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Dynamical Processes on Large Networks B. Aditya Prakash Carnegie Mellon University MMS, SIAM AN, Minneapolis, July 10, 2012

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Thanks! Jeremy Kepner David Bader John Gilbert 2

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Networks are everywhere! Human Disease Network [Barabasi 2007] Gene Regulatory Network [Decourty 2008] Facebook Network [2010] The Internet [2005]

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4 Dynamical Processes over networks are also everywhere!

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Why do we care? Online Information Diffusion Viral Marketing Epidemiology and Public Health Cyber Security Human mobility Games and Virtual Worlds Ecology

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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 6

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Why do we care? (2: Online Diffusion) Dynamical Processes over networks Celebrity Buy Versace™! Followers 7 Social Media Marketing

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Why do we care? (3: To change the world?) Dynamical Processes over networks Social networks and Collaborative Action 8

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High Impact – Multiple Settings Q. How to squash rumors faster? Q. How do opinions spread? Q. How to market better? 9 epidemic out-breaks products/viruses transmit s/w patches

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Research Theme DATA Large real-world networks & processes 10 ANALYSIS Understanding POLICY/ ACTION Managing

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Research Theme – Public Health DATA Modeling # patient transfers ANALYSIS Will an epidemic happen? POLICY/ ACTION How to control out-breaks?

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Research Theme – Social Media DATA Modeling Tweets spreading POLICY/ ACTION How to market better? ANALYSIS # cascades in future?

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In this talk 13 DATA Large real-world networks & processes Q1: How do bursts look like? Q2: How do #hashtags spread?

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In this talk 14 ANALYSIS Understanding Given propagation models: Q2: Will an epidemic happen?

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In this talk 15 Q3: How to immunize and control out-breaks better? POLICY/ ACTION Managing

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Outline Motivation Learning Models (Empirical Studies) – Rise and Fall patterns – Twitter hashtags Epidemics: what happens? (Theory) Action: Who to immunize? (Algorithms) Conclusion 16

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Meme (# of mentions in blogs) – short phrases Sourced from U.S. politics in “you can put lipstick on a pig” “yes we can” Rise and fall patterns in social media Thanks to Yasuko Matsubara for some slides

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Rise and fall patterns in social media 18 Can we find a unifying model, which includes these patterns? four classes on YouTube [Crane et al. ’08] six classes on Meme [Yang et al. ’11]

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Rise and fall patterns in social media 19 Answer: YES! We can represent all patterns by single model In SIGKDD 2012

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Main idea - SpikeM -1. Un-informed bloggers (uninformed about rumor) -2. External shock at time n b (e.g, breaking news) -3. Infection (word-of-mouth) 20 Infectiveness of a blog-post at age n: -Strength of infection (quality of news) -Decay function (how infective a blog posting is) Time n=0Time n=n b Time n=n b +1 β Power Law

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SpikeM - with periodicity Full equation of SpikeM 21 Periodicity 12pm Peak activity 3am Low activity Time n Bloggers change their activity over time (e.g., daily, weekly, yearly) Bloggers change their activity over time (e.g., daily, weekly, yearly) activity

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Details Analysis – exponential rise and power-raw fall 22 Liner-log Log-log Rise-part SI -> exponential SpikeM -> exponential Rise-part SI -> exponential SpikeM -> exponential

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Details Analysis – exponential rise and power-raw fall 23 Liner-log Log-log Fall-part SI -> exponential SpikeM -> power law Fall-part SI -> exponential SpikeM -> power law

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Tail-part forecasts 24 SpikeM can capture tail part

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“What-if” forecasting 25 e.g., given (1) first spike, (2) release date of two sequel movies (3) access volume before the release date ? ? ? ? (1) First spike(2) Release date(3) Two weeks before release

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“What-if” forecasting 26 – SpikeM can forecast not only tail-part, but also rise-part ! SpikeM can forecast upcoming spikes (1) First spike(2) Release date(3) Two weeks before release

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Outline Motivation Learning Models (Empirical Studies) – Rise and Fall patterns – Twitter hashtags Epidemics: what happens? (Theory) Action: Who to immunize? (Algorithms) Conclusion 27

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Tweets Diffusion: Problem Definition Given: – Action log of people tweeting a #hashtag – A network of users Find: – How external influence varies with #hashtags? ?? ? ? ?? 28

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Tweet Diffusion: Data Yahoo! Twitter firehose More than 750 million tweets (> 10 Tera-bytes) Test-bed of > 6000 machines – Hadoop+PIG system ver Took top 500 hashtags (by volume) in Feb 2011 Network of users: – connecting user X to user Y if X directed at least to Y (or RT-ed a tweet) 29 Under Review

