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CS 6604: Data Mining Large Networks and Time-‐series
B. Aditya Prakash Lecture #8: Epidemics: Thresholds
A fundamental ques@on Strong Virus
Epidemic?
2 CS 6604:DM Large Networks & Time-‐Series Prakash 2013
Problem Statement
Find, a condi4on under which – virus will die out exponen4ally quickly – regardless of ini4al infec4on condi4on
above (epidemic)
below (ex4nc4on)
# Infected
@me
4
Separate the regimes?
CS 6604:DM Large Networks & Time-‐Series Prakash 2013
Threshold (sta@c version)
Problem Statement § Given: – Graph G, and – Virus specs (aOack prob. etc.)
§ Find: – A condiTon for virus exTncTon/invasion
5 CS 6604:DM Large Networks & Time-‐Series Prakash 2013
Threshold: Why important?
§ AcceleraTng simulaTons § ForecasTng (‘What-‐if’ scenarios) § Design of contagion and/or topology § A great handle to manipulate the spreading – ImmunizaTon – Maximize collaboraTon …..
6 CS 6604:DM Large Networks & Time-‐Series Prakash 2013
Outline
§ Q: What is the epidemic threshold? – Background – Result and IntuiTon (StaTc Graphs) – Proof Ideas (StaTc Graphs) – Bonus: Dynamic Graphs
7 CS 6604:DM Large Networks & Time-‐Series Prakash 2013
“SIR” model: life immunity (mumps)
§ Each node in the graph is in one of three states – SuscepTble (i.e. healthy) – Infected – Removed (i.e. can’t get infected again)
8
Prob. β Prob. δ
t = 1 t = 2 t = 3 CS 6604:DM Large Networks & Time-‐Series Prakash 2013
Related Work q R. M. Anderson and R. M. May. InfecTous Diseases of Humans. Oxford University Press,
1991. q A. Barrat, M. Barthélemy, and A. Vespignani. Dynamical Processes on Complex
Networks. Cambridge University Press, 2010. q F. M. Bass. A new product growth for model consumer durables. Management Science,
15(5):215–227, 1969. q D. ChakrabarT, Y. Wang, C. Wang, J. Leskovec, and C. Faloutsos. Epidemic thresholds in
real networks. ACM TISSEC, 10(4), 2008. q D. Easley and J. Kleinberg. Networks, Crowds, and Markets: Reasoning About a Highly
Connected World. Cambridge University Press, 2010. q A. Ganesh, L. Massoulie, and D. Towsley. The effect of network topology in spread of
epidemics. IEEE INFOCOM, 2005. q Y. Hayashi, M. Minoura, and J. Matsukubo. Recoverable prevalence in growing scale-‐free
networks and the effecTve immunizaTon. arXiv:cond-‐at/0305549 v2, Aug. 6 2003. q H. W. Hethcote. The mathemaTcs of infecTous diseases. SIAM Review, 42, 2000. q H. W. Hethcote and J. A. Yorke. Gonorrhea transmission dynamics and control. Springer
Lecture Notes in BiomathemaTcs, 46, 1984. q J. O. Kephart and S. R. White. Directed-‐graph epidemiological models of computer
viruses. IEEE Computer Society Symposium on Research in Security and Privacy, 1991. q J. O. Kephart and S. R. White. Measuring and modeling computer virus prevalence. IEEE
Computer Society Symposium on Research in Security and Privacy, 1993. q R. Pastor-‐Santorras and A. Vespignani. Epidemic spreading in scale-‐free networks.
Physical Review LeOers 86, 14, 2001.
q ……… q ……… q ………
All are about either:
• Structured topologies (cliques, block-‐diagonals, hierarchies, random)
• Specific virus propaga@on models
• Sta@c graphs
9 CS 6604:DM Large Networks & Time-‐Series Prakash 2013
Outline
§ Q: What is the epidemic threshold? – Background – Result and Intui@on (Sta@c Graphs) – Proof Ideas (StaTc Graphs) – Bonus: Dynamic Graphs
10 CS 6604:DM Large Networks & Time-‐Series Prakash 2013
How should the answer look like?
§ Answer should depend on: – Graph – Virus PropagaTon Model (VPM)
§ But how?? – Graph – average degree? max. degree? diameter? – VPM – which parameters? – How to combine – linear? quadraTc? exponenTal?
