Modeling temporal networks using random itineraries

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Griselda Wilkinson
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110, 158702 (2013) arxiv:1303.4411 http://www.cpt.

1 Modeling temporal networks using random itineraries Alain Barrat CPT, Marseille, France & ISI, Turin, Italy j j i i G k k i j k i j me$ N k 2$ i j k 1$ 0$ A.B., B. Fernandez, K. Lin, L.-S. Young Phys. Rev. Lett. 110, (2013) arxiv:

2 Time-varying networks Networks= (often) dynamical entities Which dynamics? Characterization? Modeling? Consequences on dynamical phenomena? (e.g. epidemics, information propagation ) Time-varying networks: often represented by aggregated views Lack of data Convenience

3 Example: contacts in a primary school, aggregated view J. Stehlé et al. PLoS ONE 6(8):e23176 (2011)

4 Example: contacts in a primary school, dynamic view J. Stehlé et al. PLoS ONE 6(8):e23176 (2011)

5 Definition: temporal network Temporal network: T=(V,S) V=set of nodes S=set of event sequences assigned to pairs of nodes s ij 2 S : s ij = {(t s,1 ij,te,1 ij ) (ts,` ij,te,` ij )} Other representation: time-dependent adjacency matrix: a(i,j,t)= 1 <=> i and j connected at time t 5

6 Reachability issue Review Holme-Saramaki, Phys. Rep. (2012), arxiv:

7 Aggregation of temporal network w ij = Z tmax t min a(i, j, t)dt NB: enough information if underlying process is Poissonian 7 Review Holme-Saramaki, Phys. Rep. (2012), arxiv:

8 Aggregation of temporal network Temporal behavior most often non-poissonian => aggregate view hides important temporal patterns 8 Review Holme-Saramaki, Phys. Rep. (2012), arxiv:

9 Burstiness Poisson process Bursty behavior A.-L. Barabasi, Nature (2006) 9

10 Generalization of definitions to temporal networks Reachability issue => time respecting path ( journey ) => set of influence of a node => temporal connectivity (similar to case of directed graphs) Path length => concept of shortest paths Time respecting path duration => concept of fastest journey Temporal motifs Centrality measures ( ) Review Holme-Saramaki, Phys. Rep. (2012), arxiv:

11 fastest journeys shortest paths B Time t C B C Time t >t B Time t >t C A A A Fastest path= A->B->C Shortest path= A-C

12 Complex temporal characteristics burstiness non-poissonian inter-event distributions power-law temporal correlations heterogeneity of event durations single events aggregated durations (weights in aggregated networks) stationarity of statistical features daily, weekly, and organizational rhythms weight-topology correlations topology-activity correlations (e.g., school)

13 Temporal networks Generalization of concepts? Centrality/importance of a node? How to measure correlations? Temporal communities? Models for temporal networks? Representations of temporal networks? Impact of temporal features on dynamical processes?

14 Data on time-varying networks is often limited... i) access only to time-aggregated ii) access to a single sample on [0,T] data 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?

15 Here 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 timevarying network on [0,T] with realistic features (e.g., burstiness, non-trivial correlations), whose temporal aggregation coincides with G

16 Static weighted graph G w=3 w=1 w=1 w=2 w=3 w=1 w=2 w=3 w=1 w=2 w=1

17 Static weighted graph G

18 Static weighted graph G Static decomposition in itineraries/paths

19 Static weighted graph G Unfolding G as a dynamic aggregation of walks

20 Static weighted graph G Unfolding G as a dynamic accumulation of interwoven time-stamped walks

21 Practical algorithm: i) start from G and a time window [0,T]

22 Practical algorithm: ii) perform random walks on G with random initial position and time iii) with random v) store the events (i,j,t1); (j,k,t2) prescribed length iii) with random residence times at k i l=2 each step t2=t1+τ t1 iv) update the weight of each traversed link: w w-1 j

23 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)

24 Case study / Validation Broad distributions of in-degree, outdegree, weights Poisson distribution for random walk lengths Broad or Poisson distributions of residence times Inspired by: P. Bajardi et al., Dynamical patterns of cattle trade movements PLoS ONE 6(5):e19869 (2011)

25 Case study Distributions of degree, weight, strength on temporal network aggregated on different time-windows -stationary -converges to G s properties as the time-window length goes to T

26 Case study Distribution of activity and inactivity periods, for different aggregation time windows bursty, similar to empirical data similar results at the link level

27 Case study Global activity temporal correlation Null model: reshuffled events =equivalent to taking only random walks of length 1

28 Case study Local measure of temporal correlations: -categorize nodes according to their global activity (=aggregated strength) -measure activity in a ball of radius r around a node at t+τ, conditioned to the node being active at t -average over nodes in a given category Busy nodes (large strength) Non-busy nodes (small strength) Non-trivial spatio-temporal correlation patterns

29 Summary Framework aiming at creating realistic/plausible time-varying networks when -only aggregated data is available -only one instance is available Transform a static (aggregated) graph into a time-varying network by unfolding it as a series of random walks Flexible algorithms, robust phenomenology w.r.t. changes in path length, residence time distributions, etc... Future possible directions: Static network with community structure => temporal communities? Spreading trees instead of random walks as building blocks? Studies of spreading processes on top of such temporal networks Evaluation of the relevance of temporal centrality measures Random walks with memory Reference: A.B., B. Fernandez, K. Lin, L.-S. Young Modeling temporal networks using random itineraries Phys. Rev. Lett. 110, (2013) arxiv: j j i i G k k j i k i j me$ N k 2$ i j k 1$ 0$

Plan of the lecture I. INTRODUCTION II. DYNAMICAL PROCESSES. I. Networks: definitions, statistical characterization II. Real world networks

Plan of the lecture I. INTRODUCTION I. Networks: definitions, statistical characterization II. Real world networks II. DYNAMICAL PROCESSES I. Resilience, vulnerability II. Random walks III. Epidemic processes