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Influence propagation in large graphs - theorems and algorithms B. Aditya Prakash http://www.cs.cmu.edu/~badityap Christos Faloutsos http://www.cs.cmu.edu/~christos Carnegie Mellon University AUTH12 Slide 2 Networks are everywhere! Human Disease Network [Barabasi 2007] Gene Regulatory Network [Decourty 2008] Facebook Network [2010] The Internet [2005] Prakash and Faloutsos 20122AUTH '12 Slide 3 Dynamical Processes over networks are also everywhere! Prakash and Faloutsos 20123AUTH '12 Slide 4 Why do we care? Information Diffusion Viral Marketing Epidemiology and Public Health Cyber Security Human mobility Games and Virtual Worlds Ecology Social Collaboration........ Prakash and Faloutsos 20124AUTH '12 Slide 5 Why do we care? (1: Epidemiology) Dynamical Processes over networks [AJPH 2007] CDC data: Visualization of the first 35 tuberculosis (TB) patients and their 1039 contacts Diseases over contact networks Prakash and Faloutsos 20125AUTH '12 Slide 6 Why do we care? (1: Epidemiology) Dynamical Processes over networks Each circle is a hospital ~3000 hospitals More than 30,000 patients transferred [US-MEDICARE NETWORK 2005] Problem: Given k units of disinfectant, whom to immunize? Prakash and Faloutsos 20126AUTH '12 Slide 7 Why do we care? (1: Epidemiology) CURRENT PRACTICEOUR METHOD ~6x fewer! [US-MEDICARE NETWORK 2005] Prakash and Faloutsos 20127AUTH '12 Hospital-acquired inf. took 99K+ lives, cost $5B+ (all per year) Slide 8 Why do we care? (2: Online Diffusion) > 800m users, ~$1B revenue [WSJ 2010] ~100m active users > 50m users Prakash and Faloutsos 20128AUTH '12 Slide 9 Why do we care? (2: Online Diffusion) Dynamical Processes over networks Celebrity Buy Versace! Followers Social Media Marketing Prakash and Faloutsos 20129AUTH '12 Slide 10 High Impact Multiple Settings Q. How to squash rumors faster? Q. How do opinions spread? Q. How to market better? epidemic out-breaks products/viruses transmit s/w patches Prakash and Faloutsos 201210AUTH '12 Slide 11 Research Theme DATA Large real-world networks & processes ANALYSIS Understanding POLICY/ ACTION Managing Prakash and Faloutsos 201211AUTH '12 Slide 12 In this talk ANALYSIS Understanding Given propagation models: Q1: Will an epidemic happen? Prakash and Faloutsos 201212AUTH '12 Slide 13 In this talk Q2: How to immunize and control out-breaks better? POLICY/ ACTION Managing Prakash and Faloutsos 201213AUTH '12 Slide 14 Outline Motivation Epidemics: what happens? (Theory) Action: Who to immunize? (Algorithms) Prakash and Faloutsos 201214AUTH '12 Slide 15 A fundamental question Strong Virus Epidemic? Prakash and Faloutsos 201215AUTH '12 Slide 16 example (static graph) Weak Virus Epidemic? Prakash and Faloutsos 201216AUTH '12 Slide 17 Problem Statement Find, a condition under which virus will die out exponentially quickly regardless of initial infection condition above (epidemic) below (extinction) # Infected time Separate the regimes? Prakash and Faloutsos 201217AUTH '12 Slide 18 Threshold (static version) Problem Statement Given: Graph G, and Virus specs (attack prob. etc.) Find: A condition for virus extinction/invasion Prakash and Faloutsos 201218AUTH '12 Slide 19 Threshold: Why important? Accelerating simulations Forecasting (What-if scenarios) Design of contagion and/or topology A great handle to manipulate the spreading Immunization Maximize collaboration .. Prakash and Faloutsos 201219AUTH '12 Slide 20 Outline Motivation Epidemics: what happens? (Theory) Background Result (Static Graphs) Proof Ideas (Static Graphs) Bonus 1: Dynamic Graphs Bonus 2: Competing Viruses Action: Who to immunize? (Algorithms) Prakash and Faloutsos 201220AUTH '12 Slide 21 SIR model: life immunity (mumps) Each node in the graph is in one of three states Susceptible (i.e. healthy) Infected Removed (i.e. cant get infected again) Prob. Prob. t = 1t = 2t = 3 Prakash and Faloutsos 201221AUTH '12 Slide 22 Terminology: continued Other virus propagation models (VPM) SIS : susceptible-infected-susceptible, flu-like SIRS : temporary immunity, like pertussis SEIR : mumps-like, with virus incubation (E = Exposed) .. Underlying contact-network who-can-infect- whom Prakash and Faloutsos 201222AUTH '12 Slide 23 Related Work R. M. Anderson and R. M. May. Infectious Diseases of Humans. Oxford University Press, 1991. A. Barrat, M. Barthlemy, and A. Vespignani. Dynamical Processes on Complex Networks. Cambridge University Press, 2010. F. M. Bass. A new product growth for model consumer durables. Management Science, 15(5):215227, 1969. D. Chakrabarti, Y. Wang, C. Wang, J. Leskovec, and C. Faloutsos. Epidemic thresholds in real networks. ACM TISSEC, 10(4), 2008. D. Easley and J. Kleinberg. Networks, Crowds, and Markets: Reasoning About a Highly Connected World. Cambridge University Press, 2010. A. Ganesh, L. Massoulie, and D. Towsley. The effect of network topology in spread of epidemics. IEEE INFOCOM, 2005. Y. Hayashi, M. Minoura, and J. Matsukubo. Recoverable prevalence in growing scale-free networks and the effective immunization. arXiv:cond-at/0305549 v2, Aug. 6 2003. H. W. Hethcote. The mathematics of infectious diseases. SIAM Review, 42, 2000. H. W. Hethcote and J. A. Yorke. Gonorrhea transmission dynamics and control. Springer Lecture Notes in Biomathematics, 46, 1984. 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. J. O. Kephart and S. R. White. Measuring and modeling computer virus prevalence. IEEE Computer Society Symposium on Research in Security and Privacy, 1993. R. Pastor-Santorras and A. Vespignani. Epidemic spreading in scale-free networks. Physical Review Letters 86, 14, 2001. All are about either: Structured topologies (cliques, block-diagonals, hierarchies, random) Specific virus propagation models Static graphs All are about either: Structured topologies (cliques, block-diagonals, hierarchies, random) Specific virus propagation models Static graphs Prakash and Faloutsos 201223AUTH '12 Slide 24 Outline Motivation Epidemics: what happens? (Theory) Background Result (Static Graphs) Proof Ideas (Static Graphs) Bonus 1: Dynamic Graphs Bonus 2: Competing Viruses Action: Who to immunize? (Algorithms) Prakash and Faloutsos 201224AUTH '12 Slide 25 How should the answer look like? Answer should depend on: Graph Virus Propagation Model (VPM) But how?? Graph average degree? max. degree? diameter? VPM which parameters? How to combine linear? quadratic? exponential? .. Prakash and Faloutsos 201225AUTH '12 Slide 26 Static Graphs: Our Main Result Informally, For, any arbitrary topology (adjacency matrix A) any virus propagation model (VPM) in standard literature the epidemic threshold depends only 1.on the , rst eigenvalue of A, and 2.some constant, determined by the virus propagation model No epidemic if * < 1 Prakash and Faloutsos 201226AUTH '12 In Prakash+ ICDM 2011 (Selected among best papers). w/ Deepay Chakrabarti Slide 27 Our thresholds for some models s = effective strength s < 1 : below threshold Models Effective Strength (s) Threshold (tipping point) SIS, SIR, SIRS, SEIR s = . s = 1 SIV, SEIV s = . ( H.I.V. ) s = . Prakash and Faloutsos 201227AUTH '12 Slide 28 Our result: Intuition for Official definition: Let A be the adjacency matrix. Then is the root with the largest magnitude of the characteristic polynomial of A [det(A xI)]. Doesnt give much intuition! Un-official Intuition ~ # paths in the graph u u . (i, j) = # of paths i j of length k Prakash and Faloutsos 201228AUTH '12 Slide 29 Largest Eigenvalue () 2 = N = N-1 N = 1000 2= 31.67= 999 better connectivity higher Prakash and Faloutsos 201229AUTH '12 N nodes Slide 30 Examples: Simulations SIR (mumps) (a) Infection profile (b) Take-off plot PORTLAND graph: synthetic population, 31 million links, 6 million nodes Fraction of Infections Footprint Effective Strength Time ticks Prakash and Faloutsos 201230AUTH '12 Slide 31 Examples: Simulations SIRS (pertusis) Fraction of Infections Footprint Effective StrengthTime ticks (a) Infection profile (b) Take-off plot PORTLAND graph: