1. CMU SCS Mining Billion-Node Graphs Christos Faloutsos CMU

2. CMU SCS C. Faloutsos (CMU) 2 Our goal: Open source system for mining huge graphs: PEGASUS project (PEta GrAph mining System) www.cs.cmu.edu/~pegasus code and papers April '12

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

4. CMU SCS C. Faloutsos (CMU) 4 Graphs - why should we care? Internet Map [lumeta.com] Food Web [Martinez ’91] Friendship Network [Moody ’01] April '12

5. CMU SCS C. Faloutsos (CMU) 5 Graphs - why should we care? IR: bi-partite graphs (doc-terms) web: hyper-text graph... and more: D1D1 DNDN T1T1 TMTM... April '12

6. CMU SCS C. Faloutsos (CMU) 6 Graphs - why should we care? ‘viral’ marketing web-log (‘blog’) news propagation computer network security: email/IP traffic and anomaly detection.... April '12

7. CMU SCS C. Faloutsos (CMU) 7 Outline Introduction – Motivation Problem#1: Patterns in graphs –Static graphs –Weighted graphs –Time evolving graphs Problem#2: Tools Problem#3: Scalability Conclusions April '12

8. CMU SCS C. Faloutsos (CMU) 8 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? April '12

9. 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? –To spot anomalies (rarities), we have to discover patterns April '12

10. 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 –Large datasets reveal patterns/anomalies that may be invisible otherwise… April '12

11. CMU SCS C. Faloutsos (CMU) 11 Graph mining Are real graphs random? April '12

12. CMU SCS C. Faloutsos (CMU) 12 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 April '12

13. CMU SCS C. Faloutsos (CMU) 13 Solution# S.1 Power law in the degree distribution [SIGCOMM99] log(rank) log(degree) internet domains att.com ibm.com April '12

14. CMU SCS C. Faloutsos (CMU) 14 Solution# S.1 Power law in the degree distribution [SIGCOMM99] log(rank) log(degree) -0.82 internet domains att.com ibm.com April '12

15. CMU SCS C. Faloutsos (CMU) 15 Solution# S.2: Eigen Exponent E A2: power law in the eigenvalues of the adjacency matrix E = -0.48 Exponent = slope Eigenvalue Rank of decreasing eigenvalue May 2001 April '12

16. CMU SCS C. Faloutsos (CMU) 16 Solution# S.2: Eigen Exponent E [Mihail, Papadimitriou ’02]: slope is ½ of rank exponent E = -0.48 Exponent = slope Eigenvalue Rank of decreasing eigenvalue May 2001 April '12

17. CMU SCS C. Faloutsos (CMU) 17 But: How about graphs from other domains? April '12

18. CMU SCS C. Faloutsos (CMU) 18 More power laws: web hit counts [w/ A. Montgomery] Web Site Traffic in-degree (log scale) Count (log scale) Zipf users sites ``ebay’’ April '12

19. CMU SCS C. Faloutsos (CMU) 19 epinions.com who-trusts-whom [Richardson + Domingos, KDD 2001] (out) degree count trusts-2000-people user April '12

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’ April '12C. Faloutsos (CMU) 20

21. CMU SCS C. Faloutsos (CMU) 21 Outline Introduction – Motivation Problem#1: Patterns in graphs –Static graphs degree, diameter, eigen, triangles cliques –Weighted graphs –Time evolving graphs Problem#2: Tools April '12

22. CMU SCS C. Faloutsos (CMU) 22 Solution# S.3: Triangle ‘Laws’ Real social networks have a lot of triangles April '12

23. CMU SCS C. Faloutsos (CMU) 23 Solution# S.3: Triangle ‘Laws’ Real social networks have a lot of triangles –Friends of friends are friends Any patterns? April '12

24. CMU SCS C. Faloutsos (CMU) 24 Triangle Law: #S.3 [Tsourakakis ICDM 2008] ASNHEP-TH Epinions X-axis: # of participating triangles Y: count (~ pdf) April '12

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

26. CMU SCS C. Faloutsos (CMU) 26 Triangle Law: #S.4 [Tsourakakis ICDM 2008] SNReuters Epinions X-axis: degree Y-axis: mean # triangles n friends -> ~n 1.6 triangles April '12

27. CMU SCS C. Faloutsos (CMU) 27 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 April '12

28. 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? A: Yes! #triangles = 1/6 Sum ( i 3 ) (and, because of skewness (S2), we only need the top few eigenvalues! details April '12

