Description and simulation of dynamic mobility networks Éric Fleury http://perso.ens-lyon.fr/eric.fleury/ mailto://[email protected] ENS Lyon/LIP – INRIA/A4RES

Journée thématique dynamiques de graphes

E. Fleury (ENS Lyon / INRIA)

ResCom/ISC 2008

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Teams Involved

Multi Disciplinary Teams / Long Term Adventure

A4RES / INRIA Guillaume C HELIUS Éric F LEURY Antoine F RABOULET CeRBEP / Inserm / Pasteur Didier G UILLEMOT

UPMC, LIP6 Jean-Loup G UILLAUME LIRIS / INSA Céline R OBARDET ENS Lyon, Laboratoire de Physique

Odile L E M INOR

Pierre B ORGNAT

Lulla O PATOWSKI

Antoine S CHERRER

E. Fleury (ENS Lyon / INRIA)

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Teams Involved

Some references

Dynamic networks Antoine Scherrer, Pierre Borgnat, Éric Fleury, Jean-Loup Guillaume and Céline Robardet, Description and simulation of dynamic mobility networks, in Computer Network 2008. Antoine Scherrer, Pierre Borgnat, Éric Fleury, Jean-Loup Guillaume and Céline Robardet, A Methodology to Identify Characteristics of the Dynamic of Mobile Networks,in AINTEC 2008. Éric Fleury, Jean-Loup Guillaume, Céline Robardet and Antoine Scherrer, Analysis of Dynamic Sensor Networks: Power Law Then What?, in Comsware 2007.

E. Fleury (ENS Lyon / INRIA)

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Overview

Outline

1

MOSAR Project Accounting for inter-individual interactions Scenario overview

2

Dynamic Network Characterization Context and motivation Statistical analysis of snapshots of graphs Towards a global analysis of the dynamics Modeling of the dynamics

3

Conclusion

E. Fleury (ENS Lyon / INRIA)

ResCom/ISC 2008

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MOSAR Project

Outline

1

MOSAR Project Accounting for inter-individual interactions Scenario overview

2

Dynamic Network Characterization Context and motivation Statistical analysis of snapshots of graphs Towards a global analysis of the dynamics Modeling of the dynamics

3

Conclusion

E. Fleury (ENS Lyon / INRIA)

ResCom/ISC 2008

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MOSAR Project

Mastering hOSpital Antimicrobial Resistance

http://www.mosar-sic.org MOSAR is an Integrated Project supported for 5 years by the European Commission under the Life Science Health Priority of the Sixth Framework Program. Coordinated by INSERM (the French National Institute of Health and Medical research); MOSAR aims to significantly advance our knowledge regarding the control of antimicrobial resistance of bacteria responsible for major and emerging nosocomial infections.

E. Fleury (ENS Lyon / INRIA)

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MOSAR Project

A major conjoint challenge for TIC & LSH

Experiments A data collection strategy will combine for a period of 6 months on 400 actors: an individual antibiotic use; a contact monitoring; a characterization of the isolates to determine their epidemicity;

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MOSAR Project

Accounting for inter-individual interactions

Deployment of a large-scale dynamic networks

Document interactions between Medical / Nursing staff / Patients (400 people) 7/24 during 6 month long period

Document contact frequencies monitor the dynamic (inter & intra contact) characterize the interaction network

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MOSAR Project

Accounting for inter-individual interactions

First challenge Design, Deploy and Set up the WSN infrastructure different hospital departments rehabilitation, surgical and intensive care units Associate 1 sensor with each actor medical and nursing staff hospitalized patients