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Tweet Diffusion Propagation = Influence + External Developed a model – takes the previous observations into account – with parameters representing external influence Learn from previous data – EM-style alternating minimizing algorithm Group tags according to learnt params 30

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Results: External Influence vs Time time “External Effects” #nowwatching, #nowplaying, #epictweets #purpleglasses, #brits, #famouslies #oscar, #25jan #openfollow, #ihatequotes, #tweetmyjobs Can also use for Forecasting, Anomaly Detection! Bursty, external events “Word-of-mouth” Not trending Long-running tags “Word-of-mouth” 31

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Outline Motivation Learning Models (Empirical Studies) Epidemics: what happens? (Theory) Action: Who to immunize? (Algorithms) Conclusion 32

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A fundamental question Strong Virus Epidemic? 33

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example (static graph) Weak Virus Epidemic? 34

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Problem Statement Find, a condition under which – virus will die out exponentially quickly – regardless of initial infection condition above (epidemic) below (extinction) # Infected time 35 Separate the regimes?

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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 ….. 36

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Outline Motivation Epidemics: what happens? (Theory) – Background – Result and Intuition (Static Graphs) – Proof Ideas (Static Graphs) – Bonus : Competing Viruses Action: Who to immunize? (Algorithms) Learning Models: Twitter (Empirical Studies) 37

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“SIR” model: life immunity (mumps) Each node in the graph is in one of three states – Susceptible (i.e. healthy) – Infected – Removed (i.e. can’t get infected again) 38 Prob. β Prob. δ t = 1t = 2t = 3

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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’ 39

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Related Work R. M. Anderson and R. M. May. Infectious Diseases of Humans. Oxford University Press, A. Barrat, M. Barthélemy, and A. Vespignani. Dynamical Processes on Complex Networks. Cambridge University Press, F. M. Bass. A new product growth for model consumer durables. Management Science, 15(5):215–227, D. Chakrabarti, Y. Wang, C. Wang, J. Leskovec, and C. Faloutsos. Epidemic thresholds in real networks. ACM TISSEC, 10(4), D. Easley and J. Kleinberg. Networks, Crowds, and Markets: Reasoning About a Highly Connected World. Cambridge University Press, A. Ganesh, L. Massoulie, and D. Towsley. The effect of network topology in spread of epidemics. IEEE INFOCOM, Y. Hayashi, M. Minoura, and J. Matsukubo. Recoverable prevalence in growing scale-free networks and the effective immunization. arXiv:cond-at/ v2, Aug H. W. Hethcote. The mathematics of infectious diseases. SIAM Review, 42, H. W. Hethcote and J. A. Yorke. Gonorrhea transmission dynamics and control. Springer Lecture Notes in Biomathematics, 46, J. O. Kephart and S. R. White. Directed-graph epidemiological models of computer viruses. IEEE Computer Society Symposium on Research in Security and Privacy, J. O. Kephart and S. R. White. Measuring and modeling computer virus prevalence. IEEE Computer Society Symposium on Research in Security and Privacy, R. Pastor-Santorras and A. Vespignani. Epidemic spreading in scale-free networks. Physical Review Letters 86, 14, ……… 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 40

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Outline Motivation Epidemics: what happens? (Theory) – Background – Result and Intuition (Static Graphs) – Proof Ideas (Static Graphs) – Bonus : Competing Viruses Action: Who to immunize? (Algorithms) Learning Models: Twitter (Empirical Studies) 41

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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? 42 …..

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Static Graphs: Our Main Result Informally, 43 For, any arbitrary topology (adjacency matrix A) any virus propagation model (VPM) in standard literature the epidemic threshold depends only 1.on the λ, ﬁrst eigenvalue of A, and 2.some constant, determined by the virus propagation model λ λ No epidemic if λ * < 1 In Prakash+ ICDM 2011 (Selected among best papers).

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Our thresholds for some models s = effective strength s < 1 : below threshold Models Effective Strength (s) Threshold (tipping point) SIS, SIR, SIRS, SEIR s = λ. s = 1 SIV, SEIV s = λ. ( H.I.V. ) s = λ. 44

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Our result: Intuition for λ “Official” definition: Let A be the adjacency matrix. Then λ is the root with the largest magnitude of the characteristic polynomial of A [det(A – xI)]. Doesn’t give much intuition! “Un-official” Intuition λ ~ # paths in the graph 45 u u ≈. (i, j) = # of paths i j of length k

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N nodes Largest Eigenvalue (λ) λ ≈ 2λ = Nλ = N-1 46 N = 1000 λ ≈ 2λ= 31.67λ= 999 better connectivity higher λ

