Virus Propagation on Time-Varying Networks: Theory and Immunization Algorithms
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Virus Propagation on Time-Varying Networks:
Theory and Immunization Algorithms
ECML-PKDD 2010, Barcelona, Spain
B. Aditya Prakash*, Hanghang Tong* ^, Nicholas Valler+, Michalis Faloutsos+, Christos
Faloutsos**Carnegie Mellon University, Pittsburgh USA
+University of California – Riverside USA^ IBM Research, Hawthrone USA
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Two fundamental questions
Epidemic!
Strong Virus
Q1: Threshold?
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example (static graph)Weak Virus
Small infectio
nQ1:
Threshold?
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Questions…
Q2: Immunizatio
n
Which nodes to
immunize?
?
?
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Standard, static graph Simple stochastic framework
◦Virus is ‘Flu-like’ (‘SIS’) Underlying contact-network – ‘who-can-
infect-whom’◦Nodes (people/computers) ◦Edges (links between nodes)
OUR CASE:◦Changes in time – alternating behaviors!◦think day vs night
Our Framework
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‘S’ Susceptible (= healthy); ‘I’ Infected No immunity (cured nodes -> ‘S’)
Reminder: ‘Flu-like’ (SIS)
Susceptible Infected
Infected by neighbor
Cured internally
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Virus birth rate β Host cure rate δ
SIS model (continued)
Infected
Healthy
XN1
N3
N2Prob. β
Prob. β
Prob. δ
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Alternating BehaviorsDAY (e.g., work)
adjacency
matrix
8
8
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Alternating BehaviorsNIGHT (e.g., home)
adjacency
matrix
8
8
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√Our Framework √SIS epidemic model√Time varying graphs
Problem Descriptions Epidemic Threshold Immunization Conclusion
Outline
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SIS model◦ cure rate δ◦ infection rate β
Set of T arbitrary graphs
Formally, given
day
N
N night
N
N ….weekend…..
Infected
Healthy
XN1
N3
N2Prob. β
Prob. β
Prob. δ
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Find…Q1: Epidemic Threshold:Fast die-out?
Q2: Immunizationbest k? ?
?
above
below
I
t
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NO epidemic if
eig (S) = < 1
Q1: Threshold - Main result
Single number! Largest eigenvalue of the
“system matrix ”
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NO epidemic if eig (S) = < 1
S =
cure rateinfection rate
……..
adjacency matrix
N
N
day night
Details
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Synthetic◦ 100 nodes◦ - Clique - Chain
MIT Reality Mining◦ 104 mobile devices◦ September 2004 – June 2005◦ 12-hr adjacency matrices (day) (night)
Q1: Simulation experiments
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‘Take-off’ plotsSynthetic MIT Reality Mining
Footprint (# infected @ steady state)
Our threshol
d Our threshol
d
(log scale)
NO EPIDEMIC
EPIDEMIC EPIDEMIC
NO EPIDEMIC
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Time-plotsSynthetic MIT Reality Mining
log(# infected)
Time
BELOW threshold
AT threshold
ABOVE threshold
ABOVE threshold
AT threshold
BELOW threshold
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√Motivation√Our Framework
√SIS epidemic model√Time varying graphs
√Problem Descriptions√Epidemic Threshold Immunization Conclusion
Outline
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Our solution◦reduce (== )
◦goal: max ‘eigendrop’ Δ
Comparison - But : No competing policy We propose and evaluate many policies
Q2: Immunization
Δ = _before - _after
?
?
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Lower is better
OptimalGreedy-S
Greedy-DavgA
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Time-varying Graphs , SIS (flu-like) propagation model√ Q1: Epidemic Threshold - < 1
◦Only first eigen-value of system matrix!
√ Q2: Immunization Policies – max. Δ ◦Optimal◦Greedy-S◦Greedy-DavgA◦etc.
Conclusion….
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B. Aditya Prakash http://www.cs.cmu.edu/~badityap
Our threshold
Any questions?