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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

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Thank you! Ying Ding Jiawei Han Jie Tang Philip Yu KDD-MDS-2012C. Faloutsos (CMU)2

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Networks are everywhere! Human Disease Network [Barabasi 2007] Gene Regulatory Network [Decourty 2008] Facebook Network [2010] The Internet [2005] C. Faloutsos (CMU)3KDD-MDS-2012

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Dynamical Processes over networks are also everywhere! C. Faloutsos (CMU)4KDD-MDS-2012

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Why do we care? Information Diffusion Viral Marketing Epidemiology and Public Health Cyber Security Human mobility Games and Virtual Worlds Ecology Social Collaboration........ C. Faloutsos (CMU)5KDD-MDS-2012

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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 C. Faloutsos (CMU)6KDD-MDS-2012

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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? C. Faloutsos (CMU)7KDD-MDS-2012

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Why do we care? (1: Epidemiology) CURRENT PRACTICEOUR METHOD ~6x fewer! [US-MEDICARE NETWORK 2005] C. Faloutsos (CMU)8KDD-MDS-2012 Hospital-acquired inf. took 99K+ lives, cost $5B+ (all per year)

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Why do we care? (2: Online Diffusion) > 800m users, ~$1B revenue [WSJ 2010] ~100m active users > 50m users C. Faloutsos (CMU)9KDD-MDS-2012

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Why do we care? (2: Online Diffusion) Dynamical Processes over networks Celebrity Buy Versace™! Followers Social Media Marketing C. Faloutsos (CMU)10KDD-MDS-2012

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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 C. Faloutsos (CMU)11KDD-MDS-2012

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Research Theme DATA Large real-world networks & processes ANALYSIS Understanding POLICY/ ACTION Managing C. Faloutsos (CMU)12KDD-MDS-2012

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In this talk ANALYSIS Understanding Given propagation models: Q1: Will an epidemic happen? C. Faloutsos (CMU)13KDD-MDS-2012

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In this talk Q2: How to immunize and control out-breaks better? POLICY/ ACTION Managing C. Faloutsos (CMU)14KDD-MDS-2012

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Outline Part 1: anomaly detection Part 2: influence propagation Motivation Epidemics: what happens? (Theory) Action: Who to immunize? (Algorithms) C. Faloutsos (CMU)15KDD-MDS-2012

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A fundamental question Strong Virus Epidemic? C. Faloutsos (CMU)16KDD-MDS-2012

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example (static graph) Weak Virus Epidemic? C. Faloutsos (CMU)17KDD-MDS-2012

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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 Separate the regimes? C. Faloutsos (CMU)18KDD-MDS-2012

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Threshold (static version) Problem Statement Given: – Graph G, and – Virus specs (attack prob. etc.) Find: – A condition for virus extinction/invasion C. Faloutsos (CMU)19KDD-MDS-2012

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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 ….. C. Faloutsos (CMU)20KDD-MDS-2012

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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) C. Faloutsos (CMU)21KDD-MDS-2012

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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) Prob. β Prob. δ t = 1t = 2t = 3 C. Faloutsos (CMU)22KDD-MDS-2012

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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’ C. Faloutsos (CMU)23KDD-MDS-2012

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Related Work R. M. Anderson and R. M. May. Infectious Diseases of Humans. Oxford University Press, 1991. A. Barrat, M. Barthélemy, 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):215–227, 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 C. Faloutsos (CMU)24KDD-MDS-2012

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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) C. Faloutsos (CMU)25KDD-MDS-2012

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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? ….. C. Faloutsos (CMU)26KDD-MDS-2012

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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 λ, ﬁrst eigenvalue of A, and 2.some constant, determined by the virus propagation model λ λ No epidemic if λ * < 1 C. Faloutsos (CMU)27KDD-MDS-2012 In Prakash+ ICDM 2011 (Selected among best papers). w/ Deepay Chakrabarti

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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. C. Faloutsos (CMU)28KDD-MDS-2012

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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 u u ≈. (i, j) = # of paths i j of length k C. Faloutsos (CMU)29KDD-MDS-2012