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MOSAR Project

Patient room

E. Fleury (ENS Lyon / INRIA)

Scenario overview

Patient room

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MOSAR Project

Patient room

E. Fleury (ENS Lyon / INRIA)

Scenario overview

Patient room

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MOSAR Project

Patient room

E. Fleury (ENS Lyon / INRIA)

Scenario overview

Patient room

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MOSAR Project

Patient room

E. Fleury (ENS Lyon / INRIA)

Scenario overview

Patient room

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MOSAR Project

Patient room

E. Fleury (ENS Lyon / INRIA)

Scenario overview

Patient room

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MOSAR Project

Patient room

E. Fleury (ENS Lyon / INRIA)

Scenario overview

Patient room

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MOSAR Project

Patient room

E. Fleury (ENS Lyon / INRIA)

Scenario overview

Patient room

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MOSAR Project

Patient room

E. Fleury (ENS Lyon / INRIA)

Scenario overview

Patient room

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MOSAR Project

Scenario overview

Global infrastructure

Sensors Nodes Base Stations Data Base

http://perso.ens-lyon.fr/eric.fleury/Upload/Mosar/MosarEng080120.wmv

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MOSAR Project

Scenario overview

Global infrastructure

Sensors Nodes Base Stations Data Base

http://perso.ens-lyon.fr/eric.fleury/Upload/Mosar/MosarEng080120.wmv

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MOSAR Project

Scenario overview

Global infrastructure

BS

Sensors Nodes Base Stations Data Base

http://perso.ens-lyon.fr/eric.fleury/Upload/Mosar/MosarEng080120.wmv

E. Fleury (ENS Lyon / INRIA)

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MOSAR Project

Scenario overview

Global infrastructure

BS

Sensors Nodes Base Stations Data Base

BD

http://perso.ens-lyon.fr/eric.fleury/Upload/Mosar/MosarEng080120.wmv

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Dynamic Network Characterization

Outline

1

MOSAR Project Accounting for inter-individual interactions Scenario overview

2

Dynamic Network Characterization Context and motivation Statistical analysis of snapshots of graphs Towards a global analysis of the dynamics Modeling of the dynamics

3

Conclusion

E. Fleury (ENS Lyon / INRIA)

ResCom/ISC 2008

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Dynamic Network Characterization

Context and motivation

Context & Objectives MOSAR project Better understand the intrinsic characteristics / properties of dynamic networks Model / analyze interaction between node/users Describe accurately the dynamics Two central questions: Obtaining random models that reproduce “these” properties How do their functionalities constrain the structures of real network?

E. Fleury (ENS Lyon / INRIA)

ResCom/ISC 2008

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Dynamic Network Characterization

Context and motivation

Context & Objectives MOSAR project Better understand the intrinsic characteristics / properties of dynamic networks Model / analyze interaction between node/users Describe accurately the dynamics Two central questions: Obtaining random models that reproduce “these” properties How do their functionalities constrain the structures of real network?

E. Fleury (ENS Lyon / INRIA)

ResCom/ISC 2008

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Dynamic Network Characterization

Context and motivation

Context & Objectives MOSAR project Better understand the intrinsic characteristics / properties of dynamic networks Model / analyze interaction between node/users Describe accurately the dynamics Two central questions: Obtaining random models that reproduce “these” properties How do their functionalities constrain the structures of real network?

E. Fleury (ENS Lyon / INRIA)

ResCom/ISC 2008

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Dynamic Network Characterization

Context and motivation

Remark on “real” networks

Real networks play precise roles under constraints: They fulfill a function Trade-off / sympathy / efficiency Important point in distributed algorithms: efficient protocols / algorithms with a extremely restricted decentralized knowledge Computer scientists may have as important things to say as physicists on the matter

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Dynamic Network Characterization

Context and motivation

Preliminary data1

Traces are now available 41 nodes, 3 days (254 151 sec), every 120sec 820 possible links, inter contact time distribution can be compared to the one of power law Power law... What do power law really signify? Is it the ultimate argument?

