Anomaly Detection and Virus Propagation in Large Graphs Christos Faloutsos CMU

Faloutsos, Prakash, Chau, Koutra, Akoglu Thank you! Dr. Ching-Hao (Eric) Mao Prof. Kenneth Pao Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Faloutsos, Prakash, Chau, Koutra, Akoglu Outline Part 1: anomaly detection OddBall (anomaly detection) Belief Propagation Conclusions Part 2: influence propagation Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

OddBall: Spotting Anomalies in Weighted Graphs Leman Akoglu, Mary McGlohon, Christos Faloutsos Carnegie Mellon University School of Computer Science PAKDD 2010, Hyderabad, India

Faloutsos, Prakash, Chau, Koutra, Akoglu Main idea For each node, extract ‘ego-net’ (=1-step-away neighbors) Extract features (#edges, total weight, etc etc) Compare with the rest of the population Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Faloutsos, Prakash, Chau, Koutra, Akoglu What is an egonet? egonet ego Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu 6

Faloutsos, Prakash, Chau, Koutra, Akoglu Selected Features Ni: number of neighbors (degree) of ego i Ei: number of edges in egonet i Wi: total weight of egonet i λw,i: principal eigenvalue of the weighted adjacency matrix of egonet I Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu 7

Faloutsos, Prakash, Chau, Koutra, Akoglu Near-Clique/Star SOME OLD RULES Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu 8

Faloutsos, Prakash, Chau, Koutra, Akoglu Near-Clique/Star SOME OLD RULES Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu 9

Faloutsos, Prakash, Chau, Koutra, Akoglu Near-Clique/Star SOME OLD RULES Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu 10

Faloutsos, Prakash, Chau, Koutra, Akoglu Near-Clique/Star Andrew Lewis (director) SOME OLD RULES Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu 11

Faloutsos, Prakash, Chau, Koutra, Akoglu Outline Part 1: anomaly detection OddBall (anomaly detection) Belief Propagation Conclusions Part 2: influence propagation Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Faloutsos, Prakash, Chau, Koutra, Akoglu E-bay Fraud detection w/ Polo Chau & Shashank Pandit, CMU [www’07] Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Faloutsos, Prakash, Chau, Koutra, Akoglu E-bay Fraud detection Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Faloutsos, Prakash, Chau, Koutra, Akoglu E-bay Fraud detection Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

E-bay Fraud detection - NetProbe Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Faloutsos, Prakash, Chau, Koutra, Akoglu Popular press And less desirable attention: E-mail from ‘Belgium police’ (‘copy of your code?’) Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Faloutsos, Prakash, Chau, Koutra, Akoglu Outline OddBall (anomaly detection) Belief Propagation Ebay fraud Symantec malware detection Unification results Conclusions Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Polonium: Tera-Scale Graph Mining and Inference for Malware Detection PATENT PENDING Polonium: Tera-Scale Graph Mining and Inference for Malware Detection SDM 2011, Mesa, Arizona Polo Chau Machine Learning Dept Carey Nachenberg Vice President & Fellow Jeffrey Wilhelm Principal Software Engineer Adam Wright Software Engineer Prof. Christos Faloutsos Computer Science Dept

Faloutsos, Prakash, Chau, Koutra, Akoglu Polonium: The Data 60+ terabytes of data anonymously contributed by participants of worldwide Norton Community Watch program 50+ million machines 900+ million executable files Constructed a machine-file bipartite graph (0.2 TB+) 1 billion nodes (machines and files) 37 billion edges As of today, has grown to more than three times Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Faloutsos, Prakash, Chau, Koutra, Akoglu Polonium: Key Ideas Use Belief Propagation to propagate domain knowledge in machine-file graph to detect malware Use “guilt-by-association” (i.e., homophily) E.g., files that appear on machines with many bad files are more likely to be bad Scalability: handles 37 billion-edge graph Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Polonium: One-Interaction Results Ideal 84.9% True Positive Rate 1% False Positive Rate True Positive Rate % of malware correctly identified for files reported by four or more machines False Positive Rate % of non-malware wrongly labeled as malware Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Faloutsos, Prakash, Chau, Koutra, Akoglu Outline Part 1: anomaly detection OddBall (anomaly detection) Belief Propagation Ebay fraud Symantec malware detection Unification results Conclusions Part 2: influence propagation Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Unifying Guilt-by-Association Approaches: Theorems and Fast Algorithms Danai Koutra Tai-You Ke U Kang Duen Horng (Polo) Chau Hsing-Kuo Kenneth Pao Christos Faloutsos Work in collaboration with National Taiwan University ECML PKDD, 5-9 September 2011, Athens, Greece

