Temporal Networks - University of California,...
Transcript of Temporal Networks - University of California,...
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Temporal Networksaka time-varying networks, time-stamped
graphs, dynamical networks...
Network Theory and Applications
ECS 253 / MAE 253
Spring 2016
Márton Pósfai ([email protected])
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Sources
Reviews:
Courses:● ENS Lyon: Márton Karsai
● Northeastern University: Nicola Perra, Sean Cornelius, Roberta Sinatra
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Temporal Networks
● So far: static network
● Description:
AA
DD
BB
CC
Microscopic:Node, link properties(degree, centralities)
Mezoscopic:Motives, communities
Macroscopic:statistics
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Temporal Networks
● Static network: Spreading process can reach all nodes starting from A.
AA
DD
BB
CC
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Temporal Networks
● Now: time of interaction
AA
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BB
CC
1, 3
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2, 3
1
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Temporal Networks
● Now: time of interaction
t=0
AA
DD
BB
CC
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Temporal Networks
● Now: time of interaction
t=1
AA
DD
BB
CC
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Temporal Networks
● Now: time of interaction
t=2
AA
DD
BB
CC
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Temporal Networks
● Now: time of interaction
t=3
AA
DD
BB
CC
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Temporal Networks
● Now: time of interaction
t=0
Microscopic:Node, link properties(degree, centralities)
Mezoscopic:Motives, communities
Macroscopic:statistics
AA
DD
BB
CC
1, 3
2
2, 3
1
? ? ?
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When are temporal nets useful?
● Timescales: : timescale of dynamics
: timescale of changes in network
● : static approximation
Example: Internet
Branigan et al, Internet Computing, IEEE (2001)
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When are temporal nets useful?
● Timescales: : timescale of dynamics
: timescale of changes in network
● : annealed approximation
Example: PC or Mac
● Process slow enough that A meets all contacts
● Weight: how frequently they meet
P. Holme and J. Saramäki. "Temporal networks." Physics reports 519.3 (2012): 97-125.
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When are temporal nets useful?
● Timescales: : timescale of dynamics
: timescale of changes in network
● :
TEMPORAL NETWORKS
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Examples● Communication: Email, phone call, face-to-face
● Proximity: same hospital room, meet at conference, animals hanging out
● Transportation: train, flights…
● Cell biology: protein-protein, gene regulation
● ...
Email communicationJ. Tang, John, Social Network Systems. ACM, 2010.
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Examples● Communication: Email, phone call, face-to-face
● Proximity: same hospital room, meet at conference, animals hanging out
● Transportation: train, flights…
● Cell biology: protein-protein, gene regulation
● ...
Visitors at exhibit.Isella, Lorenzo, et al. Journal of theoretical biology 271.1 (2011): 166-180.
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Examples● Communication: Email, phone call, face-to-face
● Proximity: same hospital room, meet at conference, animals hanging out
● Transportation: train, flights…
● Cell biology: protein-protein, gene regulation
● ...
Temporal network of zebrasC. Tantipathananandh, et.al. (2007)
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Plan
1) Mathematical representation
2) Path- based measures of temporal
3) Temporal heterogeneity
4) Processes and null models
5) Motifs
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Mathematical Description
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Mathematical representation
● Temporal graph:
Set of vertices. Set of edges.
Set of times when edge e is active.
1. Contact sequence 2. Adjacency matrix sequence
● Interval graph:
Set of intervals when edge e is active.
P. Holme and J. Saramäki. "Temporal networks." Physics reports 519.3 (2012): 97-125.
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Aggregating in time windows
● Sequence of snapshots
Consecutive windows Sliding windows
● Lossy method● Sometimes data is not available● Convenient: Static measures on snapshots → Time series of measures
● Problem: snapshots depend on window size?● How to choose?
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Window size?
Clauset and Eagle (2012)
● MIT reality mining project: high resolution proximity data
● Snapshots: Time window:
● Adjacency correlation:
● 0 uncorrelated, 1 if the same
: set of nodes that are connected to j at x of y
Average degree Clustering coef. 1-Adjacency corr.
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Window size?
Clauset and Eagle (2012)
Time series autocorrelation for Δ=5 min
Fourrier spectrum
Driven by periodic patterns → Sampling rate should be twice the highest frequencyΔ = 4 hours
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Path-based measures
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Path-based measures
Time-respecting paths:● Takes into account the temporal order and timing of contacts.
● Optional: maximum wait time
Properties:● No reciprocity: path i→j does not imply path j→i.● No transitivity: path i→j and path j→k does not imply path i→j→k.● Time dependence: path i→j that starts at t does not imply paths at t'>t.
P. Holme, Physical Review E 71.4 (2005)
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Path-based measures
Observation window [t0,t
1]
Influence set of node i● Nodes that can be reached from node i within the observation window.● Reachability ratio f: fraction of nodes that can be reached
P. Holme, Physical Review E 71.4 (2005)
Source set of node i● Nodes from node i is reached within the observation window.
Reachability ratio● Fraction of node pairs (i,j) such that path i→j exists.
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Path-based measures
Temporal path length – Duration
● Duration = tn- t
0
Temporal distance – Latency● the shortest (fastest) path duration from i to j starting at t.
Information latency● the age of the information from j to i at t
● End of the observation window: paths become rare.
