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

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

3. Outline Introduction Propagation Model Problem and Result Simulations Proof Sketch Real Examples Conclusions Prakash et. al. 2012 3

4. 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. 2012 4

5. Who-can-Influence-whom Graph Prakash et. al. 2012 5

6. Competing Viruses - Attacks Prakash et. al. 2012 6

7. Competing Viruses - Attacks Prakash et. al. 2012 7 All attacks are Independent

8. Competing Viruses - Cure Prakash et. al. 2012 8 Abandons the Android Abandon the iPhone

9. Competing Viruses Prakash et. al. 2012 9

10. Outline Introduction Propagation Model Problem and Result Simulations Proof Sketch Real Examples Conclusions Prakash et. al. 2012 10

11. 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. 2012 11 for SIS (flu-like): model Largest Eigenvalue of the adjacency matrix Constant dependent on virus model

12. Question: What happens in the end? green: virus 1 red: virus 2 Footprint @ Steady State = ? Number of Infections Prakash et. al. 2012 12 ASSUME: Virus 1 is stronger than Virus 2

13. Question: What happens in the end? green: virus 1 red: virus 2 Number of Infections Strength ?? = Strength 2 Footprint @ Steady State Prakash et. al. 2012 13 ASSUME: Virus 1 is stronger than Virus 2

14. Answer: Winner-Takes-All green: virus 1 red: virus 2 ASSUME: Virus 1 is stronger than Virus 2 Number of Infections Prakash et. al. 2012 14

15. 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. 2012 15

16. Outline Introduction Propagation Model Problem and Result Simulations Proof Sketch Real Examples Conclusions Prakash et. al. 2012 16

17. CLIQUE: BOTH (V1 Weak, V2 Weak) Prakash et. al. 2012 17 Time-PlotPhase-Plot ASSUME: Virus 1 is stronger than Virus 2

18. CLIQUE: MIXED (V1 strong, V2 Weak) Prakash et. al. 2012 18 Time-PlotPhase-Plot ASSUME: Virus 1 is stronger than Virus 2

19. CLIQUE: ABOVE (V1 strong, V2 strong) Prakash et. al. 2012 19 Time-PlotPhase-Plot ASSUME: Virus 1 is stronger than Virus 2

20. AS-OREGON (BOTH V1 and V2 weak) Prakash et. al. 2012 20 15,429 links among 3,995 peers ASSUME: Virus 1 is stronger than Virus 2

21. AS-OREGON (MIXED V1 strong, V2 weak) Prakash et. al. 2012 21 15,429 links among 3,995 peers ASSUME: Virus 1 is stronger than Virus 2

22. AS-OREGON (ABOVE V1 strong, V2 strong) Prakash et. al. 2012 22 V2 in isolation 15,429 links among 3,995 peers ASSUME: Virus 1 is stronger than Virus 2

23. PORTLAND (ABOVE V1 strong, V2 strong) Prakash et. al. 2012 23 PORTLAND graph: synthetic population, 31 million links, 6 million nodes

24. Outline Introduction Propagation Model Problem and Result Simulations Proof Sketch ▫Clique ▫Arbitrary Graph Real Examples Conclusions Prakash et. al. 2012 24

25. Proof Sketch (clique) View as dynamical system Prakash et. al. 2012 25

26. Proof Sketch (clique) View as dynamical system Prakash et. al. 2012 26 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

27. Proof Sketch (clique) View as dynamical system Prakash et. al. 2012 27 # Androids at time t # iPhones at time t Rate of change

28. Proof Sketch (clique) View as dynamical system Prakash et. al. 2012 28 New VictimsCured

29. Proof Sketch (clique) View as dynamical system Fixed Points Prakash et. al. 2012 29 Both die out One dies out

30. Proof Sketch (clique) View as dynamical system Fixed Points Stability Conditions ▫when is each fixed point stable? Prakash et. al. 2012 30 Fixed Point V1 Weak, V2 Weak Field lines converge

31. Proof Sketch (clique) View as dynamical system Fixed Points Stability Conditions ▫when is each fixed point stable? Prakash et. al. 2012 31 V1 strong, V2 strong Only stable Fixed point

32. 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. 2012 32 *

33. Proof Sketch (clique) View as dynamical system Fixed Points Stability Conditions Prakash et. al. 2012 33 ……… Fixed PointConditionComment Both viruses below threshold V1 is above threshold and stronger than V2........Similarly………. and

34. Outline Introduction Propagation Model Problem and Result Simulations Proof Sketch Clique ▫Arbitrary Graph Real Examples Conclusions Prakash et. al. 2012 34

35. Proof Scheme – general graph View as dynamical system Prakash et. al. 2012 35 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

36. Proof Scheme – general graph View as dynamical system Prakash et. al. 2012 36 Non-linear function Explicitly gives the evolution of system

37. 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. 2012 37

38. 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. 2012 38

39. Outline Introduction Propagation Model Problem and Result Simulations Proof Sketch Real Examples Conclusions Prakash et. al. 2012 39

40. Real Examples Blu-Ray v HD-DVD [Google Search Trends data] Prakash et. al. 2012 40

41. Real Examples [Google Search Trends data] Prakash et. al. 2012 41 Facebook v MySpace

42. Real Examples [Google Search Trends data] Prakash et. al. 2012 42 Reddit v Digg

43. Outline Introduction Propagation Model Problem and Result Simulations Proof Sketch Real Examples Conclusions Prakash et. al. 2012 43

44. 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. 2012 44

45. Any Questions? B. Aditya Prakash http://www.cs.cmu.edu/~badityap 45