Problem Definition: GBA techniques ? Given: Graph; & few labeled nodes Find: labels of rest (assuming network effects) ? ? Classification problem where we assume that neighboring nodes are related Network effects – birds of a feather flock together ? Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Homophily and Heterophily NOT all methods handle heterophily BUT proposed method does! Step 1 All methods handle homophily Intensity of green related to the score of the node – birds of a feather flock together Eg. of network with heterophily: bad guys - fraudsters This relation can be either homophily or heterophily. All upcoming methods can handle homophily. Heterophily not all – proposed method CAN handle, others cannot Homophily: connected nodes are similar -> in the first step the neighbors of the green node will become green and the neighbors of the red node red. The same will happen in the second step – the color is diffused Heterophily: connected nodes are dissimilar -> In the first step the nodes connected to the green node become red and vice versa Step 2 Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Faloutsos, Prakash, Chau, Koutra, Akoglu Are they related? RWR (Random Walk with Restarts) google’s pageRank (‘if my friends are important, I’m important, too’) SSL (Semi-supervised learning) minimize the differences among neighbors BP (Belief propagation) send messages to neighbors, on what you believe about them Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Faloutsos, Prakash, Chau, Koutra, Akoglu Are they related? YES! RWR (Random Walk with Restarts) google’s pageRank (‘if my friends are important, I’m important, too’) SSL (Semi-supervised learning) minimize the differences among neighbors BP (Belief propagation) send messages to neighbors, on what you believe about them Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Correspondence of Methods Matrix Unknown known RWR [I – c AD-1] × x = (1-c)y SSL [I + a(D - A)] y FABP [I + a D - c’A] bh φh 0 1 0 1 0 1 ? 1 d1 d2 d3 final labels/ beliefs prior labels/ beliefs adjacency matrix Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Faloutsos, Prakash, Chau, Koutra, Akoglu Results: Scalability # of edges (Kronecker graphs) runtime (min) Kronecker graphs FABP is linear on the number of edges. Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Results (5): Parallelism runtime (min) % accuracy FABP ~2x faster & wins/ties on accuracy. Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Faloutsos, Prakash, Chau, Koutra, Akoglu Conclusions Anomaly detection: hand-in-hand with pattern discovery (‘anomalies’ == ‘rare patterns’) ‘OddBall’ for large graphs ‘NetProbe’ and belief propagation: exploit network effects. FaBP: fast & accurate Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Faloutsos, Prakash, Chau, Koutra, Akoglu Outline Part 1: anomaly detection OddBall (anomaly detection) Belief Propagation Conclusions Part 2: influence propagation Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

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

Networks are everywhere! Facebook Network [2010] Gene Regulatory Network [Decourty 2008] Human Disease Network [Barabasi 2007] The Internet [2005] Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Dynamical Processes over networks are also everywhere! Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Faloutsos, Prakash, Chau, Koutra, Akoglu Why do we care? Information Diffusion Viral Marketing Epidemiology and Public Health Cyber Security Human mobility Games and Virtual Worlds Ecology Social Collaboration ........ Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

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 Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

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? Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Why do we care? (1: Epidemiology) ~6x fewer! [US-MEDICARE NETWORK 2005] CURRENT PRACTICE OUR METHOD Hospital-acquired inf. took 99K+ lives, cost $5B+ (all per year) Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Why do we care? (2: Online Diffusion) > 800m users, ~$1B revenue [WSJ 2010] ~100m active users > 50m users Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Why do we care? (2: Online Diffusion) Dynamical Processes over networks Buy Versace™! Celebrity Followers Social Media Marketing Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

High Impact – Multiple Settings epidemic out-breaks Q. How to squash rumors faster? Q. How do opinions spread? Q. How to market better? products/viruses transmit s/w patches Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Research Theme ANALYSIS POLICY/ ACTION DATA Understanding Managing Large real-world networks & processes Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Faloutsos, Prakash, Chau, Koutra, Akoglu In this talk ANALYSIS Understanding Given propagation models: Q1: Will an epidemic happen? Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Faloutsos, Prakash, Chau, Koutra, Akoglu In this talk POLICY/ ACTION Managing Q2: How to immunize and control out-breaks better? Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Faloutsos, Prakash, Chau, Koutra, Akoglu Outline Part 1: anomaly detection Part 2: influence propagation Motivation Epidemics: what happens? (Theory) Action: Who to immunize? (Algorithms) Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

A fundamental question Strong Virus Epidemic? Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

example (static graph) Weak Virus Epidemic? Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Faloutsos, Prakash, Chau, Koutra, Akoglu Problem Statement above (epidemic) below (extinction) # Infected time Find, a condition under which virus will die out exponentially quickly regardless of initial infection condition Separate the regimes? Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Threshold (static version) Problem Statement Given: Graph G, and Virus specs (attack prob. etc.) Find: A condition for virus extinction/invasion Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