29. CMU SCS C. Faloutsos (CMU) 29 Triangle Law: Computations [Tsourakakis ICDM 2008] 1000x+ speed-up, >90% accuracy details April '12

30. CMU SCS Triangle counting for large graphs? Anomalous nodes in Twitter(~ 3 billion edges) [U Kang, Brendan Meeder, +, PAKDD’11] 30 April '12 30 C. Faloutsos (CMU)

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

32. 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, 21-24 June 2010. C. Faloutsos (CMU) 32 April '12

33. CMU SCS EigenSpokes Eigenvectors of adjacency matrix  equivalent to singular vectors (symmetric, undirected graph) 33 C. Faloutsos (CMU)April '12

34. CMU SCS EigenSpokes Eigenvectors of adjacency matrix  equivalent to singular vectors (symmetric, undirected graph) 34 C. Faloutsos (CMU)April '12 N N details

35. CMU SCS EigenSpokes Eigenvectors of adjacency matrix  equivalent to singular vectors (symmetric, undirected graph) 35 C. Faloutsos (CMU)April '12 N N details

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

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

38. CMU SCS EigenSpokes EE plot: Scatter plot of scores of u1 vs u2 One would expect –Many points @ origin –A few scattered ~randomly C. Faloutsos (CMU) 38 u1 u2 April '12 1 st Principal component 2 nd Principal component

39. CMU SCS EigenSpokes EE plot: Scatter plot of scores of u1 vs u2 One would expect –Many points @ origin –A few scattered ~randomly C. Faloutsos (CMU) 39 u1 u2 90 o April '12

40. CMU SCS EigenSpokes - pervasiveness Present in mobile social graph  across time and space Patent citation graph 40 C. Faloutsos (CMU)April '12

41. CMU SCS EigenSpokes - explanation Near-cliques, or near- bipartite-cores, loosely connected 41 C. Faloutsos (CMU)April '12

42. CMU SCS EigenSpokes - explanation Near-cliques, or near- bipartite-cores, loosely connected 42 C. Faloutsos (CMU)April '12

43. CMU SCS EigenSpokes - explanation Near-cliques, or near- bipartite-cores, loosely connected 43 C. Faloutsos (CMU)April '12

44. 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 44 C. Faloutsos (CMU)April '12

45. CMU SCS Bipartite Communities! magnified bipartite community patents from same inventor(s) `cut-and-paste’ bibliography! 45 C. Faloutsos (CMU)April '12

46. CMU SCS C. Faloutsos (CMU) 46 Outline Introduction – Motivation Problem#1: Patterns in graphs –Static graphs degree, diameter, eigen, triangles cliques –Weighted graphs –Time evolving graphs Problem#2: Tools April '12

47. CMU SCS C. Faloutsos (CMU) 47 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 April '12

48. CMU SCS C. Faloutsos (CMU) 48 Observation W.1: Fortification Q: How do the weights of nodes relate to degree? April '12

49. CMU SCS C. Faloutsos (CMU) 49 Observation W.1: Fortification More donors, more $ ? $10 $5 April '12 ‘Reagan’ ‘Clinton’ $7

50. CMU SCS Edges (# donors) In-weights ($) C. Faloutsos (CMU) 50 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 April '12

51. CMU SCS C. Faloutsos (CMU) 51 Outline Introduction – Motivation Problem#1: Patterns in graphs –Static graphs –Weighted graphs –Time evolving graphs Problem#2: Tools … April '12

52. CMU SCS C. Faloutsos (CMU) 52 Problem: Time evolution with Jure Leskovec (CMU -> Stanford) and Jon Kleinberg (Cornell – sabb. @ CMU) April '12

53. CMU SCS C. Faloutsos (CMU) 53 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? April '12

54. CMU SCS C. Faloutsos (CMU) 54 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 April '12

55. CMU SCS C. Faloutsos (CMU) 55 T.1 Diameter – “Patents” Patent citation network 25 years of data @1999 –2.9 M nodes –16.5 M edges time [years] diameter April '12

56. CMU SCS C. Faloutsos (CMU) 56 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) April '12

57. CMU SCS C. Faloutsos (CMU) 57 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’’ April '12

58. CMU SCS C. Faloutsos (CMU) 58 T.2 Densification – Patent Citations Citations among patents granted @1999 –2.9 M nodes –16.5 M edges Each year is a datapoint N(t) E(t) 1.66 April '12

59. CMU SCS C. Faloutsos (CMU) 59 Outline Introduction – Motivation Problem#1: Patterns in graphs –Static graphs –Weighted graphs –Time evolving graphs Problem#2: Tools … April '12