synthetic population, 31 million links, 6 million nodes Prakash and Faloutsos 201231AUTH '12 Slide 32 Outline Motivation Epidemics: what happens? (Theory) Background Result (Static Graphs) Proof Ideas (Static Graphs) Bonus 1: Dynamic Graphs Bonus 2: Competing Viruses Action: Who to immunize? (Algorithms) Prakash and Faloutsos 201232AUTH '12 Slide 33 * < 1 Graph-based Model-based General VPM structure Topology and stability See paper for full proof 33AUTH '12Prakash and Faloutsos 2012 Slide 34 Outline Motivation Epidemics: what happens? (Theory) Background Result (Static Graphs) Proof Ideas (Static Graphs) Bonus 1: Dynamic Graphs Bonus 2: Competing Viruses Action: Who to immunize? (Algorithms) Prakash and Faloutsos 201234AUTH '12 Slide 35 Dynamic Graphs: Epidemic? adjacency matrix 8 8 Alternating behaviors DAY (e.g., work) Prakash and Faloutsos 201235AUTH '12 Slide 36 adjacency matrix 8 8 Dynamic Graphs: Epidemic? Alternating behaviors NIGHT (e.g., home) Prakash and Faloutsos 201236AUTH '12 Slide 37 SIS model recovery rate infection rate Set of T arbitrary graphs Model Description day N N night N N, weekend.. Infected Healthy XN1 N3 N2 Prob. Prob. Prakash and Faloutsos 201237AUTH '12 Slide 38 Informally, NO epidemic if eig (S) = < 1 Our result: Dynamic Graphs Threshold Single number! Largest eigenvalue of The system matrix S In Prakash+, ECML-PKDD 2010 S = Prakash and Faloutsos 201238AUTH '12 Slide 39 Synthetic MIT Reality Mining log(fraction infected) Time BELOW AT ABOVE AT BELOW Infection-profile Prakash and Faloutsos 201239AUTH '12 Slide 40 Take-off plots Footprint (# infected @ steady state) Our threshold (log scale) NO EPIDEMIC EPIDEMIC NO EPIDEMIC SyntheticMIT Reality Prakash and Faloutsos 201240AUTH '12 Slide 41 Outline Motivation Epidemics: what happens? (Theory) Background Result (Static Graphs) Proof Ideas (Static Graphs) Bonus 1: Dynamic Graphs Bonus 2: Competing Viruses Action: Who to immunize? (Algorithms) Prakash and Faloutsos 201241AUTH '12 Slide 42 Competing Contagions iPhone v AndroidBlu-ray v HD-DVD 42AUTH '12Prakash and Faloutsos 2012 Biological common flu/avian flu, pneumococcal inf etc Slide 43 A simple model Modified flu-like Mutual Immunity (pick one of the two) Susceptible-Infected1-Infected2-Susceptible Virus 1 Virus 2 Prakash and Faloutsos 201243AUTH '12 Slide 44 Question: What happens in the end? green: virus 1 red: virus 2 Footprint @ Steady State = ? Number of Infections Prakash and Faloutsos 201244AUTH '12 ASSUME: Virus 1 is stronger than Virus 2 Slide 45 Question: What happens in the end? green: virus 1 red: virus 2 Number of Infections Strength ?? = Strength 2 Footprint @ Steady State 45AUTH '12Prakash and Faloutsos 2012 ASSUME: Virus 1 is stronger than Virus 2 Slide 46 Answer: Winner-Takes-All green: virus 1 red: virus 2 Number of Infections 46AUTH '12Prakash and Faloutsos 2012 ASSUME: Virus 1 is stronger than Virus 2 Slide 47 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) = / same as before! 47AUTH '12Prakash and Faloutsos 2012 In Prakash+ WWW 2012 Slide 48 Real Examples Reddit v DiggBlu-Ray v HD-DVD [Google Search Trends data] 48AUTH '12Prakash and Faloutsos 2012 Slide 49 Outline Motivation Epidemics: what happens? (Theory) Action: Who to immunize? (Algorithms) Prakash and Faloutsos 201249AUTH '12 Slide 50 ? ? Given: a graph A, virus prop. model and budget k; Find: k best nodes for immunization (removal). k = 2 ? ? Full Static Immunization Prakash and Faloutsos 201250AUTH '12 Slide 51 Outline Motivation Epidemics: what happens? (Theory) Action: Who to immunize? (Algorithms) Full Immunization (Static Graphs) Fractional Immunization Prakash and Faloutsos 201251AUTH '12 Slide 52 Challenges Given a graph A, budget k, Q1 (Metric) How to measure the shield- value for a set of nodes (S)? Q2 (Algorithm) How to find a set of k nodes with highest shield-value? Prakash and Faloutsos 201252AUTH '12 Slide 53 Proposed vulnerability measure Increasing Increasing vulnerability is the epidemic threshold SafeVulnerableDeadly Prakash