● Solution: periodic boundary, throw away end
R. K. Pan, and J. Saramäki.Physical Review E 84.1 (2011)
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Path-based measures
Strongly connected component● All node pairs are connected in both directions within T.
R. K. Pan, and J. Saramäki.Physical Review E 84.1 (2011)
Weakly connected component● All node pairs are connected in at least one direction within T.
Temporal betweenness centrality● Static: Temporal:
Etc.
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Temporal heterogeneity
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Source 1: Periodic patterns
For example, circadian rythm● My email usage
● Scientists work schedule
Wang et al, Journal of infometrics (2012)
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Source 2: Burstiness● Humans and many natural phenomena show heterogeneous temporal behavior on
the individual level.● Switching between periods of low activity and high activity bursts.● Sign of correlated temporal behavior
Earthquakes in Japan
Neuron firing
Phone calls
Karsai et al, Scientific Reports (2011)
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Source 2: Burstiness● Sign of correlated temporal behavior● Reference: Poisson process, events uncorrelated
Barabási, Nature (2005)
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Source 2: Burstiness● Measure of burstiness:
Barabási, Nature (2005)
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Possible explanation for burstiness● Executing tasks based on prioirity● L types of tasks, one each (e.g. work, family, movie watching,...)
● Each task i has a priority xi draw uniformly from [0,1]
● Each timestep one task is executed, probability of choosing i:
● And a new task is added of that type
Barabási, Nature (2005)
Random Deterministic
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Is inter-event time power-law?
Karsai et al, Scientific Reports (2012)
Phone calls Text message Email
● Is this a power-law? Definitely not exponential.
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Does burstiness matter?
L. EC Rocha, F. Liljeros, and P. Holme. PNAS 107.13 (2010)
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Processes and null models
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Structure and dynamics
L. EC Rocha, F. Liljeros, and P. Holme. PNAS 107.13 (2010)
● So far: various measures to characterize network● How does structure affect processes on the network?● Possibility:
1) Generate model networks with given parameters.
2) Run dynamics model → measure outcome.
3) Scan parameters.● Models from scratch: many parameters to set → few models, no dominant
● Instead:
1) Take empirical network.
2) Remove correlations by randomization.
3) Run dynamics model → measure outcome.
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Dynamics● Model: SI model
● Infection rate:
If connected
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Dynamics● Model: SI model on a temporal network● Simplest model of information spreading● Infection only spreads along active contacts● Infection can spread both ways● Infection rate● Single seed at t=0
● Mobile call data as underlying network
Slide adopted from Márton Karsai
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Original temporal net● We run SI model and measure
● Now what? Is this because burstiness, community structure?
Slide adopted from Márton Karsai
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Properties of the original● Bursty dynamics (BD)
Heterogeneous inter-event time distribution
● Community structure (CS)Densely connected subgroups(Any other structure beyond degree distribution)
● Link-link correlations (LL)Causality between consecutive calls
● Weight-topology correlation (WT)Strong ties within local communities, weak ties connect different communitiesWeight = total call times = strength of a node = some of adjacent link weights Onnela, J-P., et al. "Structure and tie strengths in mobile communication networks." PNAS 104.18 (2007).
Slide adopted from Márton Karsai
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Recap from community detection● Randomization to remove community structure of a static network
Original What we compare to
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Randomization 1: temp. config. model ● Degree preserved randomization to remove CS and WT
● Shuffle event times to remove BD and LL
Slide adopted from Márton Karsai
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Randomization 1: temp. config. model ● Degree preserved randomization to remove CS and WT
● Shuffle event times to remove BD and LL
● No correlation left.
→ Correlations slow thespread of information.
Slide adopted from Márton Karsai
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Randomization 2: time shuffled network ● Shuffle event times to remove BD and LL
● CS and WT remain.
Slide adopted from Márton Karsai
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Randomization 2: time shuffled network ● Shuffle event sequences to remove WT and LL
● CS and BD remain.
Slide adopted from Márton Karsai
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Randomization 3: time seq. shuffled● Shuffle event sequences to remove WT and LL
● CS and BD remain.
Slide adopted from Márton Karsai
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Randomization 3: time seq. shuffled● Shuffle event sequences to remove WT and LL
● CS and BD remain.
Slide adopted from Márton Karsai
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Rand. 4: equal link time seq. shuffled● Shuffle event sequences if they have the same weight to remove LL
● CS, WT and BD remain.
→ Multi-link processes slightlyaccelerate the spread.
Slide adopted from Márton Karsai
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Long time behavior● Distribution of complete infection time
● Evidence of effect of correlations in thelate time stage.
● Multi-link correlations have contraryeffect compared to early stage
● WT and BD are the main factorsin slowing down
Slide adopted from Márton Karsai
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Summary● Timescale of dynamics and changes in network structure comparable
→ Temporal networks
● Time respecting paths profound effect on spreading
● Temporal inhomogeneities: circadian rhythm and burstiness
● Measures more involved, computationally more difficult
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Temporal motifs
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Definitions
Slide adopted from Márton Karsai
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Detection
Slide adopted from Márton Karsai
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Algorithm
Slide adopted from Márton Karsai
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What to compare to?
Slide adopted from Márton Karsai
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Phone call network
Slide adopted from Márton Karsai
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Phone call network
Slide adopted from Márton Karsai