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CMU SCS Mining Billion-Node Graphs - Patterns and Algorithms Christos Faloutsos CMU

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CMU SCS Thank you! Henry Kautz Melissa Singkhamsack RocData'12C. Faloutsos (CMU) 2

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CMU SCS C. Faloutsos (CMU) 3 Resource Open source system for mining huge graphs: PEGASUS project (PEta GrAph mining System) code and papers RocData'12

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CMU SCS C. Faloutsos (CMU) 4 Roadmap Introduction – Motivation Problem#1: Patterns in graphs Problem#2: Tools Problem#3: Scalability Conclusions RocData'12

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CMU SCS C. Faloutsos (CMU) 5 Graphs - why should we care? Internet Map [lumeta.com] Food Web [Martinez ’91] >$10B revenue >0.5B users RocData'12

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CMU SCS C. Faloutsos (CMU) 6 Graphs - why should we care? IR: bi-partite graphs (doc-terms) web: hyper-text graph... and more: D1D1 DNDN T1T1 TMTM... RocData'12

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CMU SCS C. Faloutsos (CMU) 7 Graphs - why should we care? ‘viral’ marketing web-log (‘blog’) news propagation computer network security: /IP traffic and anomaly detection.... Subject-verb-object -> graph Many-to-many db relationship -> graph RocData'12

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CMU SCS C. Faloutsos (CMU) 8 Outline Introduction – Motivation Problem#1: Patterns in graphs –Static graphs –Weighted graphs –Time evolving graphs Problem#2: Tools Problem#3: Scalability Conclusions RocData'12

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CMU SCS C. Faloutsos (CMU) 9 Problem #1 - network and graph mining What does the Internet look like? What does FaceBook look like? What is ‘normal’/‘abnormal’? which patterns/laws hold? RocData'12

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CMU SCS C. Faloutsos (CMU) 10 Problem #1 - network and graph mining What does the Internet look like? What does FaceBook look like? What is ‘normal’/‘abnormal’? which patterns/laws hold? –To spot anomalies (rarities), we have to discover patterns RocData'12

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CMU SCS C. Faloutsos (CMU) 11 Problem #1 - network and graph mining What does the Internet look like? What does FaceBook look like? What is ‘normal’/‘abnormal’? which patterns/laws hold? –To spot anomalies (rarities), we have to discover patterns –Large datasets reveal patterns/anomalies that may be invisible otherwise… RocData'12

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CMU SCS C. Faloutsos (CMU) 12 Graph mining Are real graphs random? RocData'12

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CMU SCS C. Faloutsos (CMU) 13 Laws and patterns Are real graphs random? A: NO!! –Diameter –in- and out- degree distributions –other (surprising) patterns So, let’s look at the data RocData'12

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CMU SCS C. Faloutsos (CMU) 14 Solution# S.1 Power law in the degree distribution [SIGCOMM99] log(rank) log(degree) internet domains att.com ibm.com RocData'12

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CMU SCS C. Faloutsos (CMU) 15 Solution# S.1 Power law in the degree distribution [SIGCOMM99] log(rank) log(degree) internet domains att.com ibm.com RocData'12

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CMU SCS C. Faloutsos (CMU) 16 Solution# S.2: Eigen Exponent E A2: power law in the eigenvalues of the adjacency matrix E = Exponent = slope Eigenvalue Rank of decreasing eigenvalue May 2001 RocData'12

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CMU SCS C. Faloutsos (CMU) 17 Solution# S.2: Eigen Exponent E [Mihail, Papadimitriou ’02]: slope is ½ of rank exponent E = Exponent = slope Eigenvalue Rank of decreasing eigenvalue May 2001 RocData'12

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CMU SCS C. Faloutsos (CMU) 18 But: How about graphs from other domains? RocData'12

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CMU SCS C. Faloutsos (CMU) 19 More power laws: web hit counts [w/ A. Montgomery] Web Site Traffic in-degree (log scale) Count (log scale) Zipf users sites ``ebay’’ RocData'12