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 ….. Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Faloutsos, Prakash, Chau, Koutra, Akoglu 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) Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

“SIR” model: life immunity (mumps) Background “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. δ Prob. β t = 1 t = 2 t = 3 Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Terminology: continued Background 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’ Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Faloutsos, Prakash, Chau, Koutra, Akoglu Background Related Work All are about either: Structured topologies (cliques, block-diagonals, hierarchies, random) Specific virus propagation models Static graphs 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. ……… Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Faloutsos, Prakash, Chau, Koutra, Akoglu 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) Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

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? ….. Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Static Graphs: Our Main Result Informally, w/ Deepay Chakrabarti For, any arbitrary topology (adjacency matrix A) any virus propagation model (VPM) in standard literature the epidemic threshold depends only on the λ, first eigenvalue of A, and some constant , determined by the virus propagation model λ No epidemic if λ * < 1 In Prakash+ ICDM 2011 (Selected among best papers). Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

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 (H.I.V.) Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Our result: Intuition for λ “Official” definition: “Un-official” Intuition  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! λ ~ # paths in the graph u u ≈ . (i, j) = # of paths i  j of length k Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Largest Eigenvalue (λ) better connectivity higher λ λ ≈ 2 λ = N λ = N-1 λ ≈ 2 λ= 31.67 λ= 999 N = 1000 N nodes Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Examples: Simulations – SIR (mumps) Fraction of Infections Footprint (a) Infection profile (b) “Take-off” plot PORTLAND graph: synthetic population, 31 million links, 6 million nodes Effective Strength Time ticks Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Examples: Simulations – SIRS (pertusis) Fraction of Infections Footprint (a) Infection profile (b) “Take-off” plot PORTLAND graph: synthetic population, 31 million links, 6 million nodes Time ticks Effective Strength Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Faloutsos, Prakash, Chau, Koutra, Akoglu See paper for full proof General VPM structure Model-based λ * < 1 Dimensional arguments… Graph-based Topology and stability Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Faloutsos, Prakash, Chau, Koutra, Akoglu 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) Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Faloutsos, Prakash, Chau, Koutra, Akoglu See paper for full proof General VPM structure Model-based λ * < 1 Graph-based Topology and stability Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Faloutsos, Prakash, Chau, Koutra, Akoglu 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) Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Dynamic Graphs: Epidemic? Alternating behaviors DAY (e.g., work) adjacency matrix 8 Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Dynamic Graphs: Epidemic? Alternating behaviors NIGHT (e.g., home) adjacency matrix 8 Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Faloutsos, Prakash, Chau, Koutra, Akoglu Model Description Infected Healthy X N1 N3 N2 Prob. β Prob. δ SIS model recovery rate δ infection rate β Set of T arbitrary graphs day N night N , weekend….. Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Our result: Dynamic Graphs Threshold Informally, NO epidemic if eig (S) = < 1 Single number! Largest eigenvalue of The system matrix S Details S = In Prakash+, ECML-PKDD 2010 Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Faloutsos, Prakash, Chau, Koutra, Akoglu Infection-profile log(fraction infected) Synthetic MIT Reality Mining ABOVE ABOVE AT AT BELOW BELOW Time Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Faloutsos, Prakash, Chau, Koutra, Akoglu “Take-off” plots Footprint (# infected @ “steady state”) Synthetic MIT Reality EPIDEMIC Our threshold Our threshold EPIDEMIC NO EPIDEMIC NO EPIDEMIC (log scale) Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Faloutsos, Prakash, Chau, Koutra, Akoglu 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) Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Faloutsos, Prakash, Chau, Koutra, Akoglu Competing Contagions iPhone v Android Blu-ray v HD-DVD Biological common flu/avian flu, pneumococcal inf etc Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Faloutsos, Prakash, Chau, Koutra, Akoglu Details A simple model Modified flu-like Mutual Immunity (“pick one of the two”) Susceptible-Infected1-Infected2-Susceptible Virus 1 Virus 2 Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Question: What happens in the end? green: virus 1 red: virus 2 Number of Infections Footprint @ Steady State = ? ASSUME: Virus 1 is stronger than Virus 2 Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Question: What happens in the end? Footprint @ Steady State green: virus 1 red: virus 2 Number of Infections Strength ?? = 2 Strength ASSUME: Virus 1 is stronger than Virus 2 Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Answer: Winner-Takes-All green: virus 1 red: virus 2 Number of Infections ASSUME: Virus 1 is stronger than Virus 2 Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Our Result: Winner-Takes-All Given our model, and any graph, the weaker virus always dies-out completely Details The stronger survives only if it is above threshold Virus 1 is stronger than Virus 2, if: strength(Virus 1) > strength(Virus 2) Strength(Virus) = λ β / δ  same as before! In Prakash, Beutel, + WWW 2012 Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Faloutsos, Prakash, Chau, Koutra, Akoglu Real Examples [Google Search Trends data] Reddit v Digg Blu-Ray v HD-DVD Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Faloutsos, Prakash, Chau, Koutra, Akoglu Outline Motivation Epidemics: what happens? (Theory) Action: Who to immunize? (Algorithms) Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Full Static Immunization Given: a graph A, virus prop. model and budget k; Find: k ‘best’ nodes for immunization (removal). ? ? k = 2 ? ? Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Faloutsos, Prakash, Chau, Koutra, Akoglu Outline Motivation Epidemics: what happens? (Theory) Action: Who to immunize? (Algorithms) Full Immunization (Static Graphs) Fractional Immunization Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Faloutsos, Prakash, Chau, Koutra, Akoglu 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’? Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Proposed vulnerability measure λ λ is the epidemic threshold “Safe” “Vulnerable” “Deadly” Increasing λ Increasing vulnerability Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

