CMU SCS Mining BillionNode Graphs Patterns and Algorithms

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

CMU SCS Thank you! • Vipin Kumar • Arindam Banerjee • Jennifer Olk UMN CRAY, 2012 C. Faloutsos (CMU) 2

CMU SCS Resource Open source system for mining huge graphs: PEGASUS project (PEta Gr. Aph mining System) • www. cs. cmu. edu/~pegasus • code and papers UMN CRAY, 2012 C. Faloutsos (CMU) 3

CMU SCS Roadmap • • • Introduction – Motivation Problem#1: Patterns in graphs Problem#2: Tools Problem#3: Scalability Conclusions UMN CRAY, 2012 C. Faloutsos (CMU) 4

CMU SCS Graphs - why should we care? Food Web [Martinez ’ 91] >$10 B revenue >0. 5 B users Internet Map [lumeta. com] UMN CRAY, 2012 C. Faloutsos (CMU) 5

CMU SCS Graphs - why should we care? • IR: bi-partite graphs (doc-terms) D 1 . . . DN TM • web: hyper-text graph • . . . and more: UMN CRAY, 2012 C. Faloutsos (CMU) T 1 6

CMU SCS Graphs - why should we care? • ‘viral’ marketing • web-log (‘blog’) news propagation • computer network security: email/IP traffic and anomaly detection • . . • Subject-verb-object -> graph • Many-to-many db relationship -> graph UMN CRAY, 2012 C. Faloutsos (CMU) 7

CMU SCS Outline • Introduction – Motivation • Problem#1: Patterns in graphs – Static graphs – Weighted graphs – Time evolving graphs • Problem#2: Tools • Problem#3: Scalability • Conclusions UMN CRAY, 2012 C. Faloutsos (CMU) 8

CMU SCS Problem #1 - network and graph mining • What does the Internet look like? • What does Face. Book like? • What is ‘normal’/‘abnormal’? • which patterns/laws hold? UMN CRAY, 2012 C. Faloutsos (CMU) 9

CMU SCS Problem #1 - network and graph mining • What does the Internet look like? • What does Face. Book like? • What is ‘normal’/‘abnormal’? • which patterns/laws hold? – To spot anomalies (rarities), we have to discover patterns UMN CRAY, 2012 C. Faloutsos (CMU) 10

CMU SCS Problem #1 - network and graph mining • What does the Internet look like? • What does Face. Book 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… UMN CRAY, 2012 C. Faloutsos (CMU) 11

CMU SCS Graph mining • Are real graphs random? UMN CRAY, 2012 C. Faloutsos (CMU) 12

CMU SCS 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 UMN CRAY, 2012 C. Faloutsos (CMU) 13

CMU SCS Solution# S. 1 • Power law in the degree distribution [SIGCOMM 99] internet domains att. com log(degree) ibm. com log(rank) UMN CRAY, 2012 C. Faloutsos (CMU) 14

CMU SCS Solution# S. 1 • Power law in the degree distribution [SIGCOMM 99] internet domains att. com log(degree) ibm. com -0. 82 log(rank) UMN CRAY, 2012 C. Faloutsos (CMU) 15

CMU SCS Solution# S. 2: Eigen Exponent E Eigenvalue Exponent = slope E = -0. 48 May 2001 Rank of decreasing eigenvalue • A 2: power law in the eigenvalues of the adjacency matrix UMN CRAY, 2012 C. Faloutsos (CMU) 16

CMU SCS Solution# S. 2: Eigen Exponent E Eigenvalue Exponent = slope E = -0. 48 May 2001 Rank of decreasing eigenvalue • [Mihail, Papadimitriou ’ 02]: slope is ½ of rank exponent UMN CRAY, 2012 C. Faloutsos (CMU) 17

CMU SCS But: How about graphs from other domains? UMN CRAY, 2012 C. Faloutsos (CMU) 18

CMU SCS More power laws: • web hit counts [w/ A. Montgomery] Web Site Traffic Count (log scale) Zipf ``ebay’’ users sites in-degree (log scale) UMN CRAY, 2012 C. Faloutsos (CMU) 19

