastor-Satorras Modeling human dynamics of face-to-face interaction networks, Phys. Rev. Lett. 110, 168701 (2013) Walking probability: Agent i -> attractiveness ai
  70. 70. M. Starnini, A. Baronchelli, R. Pastor-Satorras Modeling human dynamics of face-to-face interaction networks, Phys. Rev. Lett. 110, 168701 (2013) Agents with different “attractiveness” performing random walks
  71. 71. >Generative models from data
  72. 72. • Generate static underlying structure • Generate timelines of events • known P(Δt): generate successive events with inter- event times taken from P(Δt) • known P(Δt) and P(n): assign a number of events to each link from P(n), and inter-event times from P(Δt) • known P(Δt), P(n), P(τ) • known intervals of activity or overall activity timeline • … >Surrogate temporal networks from known empirical statistics L. Gauvin et al., Sci Rep 3:3099 (2013) M. Génois et al., Nat Comm 6:8860 (2015) NB: no temporal correlations, some can be introduced (through correlated intervals of activities)
  73. 73. >Temporal networks from empirical weighted networks A. Barrat et al., Phys. Rev. Lett. 110, 158702 (2013)
  74. 74. Data on time-varying networks is often limited... i) access only to time-aggregated data ii) access to a single sample on [0,T] What was the temporal evolution? What would be the temporal evolution before 0 or after T? What could have happened for another sample? What would be a realistic temporal evolution? How to generate a realistic other sample/story?
  75. 75.     i j k     i j k     i j k     i j k     i j k G !me$ N 2$ 1$ 0$
  76. 76. i) Start from a static (weighted, directed) graph G assumed to result from the temporal aggregation of a time-varying network on a time window of known length [0,T] ii) Using random paths on G, create a time- varying network on [0,T] with realistic features (e.g., burstiness, non-trivial correlations), whose temporal aggregation coincides with G
  77. 77. w=1 w=1 w=3 w=2 w=1 w=3 w=3 w=1 w=1 w=2 w=2 Static weighted graph G
  78. 78. Static weighted graph G
  79. 79. Static weighted graph G Static decomposition in itineraries/paths
  80. 80. Static weighted graph G Unfolding G as a dynamic aggregation of walks
  81. 81. Static weighted graph G Unfolding G as a dynamic accumulation of interwoven time-stamped walks
  82. 82. Practical algorithm: i) start from G and a time window [0,T]
  83. 83. Practical algorithm: ii) perform random walks on G with random initial position and time iii) with random prescribed length i j k iii) with random residence times at each step iv) update the weight of each traversed link: w w-1 t1 t2=t1+𝜏 l=2 v) store the events (i,j,t1); (j,k,t2)
  84. 84. Practical algorithm: vi) iterate until network is empty (all w=0) vii) the temporal network is the set of all events, i.e., of all random walk steps (i,j,t) NB: generates burstiness + correlations
  85. 85. >Null Models of temporal networks
  86. 86. What are null models? • ensemble of instances of randomly built systems • that preserve some properties of the studied systems Aim: • understand which properties of the studied system are simply random, and which ones denote an underlying mechanism or organizational principle • compare measures with the known values of a random case
  87. 87. Random times keeps: • link structure • number of events per link • corresponding static correlations destroys: • global time ordering • activity timeline • burstiness • all temporal correlations for each link event: pick time at random
  88. 88. Time shuffling keeps: • link structure • number of events per link • corresponding static correlations • global time ordering and activity timeline destroys: • interevent times • all temporal correlations shuffle times of events ID1 ID2 time 1 4 2 2 3 8 1 5 12 3 4 15 ….. ID1 ID2 time 1 4 8 2 3 15 1 5 2 3 4 12 …..
