CS 6604: Data Mining Large Networks and Time- series. B. Aditya Prakash Lecture #9: Epidemics: Compe3ng Viruses

Transcription

1 CS 6604: Data Mining Large Networks and Time- series B. Aditya Prakash Lecture #9: Epidemics: Compe3ng Viruses

2 Compe<ng Contagions iphone v Android Blu- ray v HD- DVD AGack v Retreat CS 6604:DM Large Networks & Time- Series Biological common flu/avian flu, pneumococcal inf Prakash

3 A simple model Modified flu- like Mutual Immunity ( pick one of the two ) Suscep3ble- Infected1- Infected2- Suscep3ble Virus 1 Virus 2 3

4 Who- can- Influence- whom Graph 4

5 Compe<ng Viruses - AGacks β 1 β 1 β 1 5

6 Compe<ng Viruses - AGacks β 1 β 2 β 1 β 2 β 1 All ayacks are Independent 6

7 Compe<ng Viruses - Cure Abandons the Android δ 1 Abandon the iphone δ 2 7

8 Compe<ng Viruses 8

9 Interes<ng ques<ons? Remember the single- virus result 9

10 Ques<on: What happens in the end? Number of Infec3ons green: virus 1 red: virus 2 Steady State Steady State =? ASSUME: 10 Virus 1 is stronger than Virus

11 Ques<on: What happens in the end? Number of Infec3ons green: virus 1 red: virus 2 Steady State Steady State?? = Strength Strength Strength Strength 2 ASSUME: Virus 1 is stronger than Virus 11

12 Answer: Winner- Takes- All Number of Infec3ons green: virus 1 red: virus 2 ASSUME: Virus 1 is stronger than Virus 12

13 Main result: Winner- Takes- All [Prakash ] Given the 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 à same as before! 13

14 CLIQUE: BOTH (V1 Weak, V2 Weak) Time- Plot Phase- Plot ASSUME: Virus 1 is stronger than Virus 2 14

15 CLIQUE: MIXED (V1 strong, V2 Weak) Time- Plot Phase- Plot ASSUME: Virus 1 is stronger than Virus 2 15

16 CLIQUE: ABOVE (V1 strong, V2 strong) Time- Plot Phase- Plot ASSUME: Virus 1 is stronger than Virus 2 16

17 AS- OREGON (BOTH V1 and V2 weak) ASSUME: Virus 1 is stronger than Virus 2 15,429 links among 3,995 peers 17

18 AS- OREGON (MIXED V1 strong, V2 weak) ASSUME: Virus 1 is stronger than Virus 2 15,429 links among 3,995 peers 18

19 AS- OREGON (ABOVE V1 strong, V2 strong) V2 in isola3on ASSUME: Virus 1 is stronger than Virus 2 15,429 links among 3,995 peers 19

20 PORTLAND (ABOVE V1 strong, V2 strong) PORTLAND graph: synthe<c popula<on, 31 million links, 6 million nodes 20

21 Sea 20 Sea 20 ) Reddit vs Digg (Time plot) Real Examples Time (b) Facebook vs Myspace (Time plot) Time (c) Blu-ray vs HD-DVD (Time plot) Final Value 80 [Google Search Trends 60 data] Digg Search Percentage ) Reddit vs Digg (Phase plot) Facebook Search Percentage Final Value MySpace Search Percentage (e) Facebook vs Myspace (Phase plot) BluRay Search Percentage Final Value HD-DVD Search Percentage (f) Blu-ray vs HD-DVD (Phase plot) Blu- Ray v HD- DVD (a-c) Real web-search interest vs time plots for pair of competitors (see Section 5.1 for more deta responding Phase plots. As expected from our WTA result, note that the stronger rival domina er product almost dies-out. Anderson and R. M. May. Infectious Diseases of Dynamical Systems and Linear Algebra. AcademicPres ns. OxfordUniversityPress, iley. The Mathematical Theory of Infectious Diseases [18] R. A. Horn and C. R. Johnson. Topics in Matrix Analy s Applications. Gri n, London, Cambridge University Press, rathi, D. Kempe, and M. Salek. Competitive [19] D. Kempe, J. Kleinberg, and E. Tardos. Maximizing th ce maximization in social networks. WINE, spread of influence through a social network. In KDD, 2 hchandani, D. Hirshleifer, and I. Welch. A theory of [20] J. O. Kephart and S. R. White. Measuring and modelin ashion, custom, and cultural change in informational computer virus prevalence. IEEE Computer Society es. Journal of Political Economy, 100(5): , Symposium on Research in Security and Privacy, er [21] J. Kosta, Y. A. Oswald, and R. Wattenhofer. Word of 21 stillo-chavez, W. Huang, and J. Li. Competitive mouth: Rumor dissemination in social networks. 15 Int

