winner-takes-all: competing viruses or ideas on fair-play networks b. aditya prakash, alex beutel,...

Winner-takes-all: Competing Viruses or Ideas

on fair-play Networks

B. Aditya Prakash, Alex Beutel, Roni Rosenfeld, Christos Faloutsos

Carnegie Mellon University, USA

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Competing Contagions

iPhone v Android

Blu-ray v HD-DVD

Biological common flu/avian flu, pneumococcal inf etc

Prakash et. al. 2012


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Outline

•Introduction•Propagation Model•Problem and Result•Simulations•Proof Sketch•Real Examples•Conclusions


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A simple model•Modified flu-like (SIS) •Mutual Immunity (“pick one of the two”)•Susceptible-Infected1-Infected2-

Susceptible

Virus 1

Virus 2


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Who-can-Influence-whom Graph


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Competing Viruses - Attacks

1

1

1


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Competing Viruses - Attacks

1

1

22

All attacks are

Independent

1


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Competing Viruses - Cure

1

Abandons the Android

Abandon the iPhone

2


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Competing Viruses


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Outline



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Single Virus – will it “take-off”?

•Virus dies out if strength below threshold▫For almost any virus model on any graph

[Prakash+ 2011]

for SIS (flu-like): model

Largest Eigenvalue of the adjacency matrix

Constant dependent on

virus model


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Question: What happens in the end?

green: virus 1red: virus 2

Footprint @ Steady State Footprint @ Steady State= ?

Number of Infections

ASSUME: Virus 1 is stronger than Virus 2


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Question: What happens in the end?

green: virus 1red: virus 2


Strength Strength

??=

Strength Strength

2

Footprint @ Steady State

Footprint @ Steady State



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Answer: Winner-Takes-Allgreen: virus 1red: virus 2




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Our Result: Winner-Takes-All

Given our model, and any graph, the weaker virus always dies-out

completely

1. The stronger survives only if it is above threshold

2. Virus 1 is stronger than Virus 2, if: strength(Virus 1) > strength(Virus

2)3. Strength(Virus) = λ β / δ

Details


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Outline



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CLIQUE: BOTH (V1 Weak, V2 Weak)

Time-Plot Phase-Plot



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CLIQUE: MIXED (V1 strong, V2 Weak)




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CLIQUE: ABOVE (V1 strong, V2 strong)




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AS-OREGON (ABOVE V1 strong, V2 strong)

V2 in isolation

15,429 links among 3,995 peersASSUME:

Virus 1 is stronger than Virus 2


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PORTLAND (ABOVE V1 strong, V2 strong)

PORTLAND graph: synthetic population,

31 million links, 6 million nodes


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Outline

•Introduction•Propagation Model•Problem and Result•Simulations•Proof Sketch

▫Clique▫Arbitrary Graph

•Real Examples •Conclusions


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Proof Sketch (clique)

•View as dynamical system


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rate of change in Androids = rate of new additions – rate of people leaving rate of new additions = current Android users X available susceptibles

X transmissability rate people leaving = current Android users X curing rate


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# Androids at time t

# iPhones at time t

Rate of change


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New Victims

Cured


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•Fixed Points

Both die out

One dies out


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•View as dynamical system•Fixed Points•Stability Conditions

▫when is each fixed point stable?

Fixed Point

V1 Weak, V2 Weak

Field lines converge


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V1 strong, V2 strong

Only stable Fixed point


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Formally: when real parts of the eigenvalues of the Jacobian* are negative

*


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………

Fixed Point Condition Comment

Both viruses below threshold

V1 is above threshold and stronger than V2

........Similarly……….

and

and


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Outline

•Introduction•Propagation Model•Problem and Result•Simulations•Proof Sketch

• Clique▫Arbitrary Graph

•Real Examples•Conclusions


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Proof Scheme – general graph


I1

I2

SProbability vector Specifies the state of the system at time t

Details

N

i

probability of i in S…….

…….

size 3N x 1


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Non-linear functionExplicitly gives the evolution of system

Details


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Proof Scheme – general graph•View as dynamical system

•Fixed Points▫only three fixed points▫at least one has to die out at any point

▫Key Constraints: All probabilities have to be non-zero They are spreading on the same graph Used Perron-Frobenius Theorem

Details


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•Fixed Points

•Stability Conditions▫give the precise conditions for each

fixed point to be stable (attracting)▫Utilized Lyapunov Theorem

Details


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Outline



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Real Examples

Blu-Ray v HD-DVD

[Google Search Trends data]


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Real Examples[Google Search Trends data]

Facebook v MySpace


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Real Examples[Google Search Trends data]

Reddit v Digg


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Outline



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Conclusions•Competing Contagions (iPhone vs Android)▫Mutual Immunity▫Flu-like model

•Q: What happens in the end? A: Winner-takes-all

▫On any graph!

•Simulations and Case Studies on real data

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Any Questions?

B. Aditya Prakash http://www.cs.cmu.edu/~badityap

winner-takes-all: competing viruses or ideas on fair-play networks b. aditya prakash, alex beutel,...

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