  89. 89. Interval shuffling for each link: randomize sequence of inter-event durations for each link: randomize sequence of contacts a b c 𝛼 𝛽 ac 𝛽 𝛼 b keeps: • link structure • number of events per link • corresponding static correlations • link burstiness destroys: • all temporal correlations • activity timeline
  90. 90. Link-sequence shuffling Randomly exchange sequence of events of different links A A BB C C DD keeps: • link structure • distribution of link weights • global activity timeline • link burstiness destroys: • number of events per link & corresponding correlations • correlations between structure and activity • all temporal correlations NB: weight-conserving link-sequence shuffling
  91. 91. >Dynamical processes on dynamical networks: From toy models to epidemiology
  92. 92. >Random walks
  93. 93. Random walks on activity-driven network Perra et al., Phys. Rev. E (2012) @Wa (t) @t = aWa (t) + amw mhaiWa (t) + Z a0 Wa0 (t) F (a0 ) da0 N nodes W walkers Wa average number of walkers in a node of activity a w = W/N
  94. 94. Random walks on activity-driven network Perra et al., Phys. Rev. E (2012) 10 -3 10 -2 10 -1 10 0 a 10 10 2 10 3 W 10 -3 10 -2 10 -1 10 0 a 10 10 2 10 3 10 4 Wa a Wa = amw + a + mhai Stationary state: = Z aF (a) amw + a + mhai da
  95. 95. Random walks on empirical temporal networks Starnini et al., Phys Rev E (2012) arXiv:1203.2477
  96. 96. >Toy spreading processes
  97. 97. • deterministic SI process to probe the causal structure of the dynamical network • fastest paths ≠ shortest paths Toy spreading processes on temporal networks Time  t Time  t’>t Time  t’’>t’ B A C A B B A C C Fastest path= A->B->C Shortest path= A-C
  98. 98. M. Karsai et al., Small But Slow World: How Network Topology and Burstiness Slow Down Spreading, Phys. Rev. E (2011). Mobile phone data: • community structure (C) • weight-topology correlations (W) • burstiness on single links (B) • daily patterns (D) • event-event correlations between links (E) Effects of the different ingredients? Use series of null models!
  99. 99. M. Karsai et al., Small But Slow World: How Network Topology and Burstiness Slow Down Spreading, Phys. Rev. E (2011). Null models • community structure (C) • weight-topology correlations (W) • burstiness on single links (B) • daily patterns (D) • event-event correlations between links (E)
  100. 100. M. Karsai et al., Small But Slow World: How Network Topology and Burstiness Slow Down Spreading, Phys. Rev. E (2011). Kivela et al, Multiscale Analysis of Spreading in a Large Communication Network, arXiv:1112.4312 Mobile phone data • community structure (C) • weight-topology correlations (W) • burstiness on single links (B) • daily patterns (D) • event-event correlations between links (E) Burstiness slows down spread Correlations slightly favour spread
  101. 101. Rocha et al., PLOS Comp Biol (2011) • data: temporal network of sexual contacts • temporal correlations accelerate outbreaks Pan & Saramaki, PRE (2011) • data: mobile phone call network • slower spread when correlations removed Miritello et al., PRE (2011) • data: mobile phone call network • burstiness decreases transmissibility Takaguchi et al., PLOS ONE (2013) • data: contacts in a conference; email • threshold-based spreading model • burstiness accelerate spreading Rocha & Blondel, PLOS Comp Biol (2013) • model with tuneable distribution of inter-event times (no correlations) • burstiness => initial speedup, long time slowing down More results
  102. 102. Results • depend on data set • depend on spreading model • generally • burstiness slows down spreading • correlations (e.g., temporal motifs) favor spreading • role of turnover • +: effect of static patterns Still somewhat unclear picture
  103. 103. >Spreading processes
  104. 104. Epidemic models SI + + β µ + + β SIS + + β µSIR I I I I I I I I I I I R S S S S
  105. 105. >Reminder: case of static networks
