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COMPLEX NETWORKS: THEORY, METHODS, AND APPLICATIONS (2ND EDITION)

May 16-20, 2016

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- 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. • 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. Static networks set V of nodes joined by links (set E) very abstract representation very general convenient to represent many different systems
- 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. >Networks change over time
- 6. J. Stehlé et al. PLoS ONE 6(8):e23176 (2011) Example: contacts in a primary school, static view
- 7. Example: contacts in a primary school, dynamic view J. Stehlé et al. PLoS ONE 6(8):e23176 (2011)
- 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. 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. 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. 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. Representations of temporal networks Timelines of nodes
- 13. Representations of temporal networks Timelines of links (1,2) (1,3) (2,4) (4,5)
- 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. Aggregation of temporal networks Temporal network Static weighted network weight=sum/number of events Static network Contact matrix
- 16. contacts in a primary school
- 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. 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. Other representations? x (x,y) ? Distribution of weights, of contact numbers and durations,… Contact matrices of distributions
- 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. Temporality matters: reachability issue Review Holme-Saramaki, Phys. Rep. (2012), arXiv:1108.1780
- 22. >Paths in temporal networks
- 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. 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. 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. 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. 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. >Structure in temporal networks
- 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. 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. 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. Example: US airport network Stationarity of distributions
- 33. However: dynamically evolving Total traffic Average degree Example: US airport network
- 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. 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. Burstiness 36 Poisson process Bursty behavior A.-L. Barabási, Nature (2006)
- 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. Example: face-to-face contacts Inter-contact duration
- 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. Consequence of burstiness Bursty Poisson randomly chosen time Bursty timeline implies larger waiting time with higher probability => typically slower diffusion (if no correlations)
- 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. 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. Persistence of patterns Example: contacts in a high school, neighbourhood similarities between different days NB: need to compare with null model(s)
- 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. 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. Example: contacts in a high school, neighbourhood similarities between different days, vs different null models (b): respecting class structure
- 47. Null models for temporal networks Many properties and correlations many possible null models see later
- 48. >Structure: Temporal motifs
- 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. 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. 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. 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. 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. 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. >Models of temporal networks
- 56. > activity-driven model
- 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. 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. Memory-less With Memory Sun et al., EPJB (2015)
- 60. >a model of interacting agents
- 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. 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. 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. 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. >another model of interacting agents
- 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. Memory mechanisms can be incorporated separately Need all to reproduce empirical data C. Vestergaard et al., Phys. Rev. E 90, 042805 (2014) FullmodelEmpirical
- 68. >still another model of interacting agents
- 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. 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. >Generative models from data
- 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. >Temporal networks from empirical weighted networks A. Barrat et al., Phys. Rev. Lett. 110, 158702 (2013)
- 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. i j k i j k i j k i j k i j k G !me$ N 2$ 1$ 0$
- 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. 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. Static weighted graph G
- 79. Static weighted graph G Static decomposition in itineraries/paths
- 80. Static weighted graph G Unfolding G as a dynamic aggregation of walks
- 81. Static weighted graph G Unfolding G as a dynamic accumulation of interwoven time-stamped walks
- 82. Practical algorithm: i) start from G and a time window [0,T]
- 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. 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. >Null Models of temporal networks
- 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. 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. 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. 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. 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. >Dynamical processes on dynamical networks: From toy models to epidemiology
- 92. >Random walks
- 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. 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. Random walks on empirical temporal networks Starnini et al., Phys Rev E (2012) arXiv:1203.2477
- 96. >Toy spreading processes
- 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. 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. 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. 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. 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. 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. >Spreading processes
- 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. >Reminder: case of static networks
- 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. 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. 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. 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. >Spread on temporal networks
- 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. SIS model on activity-driven network Perra et al., Sci. Rep. (2012) Epidemic threshold: c = 1 m ⇣ hai + p ha2i ⌘
- 113. SIS and SIR models on activity-driven networks with memory Sun et al., EPJB (2015) SIS model: memory favors spread µ
- 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. >Epidemic threshold on temporal networks: Markov chain approach Valdano et al., PRX (2015)
- 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. 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. Valdano et al., PRX (2015) From temporal to multilayer Att0 ij = t,t0 +1 ⇣ ij + At0 ij ⌘ Markov chain approach for temporal networks
- 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. 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. 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. 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. Valdano et al., PRX (2015) Markov chain approach for temporal networks
- 124. > data-driven numerical simulations and data representations
- 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. “synopsis” of dynamic network data Temporal network Static network Contact matrix (underlying fully connected assumption + no within-class heterogeneity)
- 127. “synopsis” of dynamic network data x (x,y) + Contact matrix of distributions: -role based -takes heterogeneities into account Role-based structure Heterogeneities
- 128. + J. Stehlé et al., BMC Medicine (2011); A. Machens et al, BMC Inf Dis (2013) Evaluating representations
- 129. >A concrete example: contact patterns in a hospital
- 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. • 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. 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. + 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. SEIR simulation results Machens et al, BMC Infect. Dis.(2013) Importance of heterogeneities of contact durations
- 135. SEIR simulation results Attack rate by groups (for AR > 10%) Machens et al, BMC Infect. Dis.(2013)
- 136. > interventions
- 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. > mesoscale interventions • act at intermediate scales • avoid need to identify nodes with speciﬁc properties
- 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. An example: SEIR + school Which containment strategies? Gemmetto, Barrat, Cattuto, BMC Inf Dis (2014) Average over cases with AR > 10%
- 141. An example: SEIR + school Which containment strategies? Average over cases with AR > 10% Gemmetto, Barrat, Cattuto, BMC Inf Dis (2014)
- 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. An example: SEIR + school Which containment strategies? Cost in number of lost class-days Gemmetto, Barrat, Cattuto, BMC Inf Dis (2014)
- 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. > mesoscale interventions - 2
- 146. contacts in a primary school: • classes • correlated link activity J. Stehlé et al. PLoS ONE 6(8):e23176 (2011)
- 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. 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. L. Gauvin et al., PLoS ONE (2014)
- 150. L. Gauvin et al., PLoS ONE (2014) 10 classes 4 mixed-membership components = breaks Primary school case study
- 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. 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. 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. > Incomplete data Context: temporal contact networks
- 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. I- Evaluation of biases due to missing data II- Compensating for biases Issues
- 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. 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. 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. 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. 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. How to estimate the outcome of spreading processes from incomplete data? Do the incomplete data contain enough information?
- 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. 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. 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. Results
- 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. 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 SIR model, distribution of epidemic sizes Offices High School Conference
- 170. 170 SIS case
- 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. 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. 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. 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. Still very open field! Data Structures in data Incompleteness of data Models Processes on temporal networks …
- 176. http://complexnets2016.org

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