1 A. Chaintreau and J. Crowcroft and C. Diot and R. Gass and P. Hui and J. Scott, Impact of Human Mobility on the Design of Opportunistic Forwarding Algorithms, INFOCOM 2006 E. Fleury (ENS Lyon / INRIA)

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Dynamic Network Characterization

Context and motivation

Some remarks2 on “Power laws” Various kinds of data can be approximated by inferring a fitting curve on a log-log scale plot Quite “easy” to generate “Scale free” network does not imply deep knowledge on the intrinsic dynamic structure It is worthy to analyze dynamics of contact network Coupled arguments Graph theory / random process / data mining

2

Keller Evelyn Fox, Revisiting "scale-free" networks, BioEssays, 2006

E. Fleury (ENS Lyon / INRIA)

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Dynamic Network Characterization

Statistical analysis of snapshots of graphs

Statistical analysis of snapshots of graphs

Standard graph properties as a function of time temporal evolution of the snapshots statistical signal processing Method descriptive analysis models / simulation

E. Fleury (ENS Lyon / INRIA)

ResCom/ISC 2008

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Dynamic Network Characterization

Statistical analysis of snapshots of graphs

Standard graph properties Snapshots Gt = (V 0 , Et ) Active links: E(t) = |Et | Connected vertcices: V (t) = |{u ∈ V 0 , dGt (u) > 0}| P Average degree of connected vertices is D(t) = u∈V 0 dGt (u)/V (t) Number of connected components (maximal subgraph such as every node of the subgraph is connected to each another node): Nc (t) = |CGt | Number of triangles: T (t) = |TGt | Property #Active links #Connected vertices Avg degree #CC #Triangles E. Fleury (ENS Lyon / INRIA)

E(t) V (t) D(t) Nc (t) T (t)

Mean

I MOTE Std. Dev.

21.9 19.9 2.1 4.8 6.9

12.4 4.7 0.8 2.1 8.30

Corr. Time (s) 5200 7400 3600 5600 4700 ResCom/ISC 2008

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Dynamic Network Characterization

Statistical analysis of snapshots of graphs

Standard graph properties (cont) Probability distribution time bin of 1s t − (< X (t) >t )2 correlation time: first time where the function CX (τ ) goes to zero notes correlation times of E, V and Nc are rather large: ∼ 1h15. D and T have comparable correlation times. This suggests that these properties evolve under a common cause. E. Fleury (ENS Lyon / INRIA)

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Statistical analysis of snapshots of graphs

Dynamic Network Characterization

Dynamical characteristics (cont) 0

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Mean : 3680; α = 0.60

Contact and inter-contact durations P[X > x] ∼ cx −α . x→∞

α > 2: finite mean/variance; α < 2, infinite variance (heavy tailed). α < 1, infinite mean/variance. E. Fleury (ENS Lyon / INRIA)

ResCom/ISC 2008

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Dynamic Network Characterization

Statistical analysis of snapshots of graphs

Dynamics of links creation and deletion

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E⊕ (t) = |{e ∈ Et , e ∈ / Et−1 }|, the number of links added at time t

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Dynamic Network Characterization

Statistical analysis of snapshots of graphs

Dynamics of links creation and deletion (cont) 10

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E (t) = |{e ∈ Et−1 , e ∈ / Et }|, the number of links removed at time t Property Edge creation E⊕ (t) Edge delation E (t)

E. Fleury (ENS Lyon / INRIA)

Mean 0.15 0.15

I MOTE Std. Dev. 0.55 0.55

Corr. Time (s) 680 ∼ 12min 680∼ 12min

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Dynamic Network Characterization

Statistical analysis of snapshots of graphs

Multivariate statistics of graph properties Cross-correlations Strong influence E(t) over V (t); Nc (t) related to E(t) Less related: Nc (t) and V (t) E⊕ (t) and E (t): mostly uncorrelated

E(t) V (t) Nc (t) D(t) T (t) E⊕ (t) E (t)