A1: “Eigen-Drop”: an ideal shield value Eigen-Drop(S) Δ λ = λ - λs 9 9 Δ 9 11 10 10 1 1 6 2 4 4 8 8 2 3 7 3 7 5 5 6 Original Graph Without {2, 6} Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

(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! Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Faloutsos, Prakash, Chau, Koutra, Akoglu 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(nk2+m) n = # nodes; m = # edges In Tong, Prakash+ ICDM 2010 Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Experiment: Immunization quality Log(fraction of infected nodes) PageRank Betweeness (shortest path) Degree Lower is better Acquaintance Eigs (=HITS) NetShield Time Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Faloutsos, Prakash, Chau, Koutra, Akoglu Outline Motivation Epidemics: what happens? (Theory) Action: Who to immunize? (Algorithms) Full Immunization (Static Graphs) Fractional Immunization Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Faloutsos, Prakash, Chau, Koutra, Akoglu Fractional Immunization of Networks B. Aditya Prakash, Lada Adamic, Theodore Iwashyna (M.D.), Hanghang Tong, Christos Faloutsos Under review Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Fractional Asymmetric Immunization Drug-resistant Bacteria (like XDR-TB) Hospital Another Hospital Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Fractional Asymmetric Immunization Drug-resistant Bacteria (like XDR-TB) Hospital Another Hospital Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Fractional Asymmetric Immunization Problem: Given k units of disinfectant, how to distribute them to maximize hospitals saved? Hospital Another Hospital Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Our Algorithm “SMART-ALLOC” ~6x fewer! [US-MEDICARE NETWORK 2005] Each circle is a hospital, ~3000 hospitals More than 30,000 patients transferred CURRENT PRACTICE SMART-ALLOC Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Faloutsos, Prakash, Chau, Koutra, Akoglu Running Time Wall-Clock Time > 1 week ≈ > 30,000x speed-up! Lower is better 14 secs Simulations SMART-ALLOC Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Faloutsos, Prakash, Chau, Koutra, Akoglu Experiments Lower is better SECOND-LIFE PENN-NETWORK ~5 x ~2.5 x K = 200 K = 2000 Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Faloutsos, Prakash, Chau, Koutra, Akoglu Acknowledgements Funding Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Faloutsos, Prakash, Chau, Koutra, Akoglu References 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.) 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 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 Winner-takes-all: Competing Viruses or Ideas on fair-play networks (B. Aditya Prakash, Alex Beutel, Roni Rosenfeld, Christos Faloutsos) – In WWW 2012, Lyon On the Vulnerability of Large Graphs (Hanghang Tong, B. Aditya Prakash, Tina Eliassi- Rad and Christos Faloutsos) – In IEEE ICDM 2010, Sydney, Australia Fractional Immunization of Networks (B. Aditya Prakash, Lada Adamic, Theodore Iwashyna, Hanghang Tong, Christos Faloutsos) - Under Submission Rise and Fall Patterns of Information Diffusion: Model and Implications (Yasuko Matsubara, Yasushi Sakurai, B. Aditya Prakash, Lei Li, Christos Faloutsos) - Under Submission http://www.cs.cmu.edu/~badityap/ Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu

Propagation on Large Networks B. Aditya Prakash Christos Faloutsos Analysis Policy/Action Data Taiwan'12 Faloutsos, Prakash, Chau, Koutra, Akoglu