CMU SCS epinions. com • who-trusts-whom [Richardson + Domingos, KDD 2001] count trusts-2000 -people user (out) degree UMN CRAY, 2012 C. Faloutsos (CMU) 20

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’ UMN CRAY, 2012 C. Faloutsos (CMU) 21

CMU SCS Roadmap • Introduction – Motivation • Problem#1: Patterns in graphs – Static graphs • degree, diameter, eigen, • triangles • cliques – Weighted graphs – Time evolving graphs • Problem#2: Tools UMN CRAY, 2012 C. Faloutsos (CMU) 22

CMU SCS Solution# S. 3: Triangle ‘Laws’ • Real social networks have a lot of triangles UMN CRAY, 2012 C. Faloutsos (CMU) 23

CMU SCS Solution# S. 3: Triangle ‘Laws’ • Real social networks have a lot of triangles – Friends of friends are friends • Any patterns? UMN CRAY, 2012 C. Faloutsos (CMU) 24

CMU SCS Triangle Law: #S. 3 [Tsourakakis ICDM 2008] HEP-TH ASN Epinions UMN CRAY, 2012 X-axis: # of participating triangles Y: count (~ pdf) C. Faloutsos (CMU) 25

CMU SCS Triangle Law: #S. 3 [Tsourakakis ICDM 2008] HEP-TH ASN Epinions UMN CRAY, 2012 X-axis: # of participating triangles Y: count (~ pdf) C. Faloutsos (CMU) 26

CMU SCS Triangle Law: #S. 4 [Tsourakakis ICDM 2008] Reuters SN X-axis: degree Y-axis: mean # triangles n friends -> ~n 1. 6 triangles Epinions UMN CRAY, 2012 C. Faloutsos (CMU) 27

CMU SCS details Triangle Law: Computations [Tsourakakis ICDM 2008] But: triangles are expensive to compute (3 -way join; several approx. algos) Q: Can we do that quickly? UMN CRAY, 2012 C. Faloutsos (CMU) 28

CMU SCS details 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 ( li 3 ) (and, because of skewness (S 2) , we only need the top few eigenvalues! UMN CRAY, 2012 C. Faloutsos (CMU) 29

CMU SCS details Triangle Law: Computations [Tsourakakis ICDM 2008] UMN CRAY, 2012 1000 x+ speed-up, >90% accuracy C. Faloutsos (CMU) 30

CMU SCS Triangle counting for large graphs? Anomalous nodes in Twitter(~ 3 billion edges) [U Kang, Brendan Meeder, +, PAKDD’ 11] UMN CRAY, 2012 C. Faloutsos (CMU) 31

CMU SCS Triangle counting for large graphs? Anomalous nodes in Twitter(~ 3 billion edges) [U Kang, Brendan Meeder, +, PAKDD’ 11] UMN CRAY, 2012 C. Faloutsos (CMU) 32

CMU SCS Triangle counting for large graphs? Anomalous nodes in Twitter(~ 3 billion edges) [U Kang, Brendan Meeder, +, PAKDD’ 11] UMN CRAY, 2012 C. Faloutsos (CMU) 33

CMU SCS Triangle counting for large graphs? Anomalous nodes in Twitter(~ 3 billion edges) [U Kang, Brendan Meeder, +, PAKDD’ 11] UMN CRAY, 2012 C. Faloutsos (CMU) 34

CMU SCS Eigen. Spokes B. Aditya Prakash, Mukund Seshadri, Ashwin Sridharan, Sridhar Machiraju and Christos Faloutsos: Eigen. Spokes: Surprising Patterns and Scalable Community Chipping in Large Graphs, PAKDD 2010, Hyderabad, India, 21 -24 June 2010. UMN CRAY, 2012 C. Faloutsos (CMU) 35

CMU SCS Eigen. Spokes • Eigenvectors of adjacency matrix § equivalent to singular vectors (symmetric, undirected graph) UMN CRAY, 2012 C. Faloutsos (CMU) 36

CMU SCS Eigen. Spokes details • Eigenvectors of adjacency matrix § equivalent to singular vectors (symmetric, undirected graph) N N UMN CRAY, 2012 C. Faloutsos (CMU) 37