E(t) 1 0.85 -0.56 0.95 0.90 0.19 0.15

E. Fleury (ENS Lyon / INRIA)

V (t) 0.85 1 -0.20 0.69 0.66 0.15 0.11

Nc (t) -0.56 -0.20 1 -0.69 -0.41 -0.16 -0.15

D(t) 0.95 0.70 -0.70 1 0.86 0.20 0.16

T (t) 0.90 0.66 -0.41 0.86 1 0.15 0.10

E⊕ (t) 0.19 0.15 -0.16 0.19 0.15 1 0.03

E (t) 0.15 0.11 -0.15 0.15 0.11 0.03 1

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Statistical analysis of snapshots of graphs

Dynamic Network Characterization

Multivariate statistics of graph properties Joint distributions PXY (x, y ) = P[X = x and Y = y ] = P[X = x/Y = y ]P[X = x] variation of the # links is not constant over the # vertices 80

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Dynamic Network Characterization

Statistical analysis of snapshots of graphs

Multivariate statistics of graph properties

Link correlations Most pairs of links have a very low correlation coefficient. 4

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Number of edge couples

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E. Fleury (ENS Lyon / INRIA)

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Dynamic Network Characterization

Statistical analysis of snapshots of graphs

Multivariate statistics of graph properties

Link correlations Most pairs of links have a very low correlation coefficient. 4

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Markovian evolution 1 Correlation time link creation/deletion is small

Number of edge couples

10 8 6

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Independent from the evolution of other graph properties

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Dynamic Network Characterization

Towards a global analysis of the dynamics

Towards a global analysis of the dynamics

global properties not directly interpretable in the sequence of static graphs stability of connected components proportion of creation of triangles communities embedded in the network

E. Fleury (ENS Lyon / INRIA)

ResCom/ISC 2008

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Dynamic Network Characterization

Towards a global analysis of the dynamics

Stability of Connected Components CCs (+) & CCNs (x) 6819 CCs & 2608 CCNs (292 isolated links) strong heterogeneity: 52% of the CCs and 40% of the CCNs exist during one time step. some during 1/3 to 1/4.

most frequent CCs and CCNs are just couples of vertices 10000

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E. Fleury (ENS Lyon / INRIA)

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Dynamic Network Characterization

Towards a global analysis of the dynamics

Stability of Connected Components (cont) CCs (+) & evolution absence of stability of large CCs. No CCs (> 8 –67%) / 100s. 74% correspond to one CC appearing for one disappearing. 26% CCs appearing from scratch, disappearing completely, or even merging/splitting. 100000

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E. Fleury (ENS Lyon / INRIA)

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Dynamic Network Characterization

Towards a global analysis of the dynamics

Triangles in the graphs

I MOTE R ANDOM

P+/tri+ 44 % 10 %

P+/tri= 56 % 90 %

f+/tri+ 6% 5%

f+/tri= 94 % 95 %

links / triangles P+/tri+ : link creation → triangle f+/tri+ : innactive link → triangle 40% of link creations increase the number of triangles proportion of inactive links that would create a triangle is very low More potential links doesn not imply higher P+/tri+

E. Fleury (ENS Lyon / INRIA)

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Towards a global analysis of the dynamics

Dynamic Network Characterization

Communities in dynamic interaction networks Formal concepts maximal rectangles of true values / pattern mining S ⊆ Gt : C = {S = (V , E), |{t | S ⊆ Gt }| ≥ τ and |E| ≥ σ and S is connected}. i8 −→ g 13 −→ g 7 i30 i36 i1

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Dynamic Network Characterization

Modeling of the dynamics

Modeling of the dynamics Simulation algorithm transition model with Markovian property links e are independent state of the network links e changes with Ptr (e, Gt ) duration τ (e) since the link e has last changed its status Ingredients contact / inter contact duration distribution elaborated graph properties (E(t), V (t), NC (t), D(t)) dynamical information (triangles)

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Dynamic Network Characterization

Modeling of the dynamics

Modeling of the dynamics Input: Simulation time Output: Random Dynamic Graph foreach Simulation Time Step t do foreach link e do Ptr (e, Gt ) = TransitionProbability(e) given the state Gt ; pr = Uniform(0,1); if (pr ≤ Ptr (e)) then ChangeState(e); end end end