60. CMU SCS C. Faloutsos (CMU) 60 More on Time-evolving graphs M. McGlohon, L. Akoglu, and C. Faloutsos Weighted Graphs and Disconnected Components: Patterns and a Generator. SIG-KDD 2008 April '12

61. CMU SCS C. Faloutsos (CMU) 61 Observation T.3: NLCC behavior Q: How do NLCC’s emerge and join with the GCC? (``NLCC’’ = non-largest conn. components) –Do they continue to grow in size? – or do they shrink? – or stabilize? April '12

62. CMU SCS C. Faloutsos (CMU) 62 Observation T.3: NLCC behavior Q: How do NLCC’s emerge and join with the GCC? (``NLCC’’ = non-largest conn. components) –Do they continue to grow in size? – or do they shrink? – or stabilize? April '12

63. CMU SCS C. Faloutsos (CMU) 63 Observation T.3: NLCC behavior Q: How do NLCC’s emerge and join with the GCC? (``NLCC’’ = non-largest conn. components) –Do they continue to grow in size? – or do they shrink? – or stabilize? April '12 YES

64. CMU SCS C. Faloutsos (CMU) 64 Observation T.3: NLCC behavior After the gelling point, the GCC takes off, but NLCC’s remain ~constant (actually, oscillate). IMDB CC size Time-stamp April '12

65. CMU SCS C. Faloutsos (CMU) 65 Timing for Blogs with Mary McGlohon (CMU->Google) Jure Leskovec (CMU->Stanford) Natalie Glance (now at Google) Mat Hurst (now at MSR) [SDM’07] April '12

66. CMU SCS C. Faloutsos (CMU) 66 T.4 : popularity over time Post popularity drops-off – exponentially? lag: days after post # in links 1 2 3 @t @t + lag April '12

67. CMU SCS C. Faloutsos (CMU) 67 T.4 : popularity over time Post popularity drops-off – exponentially? POWER LAW! Exponent? # in links (log) days after post (log) April '12

68. CMU SCS C. Faloutsos (CMU) 68 T.4 : 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) April '12

69. CMU SCS April '12C. Faloutsos (CMU) 69 -1.5 slope J. G. Oliveira & A.-L. Barabási Human Dynamics: The Correspondence Patterns of Darwin and Einstein. Nature 437, 1251 (2005). [PDF]PDF

70. CMU SCS T.5: 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 April '12C. Faloutsos (CMU) 70

71. CMU SCS Probably, power law (?) April '12C. Faloutsos (CMU) 71 ??

72. CMU SCS No Power Law! April '12C. Faloutsos (CMU) 72

73. CMU SCS ‘TLaC: Lazy Contractor’ The longer a task (phonecall) has taken, The even longer it will take April '12C. Faloutsos (CMU) 73 Odds ratio= Casualties(<x): Survivors(>=x) == power law

74. CMU SCS 74 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) April '12C. Faloutsos (CMU)

75. CMU SCS Outliers: April '12C. Faloutsos (CMU) 75

76. CMU SCS C. Faloutsos (CMU) 76 Outline Introduction – Motivation Problem#1: Patterns in graphs Problem#2: Tools –OddBall (anomaly detection) –Belief Propagation –Immunization Problem#3: Scalability Conclusions April '12

77. CMU SCS OddBall: Spotting A n o m a l i e s in Weighted Graphs Leman Akoglu, Mary McGlohon, Christos Faloutsos Carnegie Mellon University School of Computer Science PAKDD 2010, Hyderabad, India

78. CMU SCS 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 C. Faloutsos (CMU) 78 April '12

79. CMU SCS What is an egonet? ego 79 egonet C. Faloutsos (CMU)April '12

80. CMU SCS Selected Features  N i : number of neighbors (degree) of ego i  E i : number of edges in egonet i  W i : total weight of egonet i  λ w,i : principal eigenvalue of the weighted adjacency matrix of egonet I 80 C. Faloutsos (CMU)April '12

81. CMU SCS Near-Clique/Star 81 April '12C. Faloutsos (CMU)

82. CMU SCS Near-Clique/Star 82 C. Faloutsos (CMU)April '12

83. CMU SCS Near-Clique/Star 83 C. Faloutsos (CMU)April '12

84. CMU SCS Andrew Lewis (director) Near-Clique/Star 84 C. Faloutsos (CMU)April '12

85. CMU SCS C. Faloutsos (CMU) 85 Outline Introduction – Motivation Problem#1: Patterns in graphs Problem#2: Tools –OddBall (anomaly detection) –Belief Propagation –Immunization Problem#3: Scalability Conclusions April '12