CMU SCS Eigen. Spokes details • Eigenvectors of adjacency matrix § equivalent to singular vectors (symmetric, undirected graph) N N UMN CRAY, 2012 C. Faloutsos (CMU) 38

CMU SCS Eigen. Spokes details • Eigenvectors of adjacency matrix § equivalent to singular vectors (symmetric, undirected graph) N N UMN CRAY, 2012 C. Faloutsos (CMU) 39

CMU SCS Eigen. Spokes details • Eigenvectors of adjacency matrix § equivalent to singular vectors (symmetric, undirected graph) N N UMN CRAY, 2012 C. Faloutsos (CMU) 40

CMU SCS Eigen. Spokes 2 nd Principal component u 2 • EE plot: • Scatter plot of scores of u 1 vs u 2 • One would expect – Many points @ origin – A few scattered ~randomly UMN CRAY, 2012 C. Faloutsos (CMU) u 1 1 st Principal component 41

CMU SCS Eigen. Spokes • EE plot: • Scatter plot of scores of u 1 vs u 2 • One would expect – Many points @ origin – A few scattered ~randomly UMN CRAY, 2012 C. Faloutsos (CMU) u 2 90 o u 1 42

CMU SCS Eigen. Spokes - pervasiveness • Present in mobile social graph § across time and space • Patent citation graph UMN CRAY, 2012 C. Faloutsos (CMU) 43

CMU SCS Eigen. Spokes - explanation Near-cliques, or nearbipartite-cores, loosely connected UMN CRAY, 2012 C. Faloutsos (CMU) 44

CMU SCS Eigen. Spokes - explanation Near-cliques, or nearbipartite-cores, loosely connected UMN CRAY, 2012 C. Faloutsos (CMU) 45

CMU SCS Eigen. Spokes - explanation Near-cliques, or nearbipartite-cores, loosely connected UMN CRAY, 2012 C. Faloutsos (CMU) 46

CMU SCS Eigen. Spokes - explanation Near-cliques, or nearbipartite-cores, loosely connected spy plot of top 20 nodes So what? § Extract nodes with high scores § high connectivity § Good “communities” UMN CRAY, 2012 C. Faloutsos (CMU) 47

CMU SCS Bipartite Communities! patents from same inventor(s) `cut-and-paste’ bibliography! magnified bipartite community UMN CRAY, 2012 C. Faloutsos (CMU) 48

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

CMU SCS Roadmap • Introduction – Motivation • Problem#1: Patterns in graphs – Static graphs • degree, diameter, eigen, • triangles • cliques – Weighted graphs – Time evolving graphs • Problem#2: Tools UMN CRAY, 2012 C. Faloutsos (CMU) 50

CMU SCS Observations on weighted graphs? • A: yes - even more ‘laws’! M. Mc. Glohon, L. Akoglu, and C. Faloutsos Weighted Graphs and Disconnected Components: Patterns and a Generator. SIG-KDD 2008 UMN CRAY, 2012 C. Faloutsos (CMU) 51

CMU SCS Observation W. 1: Fortification Q: How do the weights of nodes relate to degree? UMN CRAY, 2012 C. Faloutsos (CMU) 52

CMU SCS Observation W. 1: Fortification More donors, more $ ? $10 $5 $7 ‘Reagan’ ‘Clinton’ UMN CRAY, 2012 C. Faloutsos (CMU) 53

CMU SCS Observation W. 1: fortification: Snapshot Power Law • Weight: super-linear on in-degree • exponent ‘iw’: 1. 01 < iw < 1. 26 More donors, even more $ $10 In-weights ($) Orgs-Candidates e. g. John Kerry, $10 M received, from 1 K donors $5 Edges (# donors) UMN CRAY, 2012 C. Faloutsos (CMU) 54

CMU SCS Roadmap • Introduction – Motivation • Problem#1: Patterns in graphs – Static graphs – Weighted graphs – Time evolving graphs • Problem#2: Tools • … UMN CRAY, 2012 C. Faloutsos (CMU) 55