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Dynamic Network Characterization

Modeling of the dynamics

Ingredients I Contact distribution heavy-tailed distributions for contact PON and inter-contact POFF durations P+ (τ ): probability that one link that was OFF since τ (τ ≥ 1) is activated Q −1 PON (τ ) = P− (τ ) × τi=1 (1 − P− (i)) P (τ ) P− (τ ) = Qτ −1 ON , τ ≥ 2, P− (1) = PON (1) (1 − P (i)) − i=1 P (t) P+ (τ ) = Qτ −1 OFF , τ ≥ 2, P+ (1) = POFF (1) (1 − P (i)) + i=1

E. Fleury (ENS Lyon / INRIA)

ResCom/ISC 2008

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Dynamic Network Characterization

Modeling of the dynamics

Ingredients II

Imposed graph property distribution Rejection Sampling based on a Metropolis-Hastings algorithm new state Gt0 = {Gt+ Se (t) changed}, is accepted with probability  F (x(G0 ))

PRS (Gt , Gt0 ) = min 1, F (x(Gtt ))

F is the target PDF for the graph The total probability of transition of link e is then: Ptr (e, Gt ) = P−/+ (τ (e)) · PRS (Gt , Gt0 ).

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Dynamic Network Characterization

Modeling of the dynamics

Ingredients III

Imposed dynamics of triangles reproduce the correct dynamical transition process concerning triangles do not want to change the mean probabilities of transition The weighted probabilities are then:   P+ (τ (e)) P+/tri= for link creation without new triangle, f+/tri= Ptr (e, Gt ) =  P+ (τ (e)) P+/tri+ for link creation with a new triangle. f +/tri+

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Dynamic Network Characterization

Modeling of the dynamics

Simulation results Investigated models A: imposed empirical contact and inter-contact duration distribution only. B: imposed distributions of contact / inter-contact durations , and of number of connected components.

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−− Imote / o Model A / ∗ Model B / + Model C E. Fleury (ENS Lyon / INRIA)

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Dynamic Network Characterization

Modeling of the dynamics

Simulation results (cont) 0.12

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A: sole contact and inter-contact duration fails the number of connected vertices is strongly over-estimated the number of connected components is under-estimated

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Dynamic Network Characterization

Modeling of the dynamics

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A, B and C fail! The density of the connected components (the groups) is underestimated Links are spread uniformly in the graph

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Dynamic Network Characterization

Modeling of the dynamics

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Weighted models does not have an impact on the contact and inter-contact duration distributions the density of connected components is comparable to the experimental data

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Dynamic Network Characterization

Modeling of the dynamics

Simulation results (cont) 10

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Density of frequent connected components (τ = 7 and σ = 6) classical models fail to create dense frequent connected components the number of frequent connected subgraphs is larger in the simulated data than in the original E. Fleury (ENS Lyon / INRIA)

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Dynamic Network Characterization

Modeling of the dynamics

Simulation results (cont) 5, 13, 29, 37

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Conclusion

Outline

1

MOSAR Project Accounting for inter-individual interactions Scenario overview

2

Dynamic Network Characterization Context and motivation Statistical analysis of snapshots of graphs Towards a global analysis of the dynamics Modeling of the dynamics

3

Conclusion

E. Fleury (ENS Lyon / INRIA)

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Conclusion

Conclusion contributions rigorous / coherent set of properties (basic / advanced) probability distribution of contacts and inter contacts is only one parameter global analyses to characterize the dynamics of the graph as a whole: correlation between links stability of the connected components number of triangles evolution of communities inside the interaction networks.

simple / accurate models that generate random interaction graphs with satisfactory temporal properties.

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Conclusion

Conclusion

Futur works Introduce non-stationarity (piecewise stationary model) Dynamic community computation Trajectories of individuals as a signature Large in situ test beds to be deployed...

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