  106. 106. Transmission S (susceptible) I (infected) β Individual in state S, with k contacts, among which n infectious: in the homogeneous mixing approximation, the probability to get the infection in each time interval dt is: Proba(S I) = 1 - Proba(not to get infected by any infectious) = 1 - (1 - βdt)n ≅ β n dt (β dt << 1) ≅ β k i dt as n ~ k i for homogeneous mixing HOMOGENEOUS MIXING ASSUMPTION Hypothesis of mean-field nature: every individual sees the same density of infectious among his/her contacts, equal to the average density in the population
  107. 107. The SI model S (susceptible) I (infected) β N individuals I(t)=number of infectious, S(t)=N-I(t) number of susceptible i(t)=I(t)/N , s(t)=S(t)/N = 1- i(t) If k = <k> is the same for all individuals (homogeneous network): dI dt = S(t) ⇥ Proba(S ! I) = kS(t)i(t) di dt = ki(t)(1 i(t))
  108. 108. The SIS model N individuals I(t)=number of infectious, S(t)=N-I(t) number of susceptible i(t)=I(t)/N , s(t)=S(t)/N Homogeneous mixing S (susceptible)S (susceptible) I (infected) β µ Competition of two time scales: 1/µ and 1/(β <k>)
  109. 109. Long time limit, SIS model Stationary state: di/dt = 0 Epidemic threshold condition: Active phase Absorbing phase Finite prevalence Virus death λ=β/µ Phase diagram: hki = µ hki > µ ) i1 = 1 µ/( hki) hki < µ ) i1 = 0
  110. 110. >Spread on temporal networks
  111. 111. SIS model on activity-driven network Perra et al., Sci. Rep. (2012) Activity-based mean-field theory: It a Number of infectious nodes of activity a It+ t a = It a µIt a t + am t(Na It a) Z It a0 N da0 + t(Na It a)m Z a0 It a0 N da0 ✓t = Z aIt ada , It = Z It adaWrite equations for @t✓ = µ✓ + mhai✓ + mha2 iI @tI = µI + mhaiI + m✓ (neglecting 2nd order terms) µ > 1 m ⇣ hai + p ha2i ⌘ Condition: largest eigenvalue larger than 0 Epidemic threshold:
  112. 112. SIS model on activity-driven network Perra et al., Sci. Rep. (2012) Epidemic threshold: c = 1 m ⇣ hai + p ha2i ⌘
  113. 113. SIS and SIR models on activity-driven networks with memory Sun et al., EPJB (2015) SIS model: memory favors spread µ
  114. 114. SIS and SIR models on activity-driven networks with memory Sun et al., EPJB (2015) SIR model: memory hinders spread 0.1 0.2 0.3 0.4 0.5 0.6 0.7 b µ 0.5 1.0 1.5 2.0 R• ⇥10 1 µ =0.015 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 ⇥10 1 µ =0.005 ML WM
  115. 115. >Epidemic threshold on temporal networks: Markov chain approach Valdano et al., PRX (2015)
  116. 116. Gomez et al, EPL (2010) Markov chain approach for static networks pt i =proba of node i to be infectious at time t pt i = 1 1 (1 µ)pt 1 i Y j 1 Ajipt 1 j Quenched mean-field: • takes into account network structure • neglects dynamical correlations Epidemic threshold • develop at first order • epidemic threshold: ~p = µ~p t 1 + A~p t 1 µ > 1 ⇢(A) ⇢(A)=largest eigenvalue of A
  117. 117. Markov chain approach for temporal networks Valdano et al., PRX (2015) pt i = 1 1 (1 µ)pt 1 i Y j 1 At 1 ji pt 1 j • time-dependent adjacency matrix • mapping from temporal to multilayer
  118. 118. Valdano et al., PRX (2015) From temporal to multilayer Att0 ij = t,t0 +1 ⇣ ij + At0 ij ⌘ Markov chain approach for temporal networks
  119. 119. Valdano et al., PRX (2015) pt i = 1 1 (1 µ)pt 1 i Y j 1 At 1 ji pt 1 j • time-dependent adjacency matrix • mapping from temporal to multilayer Mtt0 ij = t,t0 +1 ⇣ (1 µ) ij + At0 ij ⌘ Att0 ij = t,t0 +1 ⇣ ij + At0 ij ⌘ pt i = 1 Y j,t0 ⇣ 1 Mtt0 ij pt0 j ⌘ Markov chain approach for temporal networks
  120. 120. Valdano et al., PRX (2015) pt i = 1 1 (1 µ)pt 1 i Y j 1 At 1 ji pt 1 j • time-dependent adjacency matrix • mapping from temporal to multilayer • matrix representation encoding both network and process (i, t) ! ↵ = Nt + i Markov chain approach for temporal networks
  121. 121. Valdano et al., PRX (2015) pt i = 1 1 (1 µ)pt 1 i Y j 1 At 1 ji pt 1 j • time-dependent adjacency matrix • mapping from temporal to multilayer • matrix representation encoding both network and process • stationary solution p↵ = 1 Y (1 M↵ p ) Markov chain approach for temporal networks