86. CMU SCS April '12C. Faloutsos (CMU) 86 E-bay Fraud detection w/ Polo Chau & Shashank Pandit, CMU [www’07]

87. CMU SCS April '12C. Faloutsos (CMU) 87 E-bay Fraud detection

88. CMU SCS April '12C. Faloutsos (CMU) 88 E-bay Fraud detection

89. CMU SCS April '12C. Faloutsos (CMU) 89 E-bay Fraud detection - NetProbe

90. CMU SCS Popular press And less desirable attention: E-mail from ‘Belgium police’ (‘copy of your code?’) April '12C. Faloutsos (CMU) 90

91. CMU SCS C. Faloutsos (CMU) 91 Outline Introduction – Motivation Problem#1: Patterns in graphs Problem#2: Tools –OddBall (anomaly detection) –Belief Propagation – antivirus app –Immunization Problem#3: Scalability Conclusions April '12

92. CMU SCS 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 Polonium: Tera-Scale Graph Mining and Inference for Malware Detection PATENT PENDING SDM 2011, Mesa, Arizona

93. CMU SCS 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 April '12 93 C. Faloutsos (CMU)

94. CMU SCS 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 April '12 94 C. Faloutsos (CMU)

95. CMU SCS Polonium: One-Interaction Results 84.9% True Positive Rate 1% False Positive Rate True Positive Rate % of malware correctly identified False Positive Rate % of non-malware wrongly labeled as malware 95 Ideal April '12C. Faloutsos (CMU)

96. CMU SCS C. Faloutsos (CMU) 96 Outline Introduction – Motivation Problem#1: Patterns in graphs Problem#2: Tools –OddBall (anomaly detection) –Belief propagation –Immunization Problem#3: Scalability -PEGASUS Conclusions April '12

97. CMU SCS Immunization and epidemic thresholds Q1: which nodes to immunize? Q2: will a virus vanish, or will it create an epidemic? April '12C. Faloutsos (CMU) 97

98. CMU SCS Q1: Immunization: ? ? Given a network, k vaccines, and the virus details Which nodes to immunize? April '12 98 C. Faloutsos (CMU)

99. CMU SCS Q1: Immunization: ? ? Given a network, k vaccines, and the virus details Which nodes to immunize? April '12 99 C. Faloutsos (CMU)

100. CMU SCS Q1: Immunization: ? ? Given a network, k vaccines, and the virus details Which nodes to immunize? April '12 100 C. Faloutsos (CMU)

101. CMU SCS Q1: Immunization: ? ? Given a network, k vaccines, and the virus details Which nodes to immunize? A: immunize the ones that maximally raise the `epidemic threshold’ [Tong+, ICDM’10] April '12 101 C. Faloutsos (CMU)

102. CMU SCS Q2: will a virus take over? Flu-like virus (no immunity, ‘SIS’) Mumps (life-time immunity, ‘SIR’) Pertussis (finite-length immunity, ‘SIRS’) April '12C. Faloutsos (CMU) 102 ? ?  : attack prob  : heal prob

103. CMU SCS Q2: will a virus take over? Flu-like virus (no immunity, ‘SIS’) Mumps (life-time immunity, ‘SIR’) Pertussis (finite-length immunity, ‘SIRS’) April '12C. Faloutsos (CMU) 103 ? ?  : attack prob  : heal prob  depends on connectivity (avg degree? Max degree? variance? Something else?

104. CMU SCS April '12C. Faloutsos (CMU) 104 Epidemic threshold  What should  depend on? avg. degree? and/or highest degree? and/or variance of degree? and/or third moment of degree? and/or diameter?

105. CMU SCS April '12C. Faloutsos (CMU) 105 Epidemic threshold [Theorem] We have no epidemic, if β/δ <τ = 1/ λ 1,A

106. CMU SCS April '12C. Faloutsos (CMU) 106 Epidemic threshold [Theorem] We have no epidemic, if β/δ <τ = 1/ λ 1,A largest eigenvalue of adj. matrix A attack prob. recovery prob. epidemic threshold Proof: [Wang+03] (for SIS=flu only)

107. CMU SCS A2: will a virus take over? For all typical virus propagation models (flu, mumps, pertussis, HIV, etc) The only connectivity measure that matters, is  1 the first eigenvalue of the adj. matrix [Prakash+, ‘10, arxiv] April '12C. Faloutsos (CMU) 107 ? ?