22 Real Examples (a) Reddit vs Digg (Time plot) (b) [Google Search Trends data] Reddit Search Percentage Final Value Digg Search Percentage (d) Reddit vs Digg (Phase plot) Facebook v MySpace (e) Figure 6: (a-c) Real web-search interest v (d-f) Corresponding Phase plots. As exp and weaker product almost dies-out. [3] R. M. Anderson and R. M. May. Infectious D Humans. OxfordUniversityPress,1991. [4] N. Bailey. The Mathematical Theory of Infect and its Applications. Gri n, London, [5] S. Bharathi, D. Kempe, and M. Salek. Compe influence maximization in social networks. WI [6] S. Bikhchandani, D. Hirshleifer, and I. Welch. fads, fashion, custom, and cultural change in i cascades. Journal of Political Economy, 100(5 October [7] C. Castillo-Chavez, W. Huang, and J. Li. Com exclusion in gonorrhea models and other 22 sexu transmitted diseases. SIAM J. Appl. Math, 56

23 Search P Search P Real Examples (a) Reddit vs Digg (Time plot) Time (b) Facebook vs Myspace (Time plot) 0 (c) Blu- 80 [Google Search Trends 60 data] Reddit Search Percentage Final Value Digg Search Percentage (d) Reddit vs Digg (Phase plot) Facebook Search Percentage Reddit v Digg Final Value BluRay Search Percentage MySpace Search Percentage (e) Facebook vs Myspace (Phase plot) (f) Blu-r Figure 6: (a-c) Real web-search interest vs time plots for pair of competitors (s (d-f) Corresponding Phase plots. As expected from our WTA result, note tha and weaker product almost dies-out. [3] R. M. Anderson and R. M. May. Infectious Diseases of Dynamical Systems an Humans. OxfordUniversityPress, [4] N. Bailey. The Mathematical Theory of Infectious Diseases [18] R. A. Horn and C. R. and its Applications. Gri n, London, Cambridge University [5] S. Bharathi, D. Kempe, and M. Salek. Competitive [19] D. Kempe, J. Kleinbe influence maximization in social networks. WINE, spread of influence thr [6] S. Bikhchandani, D. Hirshleifer, and I. Welch. A theory of [20] J. O. Kephart and S. fads, fashion, custom, and cultural change in informational computer virus preval cascades. Journal of Political Economy, 100(5): , Symposium on Resear October [21] J. Kosta, Y. A. Oswal [7] C. Castillo-Chavez, W. Huang, and J. Li. Competitive mouth: Rumor dissem exclusion in gonorrhea models and other sexually Coll. on Struct. Infor Prakash 2015 transmitted CS 6604:DM diseases. Large SIAM Networks J. Appl. & Time- Series Math, 56, [8] C. Castillo-Chavez, W. Huang, and J. Li. Competitive [22] R. Kumar, J. Novak,

24 Proof Sketch (clique) View as dynamical system 24

25 Proof Sketch (clique) View as dynamical system rate of change in Androids = rate of new addi3ons rate of people leaving rate of new addi3ons = current Android users X available suscep3bles X transmissability rate people leaving = current Android users X curing rate 25

26 Proof Sketch (clique) View as dynamical system Rate of change # Androids at <me t # iphones at <me t 26

27 Proof Sketch (clique) View as dynamical system New Vic3ms Cured 27

28 Proof Sketch (clique) View as dynamical system Fixed Points Both die out One dies out 28

29 Proof Sketch (clique) View as dynamical system Fixed Points Stability Condi3ons when is each fixed point stable? V1 Weak, V2 Weak Field lines converge Fixed Point 29

30 Proof Sketch (clique) View as dynamical system Fixed Points Stability Condi3ons when is each fixed point stable? V1 strong, V2 strong Only stable Fixed point 30

31 Proof Sketch (clique) View as dynamical system Fixed Points Stability Condi3ons when is each fixed point stable? Formally: when real parts of the eigenvalues of the Jacobian* are nega3ve * 31

32 Proof Sketch (clique) View as dynamical system Fixed Points Stability Condi3ons Fixed Point Condi<on Comment and Both viruses below threshold and V1 is above threshold and stronger than V2...Similarly. 32

33 What if the virus strengths are equal? Z I1 I 0 1 di 1 1I 1 + Z t dt = Z I2 I 0 2 di 2 2I 2 + Z t dt ) I 1 2 I2 1 (I0 2 ) 1 (I 0 1 ) 2 = e 1 2 ( )t Prakash 2015 Virus popula<on ra<o depends on the ini<al ra<o CS 6604:DM Large Networks & Time- Series 33