  122. 122. Valdano et al., PRX (2015) pt i = 1 1 (1 µ)pt 1 i Y j 1 At 1 ji pt 1 j • time-dependent adjacency matrix • mapping from temporal to multilayer • matrix representation encoding both network and process • stationary solution • epidemic threshold ⇢(M) = 1 Markov chain approach for temporal networks
  123. 123. Valdano et al., PRX (2015) Markov chain approach for temporal networks
  124. 124. > data-driven numerical simulations and data representations
  125. 125. How much detail to inform the models? Detailed dynamic network • very detailed ✔ • very realistic ✔ • takes into account individual heterogeneities of behavior ✔ • very specific (context+period), not easy to generalize ✕ Contact matrix • coarse-grained ✕ • fully connected structure ✕ • only heterogeneities between groups ✕ • very easy to generalize ✔
  126. 126. “synopsis” of dynamic network data Temporal network Static network Contact matrix (underlying fully connected assumption + no within-class heterogeneity)
  127. 127. “synopsis” of dynamic network data x (x,y) + Contact matrix of distributions: -role based -takes heterogeneities into account Role-based structure Heterogeneities
  128. 128. + J. Stehlé et al., BMC Medicine (2011); A. Machens et al, BMC Inf Dis (2013) Evaluating representations
  129. 129. >A concrete example: contact patterns in a hospital
  130. 130. 130 Network representations Construction of 3 networks: 1. Dynamic network (DYN):
 Real sequence of successive contacts 2. Heterogeneous network (HET):
 1-day aggregated network
 A—B if A and B have been in contact
 WAB = cumulative duration of the contacts A-B
 3. Homogeneous network (HOM):
 1-day aggregated network
 A—B if A and B have been in contact
 WAB = average cumulative duration Dataaggregation Networks: • Take into account network structure at the individual level • Difficult to generalize
  131. 131. • Underlying fully connected network structure • Takes into account role structure • Average temporal information, no heterogeneities within each role • Easy to generalize 4. Contact matrix
  132. 132. 5. Novel representation: Contact matrix of distributions a. Fit each role-pair distribution of weights (using negative binomials) b. Create a network in which weights are drawn from the fitted distribution (NB: including zero weights) Machens et al, BMC Infect. Dis.(2013) • Underlying realistic network structure • Takes into account role structure • Takes into account heterogeneities within each role • Easy to generalize
  133. 133. + Epidemiological model Evaluating the representations? • Evaluation of : • Extinction probability • Attack rate • Role of initial seed • Attack rate for each group • Comparison with most realistic DYN representation Machens et al, BMC Infect. Dis.(2013)
  134. 134. SEIR simulation results Machens et al, BMC Infect. Dis.(2013) Importance of heterogeneities of contact durations
  135. 135. SEIR simulation results Attack rate by groups (for AR > 10%) Machens et al, BMC Infect. Dis.(2013)
  136. 136. > interventions
  137. 137. Immunization strategies Lee et al., PLOS ONE (2012) =>inspired by “acquaintance protocol” in static networks • “Recent”: choose a node at random, immunize its most recent contact • “Weight”: choose a node at random, immunize its most frequent contact in a previous time-window Starnini et al., JTB (2012) • aggregate network on [0,T] • compare strategies • immunize nodes with highest k or BC in [0,T] • immunize random acquaintance (on [0,T]) • recent, weight strategies • vary T • find saturation of efficiency as T increases Liu et al., PRL (2014) (activity-driven network model=>analytics) • target nodes with largest activity • random neighbour (over an observation time T) of random node => take into account temporal structure