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Largest Eigenvalue (λ) λ ≈ 2λ = Nλ = N-1 N = 1000 λ ≈ 2λ= 31.67λ= 999 better connectivity higher λ C. Faloutsos (CMU)30KDD-MDS-2012 N nodes

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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 C. Faloutsos (CMU)31KDD-MDS-2012

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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 C. Faloutsos (CMU)32KDD-MDS-2012

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λ * < 1 Graph-based Model-based 33 General VPM structure Topology and stability See paper for full proof KDD-MDS-2012C. Faloutsos (CMU)

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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) C. Faloutsos (CMU)34KDD-MDS-2012

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λ * < 1 Graph-based Model-based General VPM structure Topology and stability See paper for full proof 35KDD-MDS-2012C. Faloutsos (CMU)

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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) C. Faloutsos (CMU)36KDD-MDS-2012

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Virus Propagation on Time- Varying Networks: Theory and Immunization Algorithms B. Aditya Prakash, Hanghang Tong, Nicholas Valler, Michalis Faloutsos, Christos Faloutsos in ECML-PKDD 2010 KDD-MDS-2012C. Faloutsos (CMU)37

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Dynamic Graphs: Epidemic? adjacency matrix 8 8 Alternating behaviors DAY (e.g., work) C. Faloutsos (CMU)38KDD-MDS-2012

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adjacency matrix 8 8 Dynamic Graphs: Epidemic? Alternating behaviors NIGHT (e.g., home) C. Faloutsos (CMU)39KDD-MDS-2012

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SIS model – recovery rate δ – infection rate β Set of T arbitrary graphs Model Description day N N night N N, weekend….. Infected Healthy XN1 N3 N2 Prob. β Prob. δ C. Faloutsos (CMU)40KDD-MDS-2012

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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 = C. Faloutsos (CMU)41KDD-MDS-2012

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Synthetic MIT Reality Mining log(fraction infected) Time BELOW AT ABOVE AT BELOW Infection-profile C. Faloutsos (CMU)42KDD-MDS-2012

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“Take-off” plots Footprint (# infected @ “steady state”) Our threshold (log scale) NO EPIDEMIC EPIDEMIC NO EPIDEMIC SyntheticMIT Reality C. Faloutsos (CMU)43KDD-MDS-2012

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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) C. Faloutsos (CMU)44KDD-MDS-2012

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Winner-takes-all: Competing Viruses on fair-play networks B. Aditya Prakash, Alex Beutel, Roni Rosenfeld, Christos Faloutsos, WWW 2012 KDD-MDS-2012C. Faloutsos (CMU)45

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Competing Contagions iPhone v AndroidBlu-ray v HD-DVD 46KDD-MDS-2012C. Faloutsos (CMU) 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 Virus 1 Virus 2 C. Faloutsos (CMU)47KDD-MDS-2012

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Question: What happens in the end? green: virus 1 red: virus 2 Footprint @ Steady State = ? Number of Infections C. Faloutsos (CMU)48KDD-MDS-2012 ASSUME: Virus 1 is stronger than Virus 2

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Question: What happens in the end? green: virus 1 red: virus 2 Number of Infections Strength ?? = Strength 2 Footprint @ Steady State 49KDD-MDS-2012C. Faloutsos (CMU) ASSUME: Virus 1 is stronger than Virus 2

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Answer: Winner-Takes-All green: virus 1 red: virus 2 Number of Infections 50KDD-MDS-2012C. Faloutsos (CMU) ASSUME: Virus 1 is stronger than Virus 2

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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 same as before! 51KDD-MDS-2012C. Faloutsos (CMU) In Prakash, Beutel, + WWW 2012

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More results: in KDD’12 Partially competing viruses/products; Collaborating products Interacting Viruses on a Network: Can both survive? Alex Beutel, B. Aditya Prakash, Roni Rosenfeld and Christos Faloutsos in SIGKDD 2012, Beijing KDD-MDS-2012C. Faloutsos (CMU)52

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Real Examples Reddit v DiggBlu-Ray v HD-DVD [Google Search Trends data] 53KDD-MDS-2012C. Faloutsos (CMU)