34 Proof Scheme general graph View as dynamical system N S I1 I2 Probability vector Specifies the state of the system at 3me t size 3N x 1 probability of i in S.. 34 i

35 Proof Scheme general graph View as dynamical system Fixed Points only three fixed points X X at least one has to die out at any point 1SA P 1 = 1 P 1 (7) 2SA P 2 = 2 P 2 (8) where P 1 =[p 1,1,p 2,1,...,p N,1 ] T, P 2 =[p 1,2,p 2,2,...,p N,2 ] T and S =diag(s i )=I diag( P 1 + P 2 ). 35

36 Proof Scheme general graph Fixed Points only three fixed points at least one has to die out at any point Key Constraints: All probabili3es have to be non- zero They are spreading on the same graph Used Perron- Frobenius Theorem: every non- nega<ve and irreducible matrix has a unique posi<ve eigenvalue and posi<ve eigenvector 36 X X 1SA P 1 = 1 P 1 (7) 2SA P 2 = 2 P 2 (8) where P 1 =[p 1,1,p 2,1,...,p N,1 ] T, P 2 =[p 1,2,p 2,2,...,p N,2 ] T and S =diag(s i )=I diag( P 1 + P 2 ).

37 Proof Scheme general graph Fixed Points only three fixed points at least one has to die out at any point n P 1! 0, P 2! 0o (i.e. the viruses die-out) n P 1! perron eigenvector of SA, P 2! 0o (i.e. only virus 1 survives) n P 2! perron eigenvector of SA, P 1! 0o virus 2 survives) (i.e. only All implicit equa<ons! (we don t know P_1 etc. explicitly) 37

38 Proof Scheme general graph View as dynamical system Fixed Points Stability Condi3ons (bit tricky!) give the precise condi3ons for each fixed point to be stable (ayrac3ng) U<lized Lyapunov Theorem: A matrix C is stable (has real parts of all eigenvalues <0), if C + C has all nega<ve eigenvalues 38

39 Proof Scheme general graph View as dynamical system Fixed Points Stability Condi3ons (bit tricky!) Theorem 3. The corresponding conditions for each of the fixed points to (a) exist, and (b) have stability (i.e. be a hyperbolic and stable attractor) are: 1. 1 < 1 and 2 < 1 (i.e. both are below threshold) 2. 1 > 1 and 1 > 2 (i.e. virus 1 is above threshold and virus 1 strength is greater than virus 2) 3. 2 > 1 and 2 > 1 (i.e. virus 2 is above threshold and virus 2 strength is greater than virus 1) 39

40 REMINDERS 40

41 Project Proposal Project proposal details online (see grading/policies on the web- page) DUE: Oct 13 (hard- copy in class) 3-4 pages in double- column ACM SIG format Latex preferred, but Word is also OK Should contain a sizeable survey: at least 6-8 papers, outside of the required class readings DO NOT copy abstracts (plagiarism) DO summarize, contrast, and iden3fy common threads Penalty for gramma3cal errors and spelling mistakes: DO NOT submit without a spell- check Should be reasonably self- contained Also mail me the PDF, with subject CS6604: Project Proposal 41

42 Heilmeier s Catechism What are you trying to do? Ar3culate your objec3ves using absolutely no jargon. How is it done today, and what are the limits of current prac3ce? What's new in your approach and why do you think it will be successful? Who cares? If you're successful, what difference will it make? What are the risks and the payoffs? How much will it cost? How long will it take? What are the midterm and final "exams" to check for success? George Heilmeier (Ex- DARPA director) Prakash 2015 CS 6604:DM Large Networks & Time- Series 42

43 Proposal should contain: What is the problem you are solving? Give the formal problem defini<on (in addi3on to a lay- person version). Which algorithms/techniques/models you plan to use/ develop? Be as specific as you can! How will you evaluate your method? How will you test it? How will you measure success? What data will you use (how will you get it)? Give data specifics (eg. size, format, etc.). What do you expect to accomplish by the end of the semester? (eg. novel algorithm, parallel implementa3on, etc.) You must describe what por3on of the project each team member will be expected to do. Include an expected <me- line of ac<vi<es. 43

44 PREDATOR- PREY MODEL 44

45 Lotka- Volterra Model 1910 Predator- Prey Model The prey popula3on finds ample food at all 3mes The food supply of the predator popula3on depends en3rely on the prey popula3ons The rate of change of popula3on is propor3onal to its size During the process, the environment does not change in favor of one species Different than the SI1I2S model 45