  138. 138. > mesoscale interventions • act at intermediate scales • avoid need to identify nodes with specific properties
  139. 139. An example: SEIR + school Which containment strategies? Model: -SEIR with asymptomatics -contact data as proxy for possibility of transmission inside school -when children are out of school: residual homogeneous risk of contamination by contact with population Containment strategies (suggested by the data): -detection and subsequent isolation of symptomatic individuals -whenever symptomatic individuals are detected (more than a given threshold), closure of (i) class (ii) class + most connected other class (same grade) (iii) whole school Gemmetto, Barrat, Cattuto, BMC Inf Dis (2014)
  140. 140. An example: SEIR + school Which containment strategies? Gemmetto, Barrat, Cattuto, BMC Inf Dis (2014) Average over cases with AR > 10%
  141. 141. An example: SEIR + school Which containment strategies? Average over cases with AR > 10% Gemmetto, Barrat, Cattuto, BMC Inf Dis (2014)
  142. 142. Probability that AR < 10% Average AR when AR > 10% Which containment strategies? Comparing class/grade/school closure Targeted class Targeted grade Targeted grade Targeted class Strategy (Threshold, duration) Strategy (Threshold, duration) School School Gemmetto, Barrat, Cattuto, BMC Inf Dis (2014)
  143. 143. An example: SEIR + school Which containment strategies? Cost in number of lost class-days Gemmetto, Barrat, Cattuto, BMC Inf Dis (2014)
  144. 144. Two-sided role of the data • “Simple” stylized facts, easy to generalise – more contacts within each class than between classes – which classes are most connected => from low-resolution data => suggests strategies • Complex, detailed data set – detailed sequence of contact events => high-resolution data => validation of strategies through detailed and realistic numerical simulations
  145. 145. > mesoscale interventions - 2
  146. 146. contacts in a primary school: • classes • correlated link activity J. Stehlé et al. PLoS ONE 6(8):e23176 (2011)
  147. 147. L. Gauvin et al., PLoS ONE (2014) Detection of structures: decomposition of temporal network Data: temporal network with discrete time intervals Time-dependent adjacency matrix A(i,j,t) Three-way tensor T
  148. 148. L. Gauvin et al., PLoS ONE (2014) Detection of structures: decomposition of temporal network tijk ⇡ RX r=1 airbjrckr undirected network: a=b air= membership of node i to component r ckr=timeline of component r T ⇡ ˜T = RX r=1 Sr = RX r=1 ar br cr Kruskal decomposition
  149. 149. L. Gauvin et al., PLoS ONE (2014)
  150. 150. L. Gauvin et al., PLoS ONE (2014) 10 classes 4 mixed-membership components = breaks Primary school case study
  151. 151. L. Gauvin et al., arXiv:1501.02758 Revealing latent factors of temporal networks for mesoscale intervention in epidemic spread, Intervention: removal of a component ˜T s = X r6=s SrAltered temporal network Stochastic SI model, epidemic delay ratio ⌧r = ⌧ tr j tj tj Comparison with null model=removal of randomly chosen links Components 12 and 14=1st day lunch break NB: components 12,14 carry less weight than 1 or 2
  152. 152. L. Gauvin et al., arXiv:1501.02758 Revealing latent factors of temporal networks for mesoscale intervention in epidemic spread, Intervention: removal of a component SIR model, epidemic size ratio
  153. 153. L. Gauvin et al., Revealing latent factors of temporal networks for mesoscale intervention in epidemic spread, arXiv:1501.02758 Reactive targeted intervention • SEIR • contact data as proxy for possibility of transmission inside school • when children are out of school: residual homogeneous risk of contamination by contact with population • isolation of infectious cases at the end of each day • if 2 infectious are detected: • removal of mixing components • replacement by equivalent amount of class activities
  154. 154. > Incomplete data Context: temporal contact networks