CMU SCS Problem: Time evolution • with Jure Leskovec (CMU -> Stanford) • and Jon Kleinberg (Cornell – sabb. @ CMU) UMN CRAY, 2012 C. Faloutsos (CMU) 56

CMU SCS T. 1 Evolution of the Diameter • Prior work on Power Law graphs hints at slowly growing diameter: – diameter ~ O(log N) • What is happening in real data? UMN CRAY, 2012 C. Faloutsos (CMU) 57

CMU SCS T. 1 Evolution of the Diameter • Prior work on Power Law graphs hints at slowly growing diameter: – diameter ~ O(log N) • What is happening in real data? • Diameter shrinks over time UMN CRAY, 2012 C. Faloutsos (CMU) 58

CMU SCS T. 1 Diameter – “Patents” • Patent citation network • 25 years of data • @1999 diameter – 2. 9 M nodes – 16. 5 M edges time [years] UMN CRAY, 2012 C. Faloutsos (CMU) 59

CMU SCS 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) UMN CRAY, 2012 C. Faloutsos (CMU) 60

CMU SCS 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’’ UMN CRAY, 2012 C. Faloutsos (CMU) 61

CMU SCS T. 2 Densification – Patent Citations • Citations among patents granted E(t) • @1999 – 2. 9 M nodes – 16. 5 M edges 1. 66 • Each year is a datapoint N(t) UMN CRAY, 2012 C. Faloutsos (CMU) 62

CMU SCS Roadmap • Introduction – Motivation • Problem#1: Patterns in graphs – Static graphs – Weighted graphs – Time evolving graphs • Problem#2: Tools • … UMN CRAY, 2012 C. Faloutsos (CMU) 63

CMU SCS T. 3 : popularity over time # in links 1 2 3 lag: days after post Post popularity drops-off – exponentially? @t @t + lag UMN CRAY, 2012 C. Faloutsos (CMU) 64

CMU SCS T. 3 : popularity over time # in links (log) days after post (log) Post popularity drops-off – exponentially? POWER LAW! Exponent? UMN CRAY, 2012 C. Faloutsos (CMU) 65

CMU SCS T. 3 : popularity over time # in links (log) -1. 6 days after post (log) 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 UMN CRAY, 2012 C. Faloutsos (CMU) 66

CMU SCS -1. 5 slope J. G. Oliveira & A. -L. Barabási Human Dynamics: The Correspondence Patterns of Darwin and Einstein. Nature 437, 1251 (2005). [PDF] Prob(RT > x) (log) Response time (log) UMN CRAY, 2012 C. Faloutsos (CMU) 67

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 UMN CRAY, 2012 C. Faloutsos (CMU) 68

CMU SCS Probably, power law (? ) ? ? UMN CRAY, 2012 C. Faloutsos (CMU) 69

CMU SCS No Power Law! UMN CRAY, 2012 C. Faloutsos (CMU) 70

CMU SCS ‘TLa. C: Lazy Contractor’ • The longer a task (phonecall) has taken, • The even longer it will take Odds ratio= Casualties(<x): Survivors(>=x) == power law UMN CRAY, 2012 C. Faloutsos (CMU) 71

CMU SCS 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) UMN CRAY, 2012 C. Faloutsos (CMU) 72

CMU SCS Outliers: UMN CRAY, 2012 C. Faloutsos (CMU) 73

CMU SCS Roadmap • Introduction – Motivation • Problem#1: Patterns in graphs • Problem#2: Tools – Belief Propagation – Tensors – Spike analysis • Problem#3: Scalability • Conclusions UMN CRAY, 2012 C. Faloutsos (CMU) 74

CMU SCS E-bay Fraud detection w/ Polo Chau & Shashank Pandit, CMU [www’ 07] UMN CRAY, 2012 C. Faloutsos (CMU) 75

CMU SCS E-bay Fraud detection UMN CRAY, 2012 C. Faloutsos (CMU) 76

CMU SCS E-bay Fraud detection UMN CRAY, 2012 C. Faloutsos (CMU) 77

CMU SCS E-bay Fraud detection - Net. Probe UMN CRAY, 2012 C. Faloutsos (CMU) 78

CMU SCS E-bay Fraud detection - Net. Probe Compatibility matrix F F A H 99% heterophily 99% 49% UMN CRAY, 2012 49% C. Faloutsos (CMU) 79