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CMU SCS C. Faloutsos (CMU) 20 epinions.com who-trusts-whom [Richardson + Domingos, KDD 2001] (out) degree count trusts-2000-people user RocData'12

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CMU SCS And numerous more # of sexual contacts Income [Pareto] –’80-20 distribution’ Duration of downloads [Bestavros+] Duration of UNIX jobs (‘mice and elephants’) Size of files of a user … ‘Black swans’ RocData'12C. Faloutsos (CMU) 21

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CMU SCS C. Faloutsos (CMU) 22 Roadmap Introduction – Motivation Problem#1: Patterns in graphs –Static graphs degree, diameter, eigen, triangles cliques –Weighted graphs –Time evolving graphs Problem#2: Tools RocData'12

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CMU SCS C. Faloutsos (CMU) 23 Solution# S.3: Triangle ‘Laws’ Real social networks have a lot of triangles RocData'12

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CMU SCS C. Faloutsos (CMU) 24 Solution# S.3: Triangle ‘Laws’ Real social networks have a lot of triangles –Friends of friends are friends Any patterns? RocData'12

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CMU SCS C. Faloutsos (CMU) 25 Triangle Law: #S.3 [Tsourakakis ICDM 2008] ASNHEP-TH Epinions X-axis: # of participating triangles Y: count (~ pdf) RocData'12

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CMU SCS C. Faloutsos (CMU) 26 Triangle Law: #S.3 [Tsourakakis ICDM 2008] ASNHEP-TH Epinions RocData'12 X-axis: # of participating triangles Y: count (~ pdf)

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CMU SCS C. Faloutsos (CMU) 27 Triangle Law: #S.4 [Tsourakakis ICDM 2008] SNReuters Epinions X-axis: degree Y-axis: mean # triangles n friends -> ~n 1.6 triangles RocData'12

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CMU SCS C. Faloutsos (CMU) 28 Triangle Law: Computations [Tsourakakis ICDM 2008] But: triangles are expensive to compute (3-way join; several approx. algos) Q: Can we do that quickly? details RocData'12

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CMU SCS C. Faloutsos (CMU) 29 Triangle Law: Computations [Tsourakakis ICDM 2008] But: triangles are expensive to compute (3-way join; several approx. algos) Q: Can we do that quickly? A: Yes! #triangles = 1/6 Sum ( i 3 ) (and, because of skewness (S2), we only need the top few eigenvalues! details RocData'12

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CMU SCS C. Faloutsos (CMU) 30 Triangle Law: Computations [Tsourakakis ICDM 2008] 1000x+ speed-up, >90% accuracy details RocData'12

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CMU SCS Triangle counting for large graphs? Anomalous nodes in Twitter(~ 3 billion edges) [U Kang, Brendan Meeder, +, PAKDD’11] 31 RocData'12 31 C. Faloutsos (CMU)

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CMU SCS Triangle counting for large graphs? Anomalous nodes in Twitter(~ 3 billion edges) [U Kang, Brendan Meeder, +, PAKDD’11] 32 RocData'12 32 C. Faloutsos (CMU)

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CMU SCS Triangle counting for large graphs? Anomalous nodes in Twitter(~ 3 billion edges) [U Kang, Brendan Meeder, +, PAKDD’11] 33 RocData'12 33 C. Faloutsos (CMU)

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CMU SCS Triangle counting for large graphs? Anomalous nodes in Twitter(~ 3 billion edges) [U Kang, Brendan Meeder, +, PAKDD’11] 34 RocData'12 34 C. Faloutsos (CMU)

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CMU SCS EigenSpokes B. Aditya Prakash, Mukund Seshadri, Ashwin Sridharan, Sridhar Machiraju and Christos Faloutsos: EigenSpokes: Surprising Patterns and Scalable Community Chipping in Large Graphs, PAKDD 2010, Hyderabad, India, June C. Faloutsos (CMU) 35 RocData'12

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CMU SCS EigenSpokes Eigenvectors of adjacency matrix equivalent to singular vectors (symmetric, undirected graph) 36 C. Faloutsos (CMU)RocData'12