11 ?diameterdavg δβ + ?/)( max
22 ddd avgavg δβ − ….. CS 6604:DM Large Networks & Time-‐Series Prakash 2013
Sta@c Graphs: Main Result [Prakash+,2011]
§ Informally,
§
12
For, Ø any arbitrary topology (adjacency matrix A) Ø any virus propagation model (VPM) in standard literature the epidemic threshold depends only 1. on the λ, first eigenvalue of A, and 2. some constant , determined by
the virus propagation model
λ VPMC
No
epidemic if λ * < 1 VPMCVPMC
CS 6604:DM Large Networks & Time-‐Series Prakash 2013
Our thresholds for some models
§ s = effec4ve strength § s < 1 : below threshold
Models Effec@ve Strength (s)
Threshold (@pping point)
SIS, SIR, SIRS, SEIR s = λ .
s = 1 SIV, SEIV s = λ .
(H.I.V.) s = λ .
⎟⎠
⎞⎜⎝
⎛δβ
( )⎟⎟⎠
⎞⎜⎜⎝
⎛
+θγδβγ
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛
+
+
12
221
vvvε
εββ2121 VVISI
13 CS 6604:DM Large Networks & Time-‐Series Prakash 2013
Our result: Intui@on for λ
“Official” defini@on: § Let A be the adjacency matrix. Then λ is the root with the largest magnitude of the characteristic polynomial of A [det(A – xI)].
§ Doesn’t give much intuiTon!
“Un-‐official” Intui@on J § λ ~ # paths in the graph
14
u
u≈ . kλkA
(i, j) = # of paths i à j of length k kA
CS 6604:DM Large Networks & Time-‐Series Prakash 2013
N nodes
Largest Eigenvalue (λ)
λ ≈ 2 λ = N λ = N-‐1
15 N = 1000
λ ≈ 2 λ= 31.67 λ= 999
beOer connecTvity higher λ
CS 6604:DM Large Networks & Time-‐Series Prakash 2013
Examples: Simula@ons – SIR (mumps)
(a) Infection profile (b) “Take-off” plot
PORTLAND graph 31 million links, 6 million nodes
Frac@o
n of In
fec@on
s
Footprint
Effec@ve Strength Time @cks
16 CS 6604:DM Large Networks & Time-‐Series Prakash 2013
Examples: Simula@ons – SIRS (pertusis)
Frac@o
n of In
fec@on
s
Footprint
Effec@ve Strength Time @cks (a) Infection profile (b) “Take-off” plot
PORTLAND graph 31 million links, 6 million nodes 17 CS 6604:DM Large Networks & Time-‐Series Prakash 2013
Outline
§ Q: What is the epidemic threshold? – Background – Result and IntuiTon (StaTc Graphs) – Proof Ideas (Sta@c Graphs) – Bonus: Dynamic Graphs
18 CS 6604:DM Large Networks & Time-‐Series Prakash 2013
λ * < 1 VPMC
Graph-‐based
Model-‐based
19
Proof Sketch
General VPM structure
Topology and stability
CS 6604:DM Large Networks & Time-‐Series Prakash 2013
Some trivia
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Ø first person in the US idenTfied as a healthy carrier of the pathogen associated with typhoid fever.
Ø infected some 53 people, over the course of her career as a cook!
Ø forcibly quaranTned by public health authoriTes
Prakash 2013 CS 6604:DM Large Networks & Time-‐Series
Two “Infected” States?
21
SymptomaTc
1I 2I
Sneezing
SICR: with a carrier
AsymtomaTc
Prakash 2013 CS 6604:DM Large Networks & Time-‐Series
Ingredient 1: Our generalized model
22
Endogenous Transi@ons
Suscep@ble Infected
Vigilant
Exogenous Transi@ons
Endogenous Transi@ons
Endogenous Transi@ons
Suscep@ble Infected
Vigilant
CS 6604:DM Large Networks & Time-‐Series Prakash 2013
Models and more models Model Used for
SIR Mumps
SIS Flu
SIRS Pertussis
SEIR Chicken-‐pox
……..
SICR Tuberculosis
MSIR Measles
SIV Sensor Stability
H.I.V. ……….