  155. 155. Data is often incomplete because of population sampling (not everyone takes part to the data collection) Consequences of missing data…. • Biases in measured statistical properties? • Biases in the estimation of the outcomes of data- driven models? A B dAB, real = 2 Risk of contamination A->B real >> sampled (C acts as an immunized node) dAB, sampled = 4 C
  156. 156. I- Evaluation of biases due to missing data II- Compensating for biases Issues
  157. 157. Methodology: resampling • Remove nodes (individuals) at random • Measure statistical properties of contacts between remaining individuals • Run simulations of spreading processes (SIS or SIR models) • Compare outcomes of simulations on initial and resampled temporal networks
  158. 158. Data sets: temporal networks of face-to-face proximity • Conference: 2 days, 403 individuals • Offices: 2 weeks, 92 individuals • High School: 1 week, 326 individuals
  159. 159. Impact of sampling on contact network properties and mixing patterns Random sampling keeps: • Temporal statistical properties (distributions of contact and inter-contact durations, distribution of aggregate durations, of # contacts per link) • Density of network • Contact matrices of link densities (for structured populations) known to be crucial for spreading processes
  160. 160. Impact of sampling on contact network properties and mixing patterns 0% 10% 20% 40% 2BIO1 2BIO2 2BIO3 MP MP*1 MP*2 PC PC* PSI* 2BIO1 2BIO2 2BIO3 MP MP*1 MP*2 PC PC* PSI* 2BIO1 2BIO2 2BIO3 MP MP*1 MP*2 PC PC* PSI* 2BIO1 2BIO2 2BIO3 MP MP*1 MP*2 PC PC* PSI* 2BIO1 2BIO2 2BIO3 MP MP*1 MP*2 PC PC* PSI* 2BIO1 2BIO2 2BIO3 MP MP*1 MP*2 PC PC* PSI* 2BIO1 2BIO2 2BIO3 MP MP*1 MP*2 PC PC* PSI* 2BIO1 2BIO2 2BIO3 MP MP*1 MP*2 PC PC* PSI* 0% 10% 20% 40% Offices High School Random sampling keeps: • Temporal statistical properties (distributions of contact and inter- contact durations, distribution of aggregate durations, of # contacts per link) • Density of network • Contact matrices of link densities (for structured populations) known to be crucial for spreading processes
  161. 161. Impact of sampling on simulations of spreading processes SIR model, distribution of epidemic sizes Strong underestimation of • risk of large epidemics • average epidemic size (absent individuals act as if they were immunised => herd immunisation effect) InVS Thiers13 SFHHOffices High School Conference
  162. 162. How to estimate the outcome of spreading processes from incomplete data? Do the incomplete data contain enough information?
  163. 163. Method • build surrogate contact networks, using only properties of sampled (incomplete) data • perform simulations on surrogate data sets NB: surrogate contacts, here we do not try to infer (unknown) real contacts
  164. 164. Building surrogate data sets I. Measure some properties, in incomplete data II. Add missing nodes (assumption: class/department known) III. Build links between missing and present nodes, in order to maintain the properties measured in the incomplete data IV. On each link, create a timeline of contacts respecting the statistical properties of the incomplete data V. (Check properties of reconstructed data set) VI. Perform simulation of spreading process
  165. 165. Which properties / how much detail? • Method “0”: measure density + average weight, add back nodes and links in order to keep density fixed (=>increases average degree, simplest compensation method); Poissonian timelines of contact events on links; • Method “W”: measure density + statistics of link’s weights, add back nodes and links to keep density fixed, weights on surrogate links taken from measured ones, Poissonian timelines on links; • Method “WS”: measure link density contact matrix + statistics of link’s weights, separating intra- and inter-groups links; add back nodes and links to keep contact matrix fixed, weights on surrogate links taken from measured