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CMU SCS EigenSpokes Eigenvectors of adjacency matrix equivalent to singular vectors (symmetric, undirected graph) 37 C. Faloutsos (CMU)RocData'12 N N details

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CMU SCS EigenSpokes Eigenvectors of adjacency matrix equivalent to singular vectors (symmetric, undirected graph) 38 C. Faloutsos (CMU)RocData'12 N N details

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CMU SCS EigenSpokes Eigenvectors of adjacency matrix equivalent to singular vectors (symmetric, undirected graph) 39 C. Faloutsos (CMU)RocData'12 N N details

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CMU SCS EigenSpokes Eigenvectors of adjacency matrix equivalent to singular vectors (symmetric, undirected graph) 40 C. Faloutsos (CMU)RocData'12 N N details

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CMU SCS EigenSpokes EE plot: Scatter plot of scores of u1 vs u2 One would expect –Many origin –A few scattered ~randomly C. Faloutsos (CMU) 41 u1 u2 RocData'12 1 st Principal component 2 nd Principal component

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CMU SCS EigenSpokes EE plot: Scatter plot of scores of u1 vs u2 One would expect –Many origin –A few scattered ~randomly C. Faloutsos (CMU) 42 u1 u2 90 o RocData'12

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CMU SCS EigenSpokes - pervasiveness Present in mobile social graph across time and space Patent citation graph 43 C. Faloutsos (CMU)RocData'12

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CMU SCS EigenSpokes - explanation Near-cliques, or near- bipartite-cores, loosely connected 44 C. Faloutsos (CMU)RocData'12

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CMU SCS EigenSpokes - explanation Near-cliques, or near- bipartite-cores, loosely connected 45 C. Faloutsos (CMU)RocData'12

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CMU SCS EigenSpokes - explanation Near-cliques, or near- bipartite-cores, loosely connected 46 C. Faloutsos (CMU)RocData'12

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CMU SCS EigenSpokes - explanation Near-cliques, or near- bipartite-cores, loosely connected So what? Extract nodes with high scores high connectivity Good “communities” spy plot of top 20 nodes 47 C. Faloutsos (CMU)RocData'12

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CMU SCS Bipartite Communities! magnified bipartite community patents from same inventor(s) `cut-and-paste’ bibliography! 48 C. Faloutsos (CMU)RocData'12

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CMU SCS (maybe, botnets?) Victim IPs? Botnet members? 49 C. Faloutsos (CMU)RocData'12

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CMU SCS C. Faloutsos (CMU) 50 Roadmap Introduction – Motivation Problem#1: Patterns in graphs –Static graphs degree, diameter, eigen, triangles cliques –Weighted graphs –Time evolving graphs Problem#2: Tools RocData'12

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CMU SCS C. Faloutsos (CMU) 51 Observations on weighted graphs? A: yes - even more ‘laws’! M. McGlohon, L. Akoglu, and C. Faloutsos Weighted Graphs and Disconnected Components: Patterns and a Generator. SIG-KDD 2008 RocData'12

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CMU SCS C. Faloutsos (CMU) 52 Observation W.1: Fortification Q: How do the weights of nodes relate to degree? RocData'12

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CMU SCS C. Faloutsos (CMU) 53 Observation W.1: Fortification More donors, more $ ? $10 $5 RocData'12 ‘Reagan’ ‘Clinton’ $7

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CMU SCS Edges (# donors) In-weights ($) C. Faloutsos (CMU) 54 Observation W.1: fortification: Snapshot Power Law Weight: super-linear on in-degree exponent ‘iw’: 1.01 < iw < 1.26 Orgs-Candidates e.g. John Kerry, $10M received, from 1K donors More donors, even more $ $10 $5 RocData'12

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CMU SCS C. Faloutsos (CMU) 55 Roadmap Introduction – Motivation Problem#1: Patterns in graphs –Static graphs –Weighted graphs –Time evolving graphs Problem#2: Tools … RocData'12

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CMU SCS C. Faloutsos (CMU) 56 Problem: Time evolution with Jure Leskovec (CMU -> Stanford) and Jon Kleinberg (Cornell – CMU) RocData'12

