Winner-takes-all: Competing Viruses or Ideas on fair-play Networks B. Aditya Prakash, Alex Beutel, Roni Rosenfeld, Christos Faloutsos Carnegie Mellon University, USA

Competing Contagions iPhone v AndroidBlu-ray v HD-DVD Biological common flu/avian flu, pneumococcal inf etc 2 Prakash et. al. 2012

Outline Introduction Propagation Model Problem and Result Simulations Proof Sketch Real Examples Conclusions Prakash et. al

A simple model Modified flu-like (SIS) Mutual Immunity (“pick one of the two”) Susceptible-Infected1-Infected2-Susceptible Virus 1 Virus 2 Prakash et. al

Who-can-Influence-whom Graph Prakash et. al

Competing Viruses - Attacks Prakash et. al

Competing Viruses - Attacks Prakash et. al All attacks are Independent

Competing Viruses - Cure Prakash et. al Abandons the Android Abandon the iPhone

Competing Viruses Prakash et. al

Outline Introduction Propagation Model Problem and Result Simulations Proof Sketch Real Examples Conclusions Prakash et. al

Single Virus – will it “take-off”? Virus dies out if strength below threshold ▫For almost any virus model on any graph [Prakash+ 2011] Prakash et. al for SIS (flu-like): model Largest Eigenvalue of the adjacency matrix Constant dependent on virus model

Question: What happens in the end? green: virus 1 red: virus 2 Steady State = ? Number of Infections Prakash et. al ASSUME: Virus 1 is stronger than Virus 2

Question: What happens in the end? green: virus 1 red: virus 2 Number of Infections Strength ?? = Strength 2 Steady State Prakash et. al ASSUME: Virus 1 is stronger than Virus 2

Answer: Winner-Takes-All green: virus 1 red: virus 2 ASSUME: Virus 1 is stronger than Virus 2 Number of Infections Prakash et. al

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 Prakash et. al

Outline Introduction Propagation Model Problem and Result Simulations Proof Sketch Real Examples Conclusions Prakash et. al

CLIQUE: BOTH (V1 Weak, V2 Weak) Prakash et. al Time-PlotPhase-Plot ASSUME: Virus 1 is stronger than Virus 2

CLIQUE: MIXED (V1 strong, V2 Weak) Prakash et. al Time-PlotPhase-Plot ASSUME: Virus 1 is stronger than Virus 2

CLIQUE: ABOVE (V1 strong, V2 strong) Prakash et. al Time-PlotPhase-Plot ASSUME: Virus 1 is stronger than Virus 2

AS-OREGON (BOTH V1 and V2 weak) Prakash et. al ,429 links among 3,995 peers ASSUME: Virus 1 is stronger than Virus 2

AS-OREGON (MIXED V1 strong, V2 weak) Prakash et. al ,429 links among 3,995 peers ASSUME: Virus 1 is stronger than Virus 2

AS-OREGON (ABOVE V1 strong, V2 strong) Prakash et. al V2 in isolation 15,429 links among 3,995 peers ASSUME: Virus 1 is stronger than Virus 2

PORTLAND (ABOVE V1 strong, V2 strong) Prakash et. al PORTLAND graph: synthetic population, 31 million links, 6 million nodes

Outline Introduction Propagation Model Problem and Result Simulations Proof Sketch ▫Clique ▫Arbitrary Graph Real Examples Conclusions Prakash et. al

Proof Sketch (clique) View as dynamical system Prakash et. al

Proof Sketch (clique) View as dynamical system Prakash et. al 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

Proof Sketch (clique) View as dynamical system Prakash et. al # Androids at time t # iPhones at time t Rate of change

Proof Sketch (clique) View as dynamical system Prakash et. al New VictimsCured

Proof Sketch (clique) View as dynamical system Fixed Points Prakash et. al Both die out One dies out

Proof Sketch (clique) View as dynamical system Fixed Points Stability Conditions ▫when is each fixed point stable? Prakash et. al Fixed Point V1 Weak, V2 Weak Field lines converge

Proof Sketch (clique) View as dynamical system Fixed Points Stability Conditions ▫when is each fixed point stable? Prakash et. al V1 strong, V2 strong Only stable Fixed point

Proof Sketch (clique) View as dynamical system Fixed Points Stability Conditions ▫when is each fixed point stable? Formally: when real parts of the eigenvalues of the Jacobian* are negative Prakash et. al *

Proof Sketch (clique) View as dynamical system Fixed Points Stability Conditions Prakash et. al ……… Fixed PointConditionComment Both viruses below threshold V1 is above threshold and stronger than V Similarly………. and

Outline Introduction Propagation Model Problem and Result Simulations Proof Sketch Clique ▫Arbitrary Graph Real Examples Conclusions Prakash et. al

Proof Scheme – general graph View as dynamical system Prakash et. al I1 I2 S S Probability vector Specifies the state of the system at time t N i probability of i in S ……. size 3N x 1

Proof Scheme – general graph View as dynamical system Prakash et. al Non-linear function Explicitly gives the evolution of system

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 Prakash et. al

Proof Scheme – general graph View as dynamical system Fixed Points Stability Conditions ▫give the precise conditions for each fixed point to be stable (attracting) ▫Utilized Lyapunov Theorem Prakash et. al

Outline Introduction Propagation Model Problem and Result Simulations Proof Sketch Real Examples Conclusions Prakash et. al

Real Examples Blu-Ray v HD-DVD [Google Search Trends data] Prakash et. al

Real Examples [Google Search Trends data] Prakash et. al Facebook v MySpace

Real Examples [Google Search Trends data] Prakash et. al Reddit v Digg

Outline Introduction Propagation Model Problem and Result Simulations Proof Sketch Real Examples Conclusions Prakash et. al

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 Prakash et. al

Any Questions? B. Aditya Prakash 45