46 Lotka- Volterra Model Prey ( x ): dx dt = x xy Predator ( y ): dy dt = xy y Prakash 2015 CS 6604:DM Large Networks & Time- Series 46

47 Fixed Points Ex3nc3on x = y = 0 Turns out to be unstable! Co- occurrence x = γ/δ y = α/β Non- hyperbolic fixed point Turns out the levels of the predator and prey popula3ons cycle, and oscillate around this fixed point. 47

48 L- V Model 48

49 INTERACTING VIRUSES 49

50 A simple model: SI 1 2 S Modified flu- like (SIS) Suscep3ble- Infected 1 or 2 - Suscep3ble Interac3on Factor ε Full Mutual Immunity: ε = 0 Par3al Mutual Immunity (compe33on): ε < 0 Coopera3on: ε > 0 I 1,2 & I 1 I 2 Virus Virus Prakash 2015 CS 6604:DM Large S Networks & Time- Series 50

51 Ques<on: What happens in the end? ε = 0 Winner takes all ε = 1 Co- exist independently ε = 2 Viruses cooperate Footprint (Fraction of Population) κ κ Time Footprint (Fraction of Population) 0.2 κ 1 κ i 1, Time Footprint (Fraction of Population) Time κ 1 κ 2 i 1,2 What about for 0 < ε <1? Is there a point at which both viruses can ASSUME: co- exist? CS 6604:DM Large Networks & Time- Series Virus 1 is stronger than Virus 2 Prakash

52 Answer: Yes! There is a phase transi<on 0.9 Footprint (Fraction of Population) κ 1 (Simulation) κ 2 (Simulation) i 1,2 (Simulation) κ 1 (Theory) κ 2 (Theory) i 1,2 (Theory) ε critical Interaction Factor (ε) Footprint (Fraction of Population) Time κ 1 κ 2 ASSUME: Virus 1 is stronger than Virus 2 52

53 Answer: Yes! There is a phase transi<on 0.9 Footprint (Fraction of Population) κ 1 (Simulation) κ 2 (Simulation) i 1,2 (Simulation) κ 1 (Theory) κ 2 (Theory) i 1,2 (Theory) ε critical Interaction Factor (ε) Footprint (Fraction of Population) Time κ 1 κ 2 i 1,2 ASSUME: Virus 1 is stronger than Virus 2 53

54 Answer: Yes! There is a phase transi<on 0.9 Footprint (Fraction of Population) κ 1 (Simulation) κ 2 (Simulation) i 1,2 (Simulation) κ 1 (Theory) κ 2 (Theory) i 1,2 (Theory) ε critical Interaction Factor (ε) Footprint (Fraction of Population) Time κ 1 κ 2 i 1,2 ASSUME: Virus 1 is stronger than Virus 2 54

55 Main Result: Viruses can Co- exist [Beutel+, 2012] Given the model and a fully connected graph, there exists an ε cri3cal such that for ε ε cri3cal, there is a fixed point where both viruses survive. 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) σ = N β / δ 55

56 Real Examples [Google Search Trends data] Search Quantity Time κ 1 κ 2 Hulu Blockbuster Hulu v Blockbuster 56

57 Real Examples [Google Search Trends data] Search Quantity Time κ 1 κ 2 Firefox Chrome Chrome v Firefox 57

58 Extensions To composite networks as well. Virus Propaga3on in Mul3ple Profile Networks. KDD

59 Empirical models [Myers+, 2012] Goal: To extract the compe33on and co- opera3on Specific Se ng: You are reading posts on TwiYer P(post X exposed to X, Y 1, Y 2, Y 3 ) =? Prakash 2015 CS 6604:DM Large Networks & Time- Series 59

60 The Model Assume viruses are independent: Too many parameters! (K. w 2 ) 60

61 Approach: Assume more structure First: First, assume: Prior infection prob. Interaction term (still has w 2 entries!) Second: feature- rize the contagions with topics (M is : how much contagion i belongs to s) Prakash 2015 CS 6604:DM Large Networks & Time- Series 61

62 Full Model And then: Prakash 2015 CS 6604:DM Large Networks & Time- Series 62

63 Experiments: Interac<ons TwiYer data: URLs tweets Predict if user will post URL X (Train/Test = 90/10) Their model performs beyer than baselines 63

64 Experiments: Interac<ons How P(post u2 exp u1) changes u1 is highly viral? u1 and u2 are similar/different in content? A: If u1 is not viral this boosts u2. Otherwise kills u2 if they are dissimilar. Else u1 helps u2. I 1,2 Recall: SI1I2S I 1 I Relative change in infection prob. S Prakash 2015 CS 6604:DM Large Networks & Time- Series 64