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CMU SCS C. Faloutsos (CMU) 57 T.1 Evolution of the Diameter Prior work on Power Law graphs hints at slowly growing diameter: –diameter ~ O(log N) –diameter ~ O(log log N) What is happening in real data? RocData'12

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CMU SCS C. Faloutsos (CMU) 58 T.1 Evolution of the Diameter Prior work on Power Law graphs hints at slowly growing diameter: –diameter ~ O(log N) –diameter ~ O(log log N) What is happening in real data? Diameter shrinks over time RocData'12

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CMU SCS C. Faloutsos (CMU) 59 T.1 Diameter – “Patents” Patent citation network 25 years of –2.9 M nodes –16.5 M edges time [years] diameter RocData'12

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CMU SCS C. Faloutsos (CMU) 60 T.2 Temporal Evolution of the Graphs N(t) … nodes at time t E(t) … edges at time t Suppose that N(t+1) = 2 * N(t) Q: what is your guess for E(t+1) =? 2 * E(t) RocData'12

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CMU SCS C. Faloutsos (CMU) 61 T.2 Temporal Evolution of the Graphs N(t) … nodes at time t E(t) … edges at time t Suppose that N(t+1) = 2 * N(t) Q: what is your guess for E(t+1) =? 2 * E(t) A: over-doubled! –But obeying the ``Densification Power Law’’ RocData'12

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CMU SCS C. Faloutsos (CMU) 62 T.2 Densification – Patent Citations Citations among patents –2.9 M nodes –16.5 M edges Each year is a datapoint N(t) E(t) 1.66 RocData'12

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CMU SCS C. Faloutsos (CMU) 63 Roadmap Introduction – Motivation Problem#1: Patterns in graphs –Static graphs –Weighted graphs –Time evolving graphs Problem#2: Tools … RocData'12

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CMU SCS C. Faloutsos (CMU) 64 T.3 : popularity over time Post popularity drops-off – exponentially? lag: days after post # in links 1 + lag RocData'12

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CMU SCS C. Faloutsos (CMU) 65 T.3 : popularity over time Post popularity drops-off – exponentially? POWER LAW! Exponent? # in links (log) days after post (log) RocData'12

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CMU SCS C. Faloutsos (CMU) 66 T.3 : popularity over time Post popularity drops-off – exponentially? POWER LAW! Exponent? -1.6 close to -1.5: Barabasi’s stack model and like the zero-crossings of a random walk # in links (log) -1.6 days after post (log) RocData'12

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CMU SCS RocData'12C. Faloutsos (CMU) slope J. G. Oliveira & A.-L. Barabási Human Dynamics: The Correspondence Patterns of Darwin and Einstein. Nature 437, 1251 (2005). [PDF]PDF Response time (log) Prob(RT > x) (log)

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CMU SCS T.4: duration of phonecalls Surprising Patterns for the Call Duration Distribution of Mobile Phone Users Pedro O. S. Vaz de Melo, Leman Akoglu, Christos Faloutsos, Antonio A. F. Loureiro PKDD 2010 RocData'12C. Faloutsos (CMU) 68

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CMU SCS Probably, power law (?) RocData'12C. Faloutsos (CMU) 69 ??

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CMU SCS No Power Law! RocData'12C. Faloutsos (CMU) 70

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CMU SCS ‘TLaC: Lazy Contractor’ The longer a task (phonecall) has taken, The even longer it will take RocData'12C. Faloutsos (CMU) 71 Odds ratio= Casualties(

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CMU SCS 72 Data Description Data from a private mobile operator of a large city 4 months of data 3.1 million users more than 1 billion phone records Over 96% of ‘talkative’ users obeyed a TLAC distribution (‘talkative’: >30 calls) RocData'12C. Faloutsos (CMU)

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CMU SCS Outliers: RocData'12C. Faloutsos (CMU) 73

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CMU SCS C. Faloutsos (CMU) 74 Roadmap Introduction – Motivation Problem#1: Patterns in graphs Problem#2: Tools –Belief Propagation –Tensors –Spike analysis Problem#3: Scalability Conclusions RocData'12