CMU SCS Popular press And less desirable attention: • E-mail from ‘Belgium police’ (‘copy of your code? ’) UMN CRAY, 2012 C. Faloutsos (CMU) 80

CMU SCS Roadmap • Introduction – Motivation • Problem#1: Patterns in graphs • Problem#2: Tools – Belief Propagation – Tensors – Spike analysis • Problem#3: Scalability • Conclusions UMN CRAY, 2012 C. Faloutsos (CMU) 81

CMU SCS Giga. Tensor: Scaling Tensor Analysis Up By 100 Times – Algorithms and Discoveries U Kang Evangelos Papalexakis Abhay Harpale Christos Faloutsos KDD’ 12 UMN CRAY, 2012 C. Faloutsos (CMU) 82

CMU SCS Background: Tensor • Tensors (=multi-dimensional arrays) are everywhere – Hyperlinks &anchor text [Kolda+, 05] Anchor Text 1 URL 2 1 1 1 URL 1 UMN CRAY, 2012 C. Faloutsos (CMU) 1 1 C# 1 C++ Java 83

CMU SCS Background: Tensor • Tensors (=multi-dimensional arrays) are everywhere – Sensor stream (time, location, type) – Predicates (subject, verb, object) in knowledge base “Eric Clapton plays guitar” “Barrack Obama is the president of U. S. ” UMN CRAY, 2012 (48 M) (26 M) NELL (Never Ending Language Learner) data Nonzeros =144 M (26 M) C. Faloutsos (CMU) 84

CMU SCS all I learned on tensors: from Nikos Sidiropoulos UMN CRAY, 2012 C. Faloutsos (CMU) Tamara Kolda, Sandia Labs (tensor toolbox) 85

CMU SCS Problem Definition • How to decompose a billion-scale tensor? – Corresponds to SVD in 2 D case UMN CRAY, 2012 C. Faloutsos (CMU) 86

CMU SCS Problem Definition q q q Q 1: Dominant concepts/topics? Q 2: Find synonyms to a given noun phrase? (and how to scale up: |data| > RAM) (48 M) (26 M) NELL (Never Ending Language Learner) data Nonzeros =144 M (26 M) UMN CRAY, 2012 C. Faloutsos (CMU) 87

CMU SCS Experiments • Giga. Tensor solves 100 x larger problem (K) (J) Giga. Tensor r o s n box e T ool T UMN CRAY, 2012 100 x Out of Memory C. Faloutsos (CMU) (I) Number of nonzero = I / 50 88

CMU SCS A 1: Concept Discovery • Concept Discovery in Knowledge Base UMN CRAY, 2012 C. Faloutsos (CMU) 89

CMU SCS A 1: Concept Discovery UMN CRAY, 2012 C. Faloutsos (CMU) 90

CMU SCS A 2: Synonym Discovery UMN CRAY, 2012 C. Faloutsos (CMU) 91

CMU SCS Next steps: tensors with side-info George Karypis UMN CRAY, 2012 Nikos Sidiropoulos UMN C. Faloutsos (CMU) Tom Mitchell CMU 92

CMU SCS Roadmap • Introduction – Motivation • Problem#1: Patterns in graphs • Problem#2: Tools – Belief propagation – Tensors – Spike analysis • Problem#3: Scalability -PEGASUS • Conclusions UMN CRAY, 2012 C. Faloutsos (CMU) 93

CMU SCS Rise and fall patterns in social media • Meme (# of mentions in blogs) – short phrases Sourced from U. S. politics in 2008 “you can put lipstick on a pig” “yes we can” UMN CRAY, 2012 C. Faloutsos (CMU) 94

CMU SCS Rise and fall patterns in social media • Can we find a unifying model, which includes these patterns? • four classes on You. Tube [Crane et al. ’ 08] • six classes on Meme [Yang et al. ’ 11] UMN CRAY, 2012 C. Faloutsos (CMU) 95

