CS#6604:#Data#Mining#Large# …people.cs.vt.edu/badityap/classes/cs6604-Fall13/lectures/lecture-8.pdf · Lecture’Notes’in’Biomathemacs,’46,’1984

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

example (sta@c graph) Weak Virus

Epidemic?


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


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 …..


Outline

§  Q: What is the epidemic threshold? – Background – Result and IntuiTon (StaTc Graphs) – Proof Ideas (StaTc Graphs) – Bonus: Dynamic Graphs


“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


Outline

§  Q: What is the epidemic threshold? – Background – Result and Intui@on (Sta@c Graphs) – Proof Ideas (StaTc Graphs) – Bonus: Dynamic Graphs


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


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


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


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


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


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


λ * < 1 VPMC

Graph-‐based

Model-‐based

19

Proof Sketch

General VPM structure

Topology and stability


Some trivia

20

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



Suscep@ble Infected

Vigilant


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


Our generalized model

24


Suscep@ble Infected

Vigilant


Special case: SIR

Suscep@ble Infected

Vigilant


Special case: H.I.V.

2121 VVISI

MulTple InfecTous, Vigilant states

26

“Terminal”

“Non-‐terminal”


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




Non-‐linear func@on Explicitly gives the evoluTon of system

28

size mN x 1

.

.

.

.

.

.

.

. CS 6604:DM Large Networks & Time-‐Series Prakash 2013



§  Threshold à Stability of NLDS


= 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


Fixed Point

1 1 .

0 0 .

0 0 .

State when no node is infected Q: Is it stable?


Stability for SIR

Stable under threshold

Unstable above threshold


λ * < 1 VPMC

Graph-‐based

Model-‐based

33

General VPM structure

Topology and stability

See paper for full proof


Outline

§  Q: What is the epidemic threshold? – Background – Result and IntuiTon (StaTc Graphs) – Proof Ideas (StaTc Graphs) – Bonus: Dynamic Graphs


Dynamic Graphs: Epidemic?

adjacency matrix

8

8

Alternating behaviors DAY (e.g., work)


adjacency matrix

8

8

Dynamic Graphs: Epidemic? Alternating behaviors NIGHT

(e.g., home)


§  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


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


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


“Take-‐off” plots Footprint (# infected @ “steady state”)

Our threshold

Our threshold

(log scale)

NO EPIDEMIC

EPIDEMIC

EPIDEMIC

NO EPIDEMIC

Synthe@c MIT Reality


Documents

CS#6604:#Data#Mining#Large# …people.cs.vt.edu/badityap/classes/cs6604-Fall13/lectures/lecture-8.pdf · Lecture’Notes’in’Biomathemacs,’46,’1984