2121 VVISI
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Our generalized model
24
Endogenous Transi@ons
Suscep@ble Infected
Vigilant
CS 6604:DM Large Networks & Time-‐Series Prakash 2013
Special case: SIR
Suscep@ble Infected
Vigilant
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Special case: H.I.V.
2121 VVISI
MulTple InfecTous, Vigilant states
26
“Terminal”
“Non-‐terminal”
CS 6604:DM Large Networks & Time-‐Series Prakash 2013
Ingredient 2: NLDS + Stability
§ View as a NLDS – discrete Tme – non-‐linear dynamical system (NLDS)
Probability vector Specifies the state of the system at Tme t
27
size mN x 1
.
.
.
.
.
size N (number of nodes in the graph)
.
.
.
S
I
V
CS 6604:DM Large Networks & Time-‐Series Prakash 2013
Ingredient 2: NLDS + Stability
§ View as a NLDS – discrete Tme – non-‐linear dynamical system (NLDS)
Non-‐linear func@on Explicitly gives the evoluTon of system
28
size mN x 1
.
.
.
.
.
.
.
. CS 6604:DM Large Networks & Time-‐Series Prakash 2013
Ingredient 2: NLDS + Stability
§ View as a NLDS – discrete Tme – non-‐linear dynamical system (NLDS)
§ Threshold à Stability of NLDS
29 CS 6604:DM Large Networks & Time-‐Series Prakash 2013
= probability that node i is not aOacked by any of its infecTous neighbors
Special case: SIR
size 3N x 1 I
R
S
NLDS
I
R
S
30 CS 6604:DM Large Networks & Time-‐Series Prakash 2013
Fixed Point
1 1 .
0 0 .
0 0 .
State when no node is infected Q: Is it stable?
31 CS 6604:DM Large Networks & Time-‐Series Prakash 2013
Stability for SIR
Stable under threshold
Unstable above threshold
32 CS 6604:DM Large Networks & Time-‐Series Prakash 2013
λ * < 1 VPMC
Graph-‐based
Model-‐based
33
General VPM structure
Topology and stability
See paper for full proof
CS 6604:DM Large Networks & Time-‐Series Prakash 2013
Outline
§ Q: What is the epidemic threshold? – Background – Result and IntuiTon (StaTc Graphs) – Proof Ideas (StaTc Graphs) – Bonus: Dynamic Graphs
34 CS 6604:DM Large Networks & Time-‐Series Prakash 2013
Dynamic Graphs: Epidemic?
adjacency matrix
8
8
Alternating behaviors DAY (e.g., work)
35 CS 6604:DM Large Networks & Time-‐Series Prakash 2013
adjacency matrix
8
8
Dynamic Graphs: Epidemic? Alternating behaviors NIGHT
(e.g., home)
36 CS 6604:DM Large Networks & Time-‐Series Prakash 2013
§ SIS model – recovery rate δ – infecTon rate β
§ Set of T arbitrary graphs
Model Descrip@on
day
N
N night
N
N , weekend…..
Infected
Healthy
XN1
N3
N2
Prob. β Prob. δ
37 CS 6604:DM Large Networks & Time-‐Series Prakash 2013
Obvious result
§ No epidemic if
§ BUT – Too pessimisTc!
Prakash 2013 CS 6604:DM Large Networks & Time-‐Series 38
λmaxβδ
<1
#inf.
Tme
This looks OK
§ Informally, NO epidemic if
eig (S) = < 1
Main result: Dynamic Graphs Threshold [Prakash+, 2010]
Single number! Largest eigenvalue of The system matrix S
39 CS 6604:DM Large Networks & Time-‐Series Prakash 2013
NO epidemic if eig (S) = < 1
S =
cure rate
infec@on rate
……..
adjacency matrix
N
N
day night Infected
Healthy
XN1
N3
N2 Prob. β
Prob. δ
Prakash 2013 CS 6604:DM Large Networks & Time-‐Series 40
Synthe@c MIT Reality Mining
log(frac4on infected)
Time
BELOW
AT
ABOVE ABOVE
AT
BELOW
Infec@on-‐profile
41 CS 6604:DM Large Networks & Time-‐Series Prakash 2013
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