CMU SCS Rise and fall patterns in social media • Answer: YES! • We can represent all patterns by single model In Matsubara+ SIGKDD 2012 UMN CRAY, 2012 C. Faloutsos (CMU) 96

CMU SCS Main idea - Spike. M - 1. Un-informed bloggers (uninformed about rumor) - 2. External shock at time nb (e. g, breaking news) - 3. Infection (word-of-mouth) Time n=0 Time n=nb β Time n=nb+1 Infectiveness of a blog-post at age n: - Strength of infection (quality of news) - 2012 Decay function UMN CRAY, C. Faloutsos (CMU) 97

CMU SCS Main idea - Spike. M - 1. Un-informed bloggers (uninformed about rumor) - 2. External shock at time nb (e. g, breaking news) - 3. Infection (word-of-mouth) Time n=0 Time n=nb β Time n=nb+1 Infectiveness of a blog-post at age n: - Strength of infection (quality of news) - 2012 Decay function UMN CRAY, C. Faloutsos (CMU) 98

Details CMU SCS Spike. M - with periodicity • Full equation of Spike. M Periodicity noon Bloggers change their activity over time activity (e. g. , daily, weekly, yearly) Peak 3 am Dip Time n UMN CRAY, 2012 C. Faloutsos (CMU) 99

CMU SCS Details • Analysis – exponential rise and power-raw fall Rise-part Lin-log SI -> exponential Spike. M -> exponential Log-log UMN CRAY, 2012 C. Faloutsos (CMU) 100

CMU SCS Details • Analysis – exponential rise and power-raw fall Fall-part Lin-log SI -> exponential Spike. M -> power law UMN CRAY, 2012 C. Faloutsos (CMU) Log-log 101

CMU SCS Tail-part forecasts • Spike. M can capture tail part UMN CRAY, 2012 C. Faloutsos (CMU) 102

CMU SCS “What-if” forecasting (1) First spike (2) Release date (3) Two weeks before release ? ? e. g. , given (1) first spike, (2) release date of two sequel movies (3) access volume before the release date UMN CRAY, 2012 C. Faloutsos (CMU) 103

CMU SCS “What-if” forecasting –Spike. M can forecast not only tail-part, but also rise-part! (1) First spike (2) Release date (3) Two weeks before release • Spike. M can forecast shape of upcoming spikes UMN CRAY, 2012 C. Faloutsos (CMU) 104

CMU SCS Roadmap • Introduction – Motivation • Problem#1: Patterns in graphs • Problem#2: Tools – Belief propagation – Spike analysis – Tensors • Problem#3: Scalability -PEGASUS • Conclusions UMN CRAY, 2012 C. Faloutsos (CMU) 105

CMU SCS 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: 5 Pb 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) http: //hadoop. apache. org/ UMN CRAY, 2012 C. Faloutsos (CMU) 106

CMU SCS Roadmap – Algorithms & results Degree Distr. Pagerank Diameter/ANF Conn. Comp Triangles Visualization UMN CRAY, 2012 Centralized Hadoop/PEG ASUS old old old HERE old done started C. Faloutsos (CMU) 107

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~1 B) • Our HADI: linear on E (~10 B) – Near-linear scalability wrt # machines – Several optimizations -> 5 x faster UMN CRAY, 2012 C. Faloutsos (CMU) 108

CMU SCS Count ? ? 19+ [Barabasi+] ~1999, ~1 M nodes Radius UMN CRAY, 2012 C. Faloutsos (CMU) 109

CMU SCS ? ? Count ? ? 19+ [Barabasi+] ~1999, ~1 M nodes Radius Yahoo. Web graph (120 Gb, 1. 4 B nodes, 6. 6 B edges) • Largest publicly available graph ever studied. UMN CRAY, 2012 C. Faloutsos (CMU) 110

CMU SCS Count 14 (dir. ) ? ? ~7 (undir. ) 19+? [Barabasi+] Radius Yahoo. Web graph (120 Gb, 1. 4 B nodes, 6. 6 B edges) • Largest publicly available graph ever studied. UMN CRAY, 2012 C. Faloutsos (CMU) 111