and Faloutsos 201253AUTH '12 Slide 54 1 9 10 3 4 5 7 8 6 2 9 1 11 10 3 4 5 6 7 8 2 9 Original GraphWithout {2, 6} Eigen-Drop(S) = - s Eigen-Drop(S) = - s A1: Eigen-Drop: an ideal shield value Prakash and Faloutsos 201254AUTH '12 Slide 55 (Q2) - Direct Algorithm too expensive! Immunize k nodes which maximize S = argmax Combinatorial! Complexity: Example: 1,000 nodes, with 10,000 edges It takes 0.01 seconds to compute It takes 2,615 years to find 5-best nodes ! Prakash and Faloutsos 201255AUTH '12 Slide 56 A2: Our Solution Part 1: Shield Value Carefully approximate Eigen-drop ( ) Matrix perturbation theory Part 2: Algorithm Greedily pick best node at each step Near-optimal due to submodularity NetShield (linear complexity) O(nk 2 +m) n = # nodes; m = # edges Prakash and Faloutsos 201256AUTH '12 In Tong, Prakash+ ICDM 2010 Slide 57 Experiment: Immunization quality Log(fraction of infected nodes) NetShield Degree PageRank Eigs (=HITS) Acquaintance Betweeness (shortest path) Lower is better Time Prakash and Faloutsos 201257AUTH '12 Slide 58 Outline Motivation Epidemics: what happens? (Theory) Action: Who to immunize? (Algorithms) Full Immunization (Static Graphs) Fractional Immunization Prakash and Faloutsos 201258AUTH '12 Slide 59 Fractional Immunization of Networks B. Aditya Prakash, Lada Adamic, Theodore Iwashyna (M.D.), Hanghang Tong, Christos Faloutsos Under review Prakash and Faloutsos 201259AUTH '12 Slide 60 Fractional Asymmetric Immunization Hospital Another Hospital Drug-resistant Bacteria (like XDR-TB) Prakash and Faloutsos 201260AUTH '12 Slide 61 Fractional Asymmetric Immunization Hospital Another Hospital Drug-resistant Bacteria (like XDR-TB) Prakash and Faloutsos 201261AUTH '12 Slide 62 Fractional Asymmetric Immunization Hospital Another Hospital Problem: Given k units of disinfectant, how to distribute them to maximize hospitals saved? Prakash and Faloutsos 201262AUTH '12 Slide 63 Our Algorithm SMART- ALLOC CURRENT PRACTICESMART-ALLOC [US-MEDICARE NETWORK 2005] Each circle is a hospital, ~3000 hospitals More than 30,000 patients transferred ~6x fewer! Prakash and Faloutsos 201263AUTH '12 Slide 64 Running Time SimulationsSMART-ALLOC > 1 week 14 secs > 30,000x speed-up! Wall-Clock Time Lower is better Prakash and Faloutsos 201264AUTH '12 Slide 65 Experiments K = 200K = 2000 PENN-NETWORK SECOND-LIFE ~5 x ~2.5 x Lower is better Prakash and Faloutsos 201265AUTH '12 Slide 66 Acknowledgements Funding Prakash and Faloutsos 201266AUTH '12 Slide 67 References 1. Threshold Conditions for Arbitrary Cascade Models on Arbitrary Networks (B. Aditya Prakash, Deepayan Chakrabarti, Michalis Faloutsos, Nicholas Valler, Christos Faloutsos) - In IEEE ICDM 2011, Vancouver (Invited to KAIS Journal Best Papers of ICDM.) 2. Virus Propagation on Time-Varying Networks: Theory and Immunization Algorithms (B. Aditya Prakash, Hanghang Tong, Nicholas Valler, Michalis Faloutsos and Christos Faloutsos) In ECML-PKDD 2010, Barcelona, Spain 3. Epidemic Spreading on Mobile Ad Hoc Networks: Determining the Tipping Point (Nicholas Valler, B. Aditya Prakash, Hanghang Tong, Michalis Faloutsos and Christos Faloutsos) In IEEE NETWORKING 2011, Valencia, Spain 4. Winner-takes-all: Competing Viruses or Ideas on fair-play networks (B. Aditya Prakash, Alex Beutel, Roni Rosenfeld, Christos Faloutsos) In WWW 2012, Lyon 5. On the Vulnerability of Large Graphs (Hanghang Tong, B. Aditya Prakash, Tina Eliassi- Rad and Christos Faloutsos) In IEEE ICDM 2010, Sydney, Australia 6. Fractional Immunization of Networks (B. Aditya Prakash, Lada Adamic, Theodore Iwashyna, Hanghang Tong, Christos Faloutsos) - Under Submission 7. Rise and Fall Patterns of Information Diffusion: Model and Implications (Yasuko Matsubara, Yasushi Sakurai, B. Aditya Prakash, Lei Li, Christos Faloutsos) - Under Submission 67 http://www.cs.cmu.edu/~badityap/ AUTH '12Prakash and Faloutsos 2012 Slide 68 Analysis Policy/Action Data Propagation on Large Networks B. Aditya Prakash Christos Faloutsos 68AUTH '12Prakash and Faloutsos 2012