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CMU SCS RocData'12C. Faloutsos (CMU) 75 E-bay Fraud detection w/ Polo Chau & Shashank Pandit, CMU [www’07]

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CMU SCS RocData'12C. Faloutsos (CMU) 76 E-bay Fraud detection

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CMU SCS RocData'12C. Faloutsos (CMU) 77 E-bay Fraud detection

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CMU SCS RocData'12C. Faloutsos (CMU) 78 E-bay Fraud detection - NetProbe

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CMU SCS RocData'12C. Faloutsos (CMU) 79 E-bay Fraud detection - NetProbe FAH F99% A H49% Compatibility matrix heterophily

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CMU SCS Popular press And less desirable attention: from ‘Belgium police’ (‘copy of your code?’) RocData'12C. Faloutsos (CMU) 80

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CMU SCS C. Faloutsos (CMU) 81 Roadmap Introduction – Motivation Problem#1: Patterns in graphs Problem#2: Tools –Belief Propagation –Tensors –Spike analysis Problem#3: Scalability Conclusions RocData'12

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CMU SCS GigaTensor: Scaling Tensor Analysis Up By 100 Times – Algorithms and Discoveries U Kang Christos Faloutsos KDD’12 Evangelos Papalexakis Abhay Harpale RocData'12 82 C. Faloutsos (CMU)

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CMU SCS Background: Tensor Tensors (=multi-dimensional arrays) are everywhere –Hyperlinks &anchor text [Kolda+,05] URL 1 URL 2 Anchor Text Java C++ C# RocData'12 83 C. Faloutsos (CMU)

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CMU SCS Background: Tensor Tensors (=multi-dimensional arrays) are everywhere –Sensor stream (time, location, type) –Predicates (subject, verb, object) in knowledge base “Barrack Obama is the president of U.S.” “Eric Clapton plays guitar” (26M) (48M) NELL (Never Ending Language Learner) data Nonzeros =144M RocData'12 84 C. Faloutsos (CMU)

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CMU SCS Problem Definition How to decompose a billion-scale tensor? –Corresponds to SVD in 2D case RocData'12 85 C. Faloutsos (CMU)

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CMU SCS Problem Definition Q1: Dominant concepts/topics? Q2: Find synonyms to a given noun phrase? (and how to scale up: |data| > RAM) (26M) (48M) NELL (Never Ending Language Learner) data Nonzeros =144M RocData'12 86 C. Faloutsos (CMU)

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CMU SCS Experiments GigaTensor solves 100x larger problem Number of nonzero = I / 50 (J) (I) (K) GigaTensor Tensor Toolbox Out of Memory 100x RocData'12 87 C. Faloutsos (CMU)

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CMU SCS A1: Concept Discovery Concept Discovery in Knowledge Base RocData'12 88 C. Faloutsos (CMU)

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CMU SCS A1: Concept Discovery RocData'12 89 C. Faloutsos (CMU)

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CMU SCS A2: Synonym Discovery RocData'12 90 C. Faloutsos (CMU)

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CMU SCS C. Faloutsos (CMU) 91 Roadmap Introduction – Motivation Problem#1: Patterns in graphs Problem#2: Tools –Belief propagation –Tensors –Spike analysis Problem#3: Scalability -PEGASUS Conclusions RocData'12

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CMU SCS 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 C. Faloutsos (CMU)RocData'12

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CMU SCS Rise and fall patterns in social media 93 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] C. Faloutsos (CMU)RocData'12

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CMU SCS Rise and fall patterns in social media 94 Answer: YES! We can represent all patterns by single model C. Faloutsos (CMU)RocData'12 In Matsubara+ SIGKDD 2012

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CMU SCS 95 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) Time n=0Time n=n b β C. Faloutsos (CMU)RocData'12 Infectiveness of a blog-post at age n: -Strength of infection (quality of news) -Decay function Time n=n b +1