CMU SCS Count 14 (dir. ) ? ? ~7 (undir. ) 19+? [Barabasi+] Radius Yahoo. Web graph (120 Gb, 1. 4 B nodes, 6. 6 B edges) • 7 degrees of separation (!) • Diameter: shrunk UMN CRAY, 2012 C. Faloutsos (CMU) 112

CMU SCS Count ? ? ~7 (undir. ) Radius Yahoo. Web graph (120 Gb, 1. 4 B nodes, 6. 6 B edges) Q: Shape? UMN CRAY, 2012 C. Faloutsos (CMU) 113

CMU SCS Yahoo. Web graph (120 Gb, 1. 4 B nodes, 6. 6 B edges) • effective diameter: surprisingly small. • Multi-modality (? !) UMN CRAY, 2012 C. Faloutsos (CMU) 114

CMU SCS Radius Plot of GCC of Yahoo. Web. UMN CRAY, 2012 C. Faloutsos (CMU) 115

CMU SCS Yahoo. Web graph (120 Gb, 1. 4 B nodes, 6. 6 B edges) • effective diameter: surprisingly small. • Multi-modality: probably mixture of cores. UMN CRAY, 2012 C. Faloutsos (CMU) 116

CMU SCS Conjecture: DE EN ~7 BR Yahoo. Web graph (120 Gb, 1. 4 B nodes, 6. 6 B edges) • effective diameter: surprisingly small. • Multi-modality: probably mixture of cores. UMN CRAY, 2012 C. Faloutsos (CMU) 117

CMU SCS Conjecture: ~7 Yahoo. Web graph (120 Gb, 1. 4 B nodes, 6. 6 B edges) • effective diameter: surprisingly small. • Multi-modality: probably mixture of cores. UMN CRAY, 2012 C. Faloutsos (CMU) 118

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

CMU SCS Roadmap – Algorithms & results Degree Distr. Pagerank Diameter/ANF Conn. Comp Triangles Visualization UMN CRAY, 2012 Centralized Hadoop/PEG ASUS old old old HERE started C. Faloutsos (CMU) 120

CMU SCS Generalized Iterated Matrix Vector Multiplication (GIMV) PEGASUS: A Peta-Scale Graph Mining System - Implementation and Observations. U Kang, Charalampos E. Tsourakakis, and Christos Faloutsos. (ICDM) 2009, Miami, Florida, USA. Best Application Paper (runner-up). UMN CRAY, 2012 C. Faloutsos (CMU) 121

CMU SCS details Generalized Iterated Matrix Vector Multiplication (GIMV) • Page. Rank • proximity (RWR) • Diameter • Connected components • (eigenvectors, • Belief Prop. • …) UMN CRAY, 2012 C. Faloutsos (CMU) Matrix – vector Multiplication (iterated) 122

CMU SCS Example: GIM-V At Work • Connected Components – 4 observations: Count Size UMN CRAY, 2012 C. Faloutsos (CMU) 123

CMU SCS Example: GIM-V At Work • Connected Components Count 1) 10 K x larger than next Size UMN CRAY, 2012 C. Faloutsos (CMU) 124

CMU SCS Example: GIM-V At Work • Connected Components Count 2) ~0. 7 B singleton nodes Size UMN CRAY, 2012 C. Faloutsos (CMU) 125

CMU SCS Example: GIM-V At Work • Connected Components Count 3) SLOPE! Size UMN CRAY, 2012 C. Faloutsos (CMU) 126

CMU SCS Example: GIM-V At Work • Connected Components Count 300 -size cmpt X 500. 1100 -size cmpt Why? X 65. Why? 4) Spikes! Size UMN CRAY, 2012 C. Faloutsos (CMU) 127

CMU SCS Example: GIM-V At Work • Connected Components Count suspicious financial-advice sites (not existing now) UMN CRAY, 2012 Size C. Faloutsos (CMU) 128

CMU SCS GIM-V At Work • Connected Components over Time • Linked. In: 7. 5 M nodes and 58 M edges Stable tail slope after the gelling point UMN CRAY, 2012 C. Faloutsos (CMU) 129