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Outline Motivation Epidemics: what happens? (Theory) Action: Who to immunize? (Algorithms) C. Faloutsos (CMU)54KDD-MDS-2012

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On the Vulnerability of Large Graphs Hanghang Tong, B. Aditya Prakash, Charalampos Tsourakakis, Tina Eliassi-Rad, Christos Faloutsos, Duen Horng (Polo) Chau, ICDM 2010 KDD-MDS-2012C. Faloutsos (CMU)55

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? ? Given: a graph A, virus prop. model and budget k; Find: k ‘best’ nodes for immunization (removal). k = 2 ? ? Full Static Immunization C. Faloutsos (CMU)56KDD-MDS-2012

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Outline Motivation Epidemics: what happens? (Theory) Action: Who to immunize? (Algorithms) – Full Immunization (Static Graphs) – Fractional Immunization C. Faloutsos (CMU)57KDD-MDS-2012

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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’? C. Faloutsos (CMU)58KDD-MDS-2012

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Proposed vulnerability measure λ Increasing λ Increasing vulnerability λ is the epidemic threshold “Safe”“Vulnerable”“Deadly” C. Faloutsos (CMU)59KDD-MDS-2012

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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) Δ λ = λ - λ s Eigen-Drop(S) Δ λ = λ - λ s Δ A1: “Eigen-Drop”: an ideal shield value C. Faloutsos (CMU)60KDD-MDS-2012

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(Q2) - Direct Algorithm too expensive! Immunize k nodes which maximize Δ λ S = argmax Δ λ Combinatorial! Complexity: – Example: 1,000 nodes, with 10,000 edges It takes 0.01 seconds to compute λ It takes 2,615 years to find 5-best nodes ! C. Faloutsos (CMU)61KDD-MDS-2012

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A2: Our Solution Part 1: Shield Value – Carefully approximate Eigen-drop (Δ λ) – 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 C. Faloutsos (CMU)62KDD-MDS-2012 In Tong, Prakash+ ICDM 2010

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Experiment: Immunization quality Log(fraction of infected nodes) NetShield Degree PageRank Eigs (=HITS) Acquaintance Betweeness (shortest path) Lower is better Time C. Faloutsos (CMU)63KDD-MDS-2012

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Outline Motivation Epidemics: what happens? (Theory) Action: Who to immunize? (Algorithms) – Full Immunization (Static Graphs) – Fractional Immunization C. Faloutsos (CMU)64KDD-MDS-2012

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Fractional Immunization of Networks B. Aditya Prakash, Lada Adamic, Theodore Iwashyna (M.D.), Hanghang Tong, Christos Faloutsos Under review C. Faloutsos (CMU)65KDD-MDS-2012

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Fractional Asymmetric Immunization Hospital Another Hospital Drug-resistant Bacteria (like XDR-TB) C. Faloutsos (CMU)66KDD-MDS-2012

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Fractional Asymmetric Immunization Hospital Another Hospital Drug-resistant Bacteria (like XDR-TB) C. Faloutsos (CMU)67KDD-MDS-2012

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Fractional Asymmetric Immunization Hospital Another Hospital Problem: Given k units of disinfectant, how to distribute them to maximize hospitals saved? C. Faloutsos (CMU)68KDD-MDS-2012

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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! C. Faloutsos (CMU)69KDD-MDS-2012

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Running Time ≈ SimulationsSMART-ALLOC > 1 week 14 secs > 30,000x speed-up! Wall-Clock Time Lower is better C. Faloutsos (CMU)70KDD-MDS-2012

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Experiments K = 200K = 2000 PENN-NETWORK SECOND-LIFE ~5 x ~2.5 x Lower is better C. Faloutsos (CMU)71KDD-MDS-2012

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Acknowledgements Funding C. Faloutsos (CMU)72KDD-MDS-2012

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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 73 http://www.cs.cmu.edu/~badityap/ KDD-MDS-2012C. Faloutsos (CMU)

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Analysis Policy/Action Data Propagation on Large Networks B. Aditya Prakash Christos Faloutsos 74KDD-MDS-2012C. Faloutsos (CMU)

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