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CMU SCS Un-informed bloggers (uninformed about rumor) -2. External shock at time n b (e.g, breaking news) -3. Infection (word-of-mouth) Time n=0Time n=n b β C. Faloutsos (CMU)RocData'12 Infectiveness of a blog-post at age n: -Strength of infection (quality of news) -Decay function Time n=n b +1 Main idea - SpikeM

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CMU SCS SpikeM - with periodicity Full equation of SpikeM 97 Periodicity noon Peak 3am Dip 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 C. Faloutsos (CMU)RocData'12

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CMU SCS Details Analysis – exponential rise and power-raw fall 98 Lin-log Log-log Rise-part SI -> exponential SpikeM -> exponential Rise-part SI -> exponential SpikeM -> exponential C. Faloutsos (CMU)RocData'12

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CMU SCS Details Analysis – exponential rise and power-raw fall 99 Lin-log Log-log Fall-part SI -> exponential SpikeM -> power law Fall-part SI -> exponential SpikeM -> power law C. Faloutsos (CMU)RocData'12

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CMU SCS Tail-part forecasts 100 SpikeM can capture tail part C. Faloutsos (CMU)RocData'12

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CMU SCS “What-if” forecasting 101 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 C. Faloutsos (CMU)RocData'12

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CMU SCS “What-if” forecasting 102 –SpikeM can forecast not only tail-part, but also rise-part ! SpikeM can forecast shape of upcoming spikes (1) First spike (2) Release date (3) Two weeks before release C. Faloutsos (CMU)RocData'12

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CMU SCS C. Faloutsos (CMU) 103 Roadmap Introduction – Motivation Problem#1: Patterns in graphs Problem#2: Tools –Belief propagation –Spike analysis –Tensors Problem#3: Scalability -PEGASUS Conclusions RocData'12

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CMU SCS RocData'12C. Faloutsos (CMU) 104 Scalability Google: > 450,000 processors in clusters of ~2000 processors each [ Barroso, Dean, Hölzle, “Web Search for a Planet: The Google Cluster Architecture” IEEE Micro 2003 ] Yahoo: 5Pb of data [Fayyad, KDD’07] Problem: machine failures, on a daily basis How to parallelize data mining tasks, then? A: map/reduce – hadoop (open-source clone)

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CMU SCS C. Faloutsos (CMU) 105 CentralizedHadoop/PEG ASUS Degree Distr. old Pagerank old Diameter/ANF old HERE Conn. Comp old HERE TrianglesdoneHERE Visualizationstarted Roadmap – Algorithms & results RocData'12

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CMU SCS HADI for diameter estimation Radius Plots for Mining Tera-byte Scale Graphs U Kang, Charalampos Tsourakakis, Ana Paula Appel, Christos Faloutsos, Jure Leskovec, SDM’10 Naively: diameter needs O(N**2) space and up to O(N**3) time – prohibitive (N~1B) Our HADI: linear on E (~10B) –Near-linear scalability wrt # machines –Several optimizations -> 5x faster C. Faloutsos (CMU) 106 RocData'12

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CMU SCS ???? 19+ [Barabasi+] 107 C. Faloutsos (CMU) Radius Count RocData'12 ~1999, ~1M nodes

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CMU SCS YahooWeb graph (120Gb, 1.4B nodes, 6.6 B edges) Largest publicly available graph ever studied. ???? 19+ [Barabasi+] 108 C. Faloutsos (CMU) Radius Count RocData'12 ?? ~1999, ~1M nodes

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CMU SCS YahooWeb graph (120Gb, 1.4B nodes, 6.6 B edges) Largest publicly available graph ever studied. ???? 19+? [Barabasi+] 109 C. Faloutsos (CMU) Radius Count RocData'12 14 (dir.) ~7 (undir.)

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CMU SCS YahooWeb graph (120Gb, 1.4B nodes, 6.6 B edges) 7 degrees of separation (!) Diameter: shrunk ???? 19+? [Barabasi+] 110 C. Faloutsos (CMU) Radius Count RocData'12 14 (dir.) ~7 (undir.)

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CMU SCS YahooWeb graph (120Gb, 1.4B nodes, 6.6 B edges) Q: Shape? ???? 111 C. Faloutsos (CMU) Radius Count RocData'12 ~7 (undir.)