CMU SCS Roadmap • • • Introduction – Motivation Problem#1: Patterns in graphs Problem#2: Tools Problem#3: Scalability Conclusions UMN CRAY, 2012 C. Faloutsos (CMU) 130

CMU SCS OVERALL CONCLUSIONS – low level: • Several new patterns (fortification, triangle -laws, conn. components, etc) • New tools: – belief propagation, giga. Tensor, etc • Scalability: PEGASUS / hadoop UMN CRAY, 2012 C. Faloutsos (CMU) 131

CMU SCS OVERALL CONCLUSIONS – high level • BIG DATA: Large datasets reveal patterns/outliers that are invisible otherwise UMN CRAY, 2012 C. Faloutsos (CMU) 132

CMU SCS Theory & Algo. Comp. Systems ML & Stats. Biology Graph Analytics Physics Social Science Econ. 133 UMN CRAY, 2012 C. Faloutsos (CMU)

CMU SCS References • Leman Akoglu, Christos Faloutsos: RTG: A Recursive Realistic Graph Generator Using Random Typing. ECML/PKDD (1) 2009: 13 -28 • Deepayan Chakrabarti, Christos Faloutsos: Graph mining: Laws, generators, and algorithms. ACM Comput. Surv. 38(1): (2006) UMN CRAY, 2012 C. Faloutsos (CMU) 134

CMU SCS 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 P 2 P Networks. INFOCOM 2007: 1316 -1324 UMN CRAY, 2012 C. Faloutsos (CMU) 135

CMU SCS References • Christos Faloutsos, Tamara G. Kolda, Jimeng Sun: Mining large graphs and streams using matrix and tensor tools. Tutorial, SIGMOD Conference 2007: 1174 UMN CRAY, 2012 C. Faloutsos (CMU) 136

CMU SCS References • T. G. Kolda and J. Sun. Scalable Tensor Decompositions for Multi-aspect Data Mining. In: ICDM 2008, pp. 363 -372, December 2008. UMN CRAY, 2012 C. Faloutsos (CMU) 137

CMU SCS 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: 133 -145 UMN CRAY, 2012 C. Faloutsos (CMU) 138

CMU SCS References • Jimeng Sun, Yinglian Xie, Hui Zhang, Christos Faloutsos. Less is More: Compact Matrix Decomposition for Large Sparse Graphs, SDM, Minneapolis, Minnesota, Apr 2007. • Jimeng Sun, Spiros Papadimitriou, Philip S. Yu, and Christos Faloutsos, Graph. Scope: Parameterfree Mining of Large Time-evolving Graphs ACM SIGKDD Conference, San Jose, CA, August 2007 UMN CRAY, 2012 C. Faloutsos (CMU) 139

CMU SCS References • Jimeng Sun, Dacheng Tao, Christos Faloutsos: Beyond streams and graphs: dynamic tensor analysis. KDD 2006: 374383 UMN CRAY, 2012 C. Faloutsos (CMU) 140

CMU SCS 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 UMN CRAY, 2012 C. Faloutsos (CMU) 141

CMU SCS References • Hanghang Tong, Christos Faloutsos, Brian Gallagher, Tina Eliassi-Rad: Fast best-effort pattern matching in large attributed graphs. KDD 2007: 737 -746 UMN CRAY, 2012 C. Faloutsos (CMU) 142

CMU SCS Project info www. cs. cmu. edu/~pegasus Thanks to: NSF IIS-0705359, IIS-0534205, CTA-INARC; Yahoo (M 45), LLNL, IBM, SPRINT, UMN CRAY, 2012 C. Faloutsos (CMU) 143 Google, INTEL, HP, i. Lab

CMU SCS Cast Akoglu, Leman Mc. Glohon, Mary UMN CRAY, 2012 Beutel, Alex Prakash, Aditya Chau, Polo Kang, U Papalexakis, Vagelis C. Faloutsos (CMU) Koutra, Danae Tong, Hanghang 144

CMU SCS OVERALL CONCLUSIONS – high level • BIG DATA: Large datasets reveal patterns/outliers that are invisible otherwise UMN CRAY, 2012 C. Faloutsos (CMU) 145