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Examples: Simulations – SIRS (pertusis) Fraction of Infections Footprint Effective StrengthTime ticks (a) Infection profile (b) “Take-off” plot PORTLAND graph 31 million links, 6 million nodes 47

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Outline Motivation Epidemics: what happens? (Theory) – Background – Result and Intuition (Static Graphs) – Proof Ideas (Static Graphs) – Bonus : Competing Viruses Action: Who to immunize? (Algorithms) Learning Models: Twitter (Empirical Studies) 48

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λ * < 1 Graph-based Model-based 49 General VPM structure Topology and stability See paper for full proof

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Outline Motivation Epidemics: what happens? (Theory) – Background – Result and Intuition (Static Graphs) – Proof Ideas (Static Graphs) – Bonus : Competing Viruses Action: Who to immunize? (Algorithms) Learning Models: Twitter (Empirical Studies) 50

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Competing Contagions 51 iPhone v AndroidBlu-ray v HD-DVD Biological common flu/avian flu, pneumococcal inf etc

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A simple model Modified flu-like Mutual Immunity (“pick one of the two”) Susceptible-Infected1-Infected2-Susceptible 52 Virus 1 Virus 2

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Question: What happens in the end? 53 green: virus 1 red: virus 2 Steady State = ? Number of Infections ASSUME: Virus 1 is stronger than Virus 2

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Question: What happens in the end? 54 green: virus 1 red: virus 2 Number of Infections ASSUME: Virus 1 is stronger than Virus 2 Strength ?? = Strength 2 Steady State

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Answer: Winner-Takes-All 55 green: virus 1 red: virus 2 ASSUME: Virus 1 is stronger than Virus 2 Number of Infections

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Our Result: Winner-Takes-All 56 In Prakash+ WWW 2012 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 same as before!

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Real Examples 57 Reddit v DiggBlu-Ray v HD-DVD [Google Search Trends data]

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Outline Motivation Learning Models (Empirical Studies) Epidemics: what happens? (Theory) Action: Who to immunize? (Algorithms) Conclusion 58

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59 Fractional Immunization of Networks B. Aditya Prakash, Lada Adamic, Theodore Iwashyna (M.D.), Hanghang Tong, Christos Faloutsos Under Review

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Fractional Asymmetric Immunization Hospital Another Hospital 60 Drug-resistant Bacteria (like XDR-TB)

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Fractional Asymmetric Immunization Hospital Another Hospital 61 Problem: Given k units of disinfectant, how to distribute them to maximize hospitals saved?

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Proposed vulnerability measure λ Increasing λ Increasing vulnerability λ is the epidemic threshold “Safe”“Vulnerable”“Deadly” 62

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Our Algorithm “SMART-ALLOC” CURRENT PRACTICESMART-ALLOC [US-MEDICARE NETWORK 2005] 63 Each circle is a hospital, ~3000 hospitals More than 30,000 patients transferred ~6x fewer!

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Running Time 64 ≈ SimulationsSMART-ALLOC > 1 week 14 secs > 30,000x speed-up! Wall-Clock Time Lower is better

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Conclusions Bursts: SpikeM model – Exponential growth, Power-law decay Twitter: Hashtags – Propagation = External + Internal Epidemic Threshold – It’s the Eigenvalue Fast Immunization – Max. drop in eigenvalue, linear-time near-optimal algorithm 65

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References 1. Interacting Viruses on a Network: Can both survive? (Alex Beutel, B. Aditya Prakash, Roni Rosenfeld, Christos Faloutsos) – In SIGKDD 2012, Beijing 2. Rise and Fall Patterns of Information Diffusion: Model and Implications (Yasuko Matsubara, Yasushi Sakurai, B. Aditya Prakash, Lei Li, Christos Faloutsos) – In SIGKDD 2012, Beijing 3. Winner-takes-all: Competing Viruses or Ideas on fair-play networks (B. Aditya Prakash, Alex Beutel, Roni Rosenfeld, Christos Faloutsos) – In WWW 2012, Lyon 4. 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.) 5. 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 6. On the Vulnerability of Large Graphs (Hanghang Tong, B. Aditya Prakash, Tina Eliassi-Rad and Christos Faloutsos) – In IEEE ICDM 2010, Sydney, Australia 7. 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 8. Fractional Immunization of Networks (B. Aditya Prakash, Lada Adamic, Theodore Iwashyna, Hanghang Tong, Christos Faloutsos) – Under Review 9. How much of Twitter is Influence? (B. Aditya Prakash, Deepayan Chakrabarti, Kunal Punera) – Under Review

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Acknowledgements Funding 67

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Analysis Policy/Action Data Dynamical Processes on Large Networks B. Aditya Prakash 68

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