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CMU SCS 112 C. Faloutsos (CMU) YahooWeb graph (120Gb, 1.4B nodes, 6.6 B edges) effective diameter: surprisingly small. Multi-modality (?!) RocData'12

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CMU SCS Radius Plot of GCC of YahooWeb. 113 C. Faloutsos (CMU)RocData'12

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CMU SCS 114 C. Faloutsos (CMU) YahooWeb graph (120Gb, 1.4B nodes, 6.6 B edges) effective diameter: surprisingly small. Multi-modality: probably mixture of cores. RocData'12

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CMU SCS 115 C. Faloutsos (CMU) YahooWeb graph (120Gb, 1.4B nodes, 6.6 B edges) effective diameter: surprisingly small. Multi-modality: probably mixture of cores. RocData'12 EN ~7 Conjecture: DE BR

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CMU SCS 116 C. Faloutsos (CMU) YahooWeb graph (120Gb, 1.4B nodes, 6.6 B edges) effective diameter: surprisingly small. Multi-modality: probably mixture of cores. RocData'12 ~7 Conjecture:

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CMU SCS Running time - Kronecker and Erdos-Renyi Graphs with billions edges. details

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CMU SCS C. Faloutsos (CMU) 118 CentralizedHadoop/PEG ASUS Degree Distr. old Pagerank old Diameter/ANF old HERE Conn. Comp old HERE Triangles HERE Visualizationstarted Roadmap – Algorithms & results RocData'12

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CMU SCS Generalized Iterated Matrix Vector Multiplication (GIMV) C. Faloutsos (CMU) 119 PEGASUS: A Peta-Scale Graph Mining System - Implementation and ObservationsSystem - Implementation and Observations. U Kang, Charalampos E. Tsourakakis, and Christos Faloutsos. (ICDM) 2009, Miami, Florida, USA. Best Application Paper (runner-up).ICDM RocData'12

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CMU SCS Generalized Iterated Matrix Vector Multiplication (GIMV) C. Faloutsos (CMU) 120 PageRank proximity (RWR) Diameter Connected components (eigenvectors, Belief Prop. … ) Matrix – vector Multiplication (iterated) RocData'12 details

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CMU SCS 121 Example: GIM-V At Work Connected Components – 4 observations: Size Count C. Faloutsos (CMU) RocData'12

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CMU SCS 122 Example: GIM-V At Work Connected Components Size Count C. Faloutsos (CMU) RocData'12 1) 10K x larger than next

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CMU SCS 123 Example: GIM-V At Work Connected Components Size Count C. Faloutsos (CMU) RocData'12 2) ~0.7B singleton nodes

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CMU SCS 124 Example: GIM-V At Work Connected Components Size Count C. Faloutsos (CMU) RocData'12 3) SLOPE!

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CMU SCS 125 Example: GIM-V At Work Connected Components Size Count 300-size cmpt X 500. Why? 1100-size cmpt X 65. Why? C. Faloutsos (CMU) RocData'12 4) Spikes!

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CMU SCS 126 Example: GIM-V At Work Connected Components Size Count suspicious financial-advice sites (not existing now) C. Faloutsos (CMU) RocData'12

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CMU SCS 127 GIM-V At Work Connected Components over Time LinkedIn: 7.5M nodes and 58M edges Stable tail slope after the gelling point C. Faloutsos (CMU)RocData'12

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CMU SCS C. Faloutsos (CMU) 128 Roadmap Introduction – Motivation Problem#1: Patterns in graphs Problem#2: Tools Problem#3: Scalability Conclusions RocData'12

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CMU SCS C. Faloutsos (CMU) 129 OVERALL CONCLUSIONS – low level: Several new patterns (fortification, triangle-laws, conn. components, etc) New tools: –belief propagation, gigaTensor, etc Scalability: PEGASUS / hadoop RocData'12

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CMU SCS C. Faloutsos (CMU) 130 OVERALL CONCLUSIONS – high level BIG DATA: Large datasets reveal patterns/outliers that are invisible otherwise RocData'12