ones, separating intra-group and inter-group links, Poissonian timelines on links; • Method “WT”: measure density + statistics of numbers of contacts per link and of contact and inter-contact durations; add back nodes and links as in method “0”; for each surrogate link, create a surrogate timeline by extracting a number of contacts at random and alternate contact and inter-contact durations; • Method “WST”: measure link density contact matrix + statistics of numbers of contacts per link and of contact and inter-contact durations, separating intra- and inter-groups links; add back nodes and surrogate links as in “WS”, create surrogate timelines as in “WT”. Increasinginformation
  166. 166. Results
  167. 167. SIR model, distribution of epidemic sizes (offices) 0.0 0.2 0.4 0.6 0.8 1.0 Epidemic size 10 -1 10 0 10 1 10 2 PDF 0 0.0 0.2 0.4 0.6 0.8 1.0 Epidemic size 10 -1 10 0 10 1 10 2 PDF sample 0.0 0.2 0.4 0.6 0.8 1.0 Epidemic size 10 -1 10 0 10 1 10 2 PDF W 0.0 0.2 0.4 0.6 0.8 1.0 Epidemic size 10 -1 10 0 10 1 10 2 PDF WS 0.0 0.2 0.4 0.6 0.8 1.0 Epidemic size 10 -1 10 0 10 1 10 2 PDF WT 0.0 0.2 0.4 0.6 0.8 1.0 Epidemic size 10 -1 10 0 10 1 10 2 PDF WST
  168. 168. SIR model, distribution of epidemic sizes (high school) 0.0 0.2 0.4 0.6 0.8 1.0 Epidemic size 10 -1 10 0 10 1 10 2 PDF 0.0 0.2 0.4 0.6 0.8 1.0 Epidemic size 10 -1 10 0 10 1 10 2 PDF 0.0 0.2 0.4 0.6 0.8 1.0 Epidemic size 10 -1 10 0 10 1 10 2 PDF WST
  169. 169. 169 SIR model, distribution of epidemic sizes Offices High School Conference
  170. 170. 170 SIS case
  171. 171. InVS Thiers13 SFHH Limitations Very low sampling rate => worse estimation of contact patterns I- Larger deviations Offices High School Conference II- Reconstruction procedure fails (not enough statistics in sampled data)
  172. 172. Limitations Systematic overestimation of •maximal size of large epidemics •probability and average size of large outbreaks Due to correlations (structure and/or temporal) ? •present in data •not reproduced in surrogate data To test this hypothesis: •use reshuffled data (structurally or temporally) in which correlations are destroyed, •perform resampling and construction of surrogate data, •simulate spreading.
  173. 173. Using structurally reshuffled data => overestimation due to the presence of small-scale structures in real data, difficult or even impossible to reproduce in surrogate data 0.0 0.2 0.4 0.6 0.8 1.0 Epidemic size 10 -1 10 0 10 1 10 2 PDF Percentage of removed nodes 0 % 10 % 20 % 30 % 40 % 0.0 0.2 0.4 0.6 0.8 1.0 Epidemic size 10 -2 10 -1 10 0 10 1 10 2 PDF 0.0 0.2 0.4 0.6 0.8 1.0 Epidemic size 10 -1 10 0 10 1 10 2 PDF 0.0 0.2 0.4 0.6 0.8 1.0 Epidemic size 10 -1 10 0 10 1 10 2 PDF 0.0 0.2 0.4 0.6 0.8 1.0 Epidemic size 10 -1 10 0 10 1 10 2 PDF 0.0 0.2 0.4 0.6 0.8 1.0 Epidemic size 10 -1 10 0 10 1 10 2 PDF Offices, shuffled High School, shuffled Conference, shuffled reconstructed sampled
  174. 174. Using incomplete sensor data • Population sampling – can strongly underestimate the outcome of spreading processes – however, maintains important properties of the temporal contact network (distribution of contact/inter-contact durations, of aggregate durations, of # of contacts per link), and of the aggregated network (overall network density, structure of contact matrix) • Possible estimation of the real outcome of simulations of spreading processes using surrogate contacts for the missing individuals – measure properties of sampled data – build surrogate links and contacts that respect the measured statistics – leads to a small overestimation of the outcome due to non captured correlations M. Génois, C. Vestergaard, C. Cattuto, A. Barrat arXiv:1503.04066, Nat. Comm. 6:8860 (2015)
  175. 175. Still very open field! Data Structures in data Incompleteness of data Models Processes on temporal networks …
  176. 176. http://complexnets2016.org

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