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CMU SCS C. Faloutsos (CMU) 131 References Leman Akoglu, Christos Faloutsos: RTG: A Recursive Realistic Graph Generator Using Random Typing. ECML/PKDD (1) 2009: Deepayan Chakrabarti, Christos Faloutsos: Graph mining: Laws, generators, and algorithms. ACM Comput. Surv. 38(1): (2006) RocData'12

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CMU SCS C. Faloutsos (CMU) 132 References Deepayan Chakrabarti, Yang Wang, Chenxi Wang, Jure Leskovec, Christos Faloutsos: Epidemic thresholds in real networks. ACM Trans. Inf. Syst. Secur. 10(4): (2008) Deepayan Chakrabarti, Jure Leskovec, Christos Faloutsos, Samuel Madden, Carlos Guestrin, Michalis Faloutsos: Information Survival Threshold in Sensor and P2P Networks. INFOCOM 2007: RocData'12

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CMU SCS C. Faloutsos (CMU) 133 References Christos Faloutsos, Tamara G. Kolda, Jimeng Sun: Mining large graphs and streams using matrix and tensor tools. Tutorial, SIGMOD Conference 2007: 1174 RocData'12

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CMU SCS C. Faloutsos (CMU) 134 References T. G. Kolda and J. Sun. Scalable Tensor Decompositions for Multi-aspect Data Mining. In: ICDM 2008, pp , December RocData'12

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CMU SCS C. Faloutsos (CMU) 135 References Jure Leskovec, Jon Kleinberg and Christos Faloutsos Graphs over Time: Densification Laws, Shrinking Diameters and Possible Explanations, KDD 2005 (Best Research paper award). Jure Leskovec, Deepayan Chakrabarti, Jon M. Kleinberg, Christos Faloutsos: Realistic, Mathematically Tractable Graph Generation and Evolution, Using Kronecker Multiplication. PKDD 2005: RocData'12

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CMU SCS C. Faloutsos (CMU) 136 References Jimeng Sun, Yinglian Xie, Hui Zhang, Christos Faloutsos. Less is More: Compact Matrix Decomposition for Large Sparse Graphs, SDM, Minneapolis, Minnesota, Apr Jimeng Sun, Spiros Papadimitriou, Philip S. Yu, and Christos Faloutsos, GraphScope: Parameter- free Mining of Large Time-evolving Graphs ACM SIGKDD Conference, San Jose, CA, August 2007 RocData'12

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CMU SCS References Jimeng Sun, Dacheng Tao, Christos Faloutsos: Beyond streams and graphs: dynamic tensor analysis. KDD 2006: RocData'12C. Faloutsos (CMU) 137

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CMU SCS C. Faloutsos (CMU) 138 References Hanghang Tong, Christos Faloutsos, and Jia-Yu Pan, Fast Random Walk with Restart and Its Applications, ICDM 2006, Hong Kong. Hanghang Tong, Christos Faloutsos, Center-Piece Subgraphs: Problem Definition and Fast Solutions, KDD 2006, Philadelphia, PA RocData'12

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CMU SCS C. Faloutsos (CMU) 139 References Hanghang Tong, Christos Faloutsos, Brian Gallagher, Tina Eliassi-Rad: Fast best-effort pattern matching in large attributed graphs. KDD 2007: RocData'12

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CMU SCS C. Faloutsos (CMU) 140 Project info RocData'12 Thanks to: NSF IIS , IIS , CTA-INARC ; Yahoo (M45), LLNL, IBM, SPRINT, Google, INTEL, HP, iLab

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CMU SCS C. Faloutsos (CMU) 141 Cast Akoglu, Leman Chau, Polo Kang, U McGlohon, Mary Tong, Hanghang Prakash, Aditya RocData'12 Koutra, Danae Beutel, Alex Papalexakis, Vagelis

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CMU SCS C. Faloutsos (CMU) 142 OVERALL CONCLUSIONS – high level BIG DATA: Large datasets reveal patterns/outliers that are invisible otherwise RocData'12

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