108. CMU SCS 108 C. Faloutsos (CMU)April '12 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.

109. CMU SCS A2: will a virus take over? April '12C. Faloutsos (CMU) 109 Fraction of infected Time ticks Below: exp. extinction Above: take-over Graph: Portland, OR 31M links 1.5M nodes

110. CMU SCS Q1: Immunization: ? ? Given a network, k vaccines, and the virus details Which nodes to immunize? A: immunize the ones that maximally raise the `epidemic threshold’ [Tong+, ICDM’10] April '12 110 C. Faloutsos (CMU)

111. CMU SCS Q1: Immunization: ? ? Given a network, k vaccines, and the virus details Which nodes to immunize? A: immunize the ones that maximally raise the `epidemic threshold’ [Tong+, ICDM’10] April '12 111 C. Faloutsos (CMU) Max eigen-drop  for any virus!

112. CMU SCS C. Faloutsos (CMU) 112 Outline Introduction – Motivation Problem#1: Patterns in graphs Problem#2: Tools –OddBall (anomaly detection) –Belief propagation –Immunization –Visualization Problem#3: Scalability -PEGASUS Conclusions April '12

113. CMU SCS Apolo Making Sense of Large Network Data: Combining Rich User Interaction & Machine Learning CHI 2011, Vancouver, Canada Polo ChauProf. Niki Kittur Prof. Jason Hong Prof. Christos Faloutsos April '12 113 C. Faloutsos (CMU)

114. CMU SCS Main Ideas of Apolo Provides a mixed-initiative approach (ML + HCI) to help users interactively explore large graphs Users start with small subgraph, then iteratively expand: 1.User specifies exemplars 2.Belief Propagation to find other relevant nodes User study showed Apolo outperformed Google Scholar in making sense of citation network data April '12 114 C. Faloutsos (CMU)

115. CMU SCS Exemplars Rest of the nodes are considered relevant (by BP); relevance indicated by color saturation. Note that BP supports multiple groups April '12 115 C. Faloutsos (CMU)

116. CMU SCS C. Faloutsos (CMU) 116 Outline Introduction – Motivation Problem#1: Patterns in graphs Problem#2: Tools –OddBall (anomaly detection) –Belief propagation –Immunization Problem#3: Scalability -PEGASUS Conclusions April '12

117. CMU SCS April '12C. Faloutsos (CMU) 117 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) http://hadoop.apache.org/ http://hadoop.apache.org/

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

119. 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) 119 April '12

120. CMU SCS ???? 19+ [Barabasi+] 120 C. Faloutsos (CMU) Radius Count April '12 ~1999, ~1M nodes

121. CMU SCS YahooWeb graph (120Gb, 1.4B nodes, 6.6 B edges) Largest publicly available graph ever studied. ???? 19+ [Barabasi+] 121 C. Faloutsos (CMU) Radius Count April '12 ?? ~1999, ~1M nodes

122. CMU SCS YahooWeb graph (120Gb, 1.4B nodes, 6.6 B edges) Largest publicly available graph ever studied. ???? 19+? [Barabasi+] 122 C. Faloutsos (CMU) Radius Count April '12 14 (dir.) ~7 (undir.)

123. CMU SCS YahooWeb graph (120Gb, 1.4B nodes, 6.6 B edges) 7 degrees of separation (!) Diameter: shrunk ???? 19+? [Barabasi+] 123 C. Faloutsos (CMU) Radius Count April '12 14 (dir.) ~7 (undir.)

124. CMU SCS YahooWeb graph (120Gb, 1.4B nodes, 6.6 B edges) Q: Shape? ???? 124 C. Faloutsos (CMU) Radius Count April '12 ~7 (undir.)

125. CMU SCS 125 C. Faloutsos (CMU) YahooWeb graph (120Gb, 1.4B nodes, 6.6 B edges) effective diameter: surprisingly small. Multi-modality (?!) April '12

126. CMU SCS Radius Plot of GCC of YahooWeb. 126 C. Faloutsos (CMU)April '12

127. CMU SCS 127 C. Faloutsos (CMU) YahooWeb graph (120Gb, 1.4B nodes, 6.6 B edges) effective diameter: surprisingly small. Multi-modality: probably mixture of cores. April '12

128. CMU SCS 128 C. Faloutsos (CMU) YahooWeb graph (120Gb, 1.4B nodes, 6.6 B edges) effective diameter: surprisingly small. Multi-modality: probably mixture of cores. April '12 EN ~7 Conjecture: DE BR

129. CMU SCS 129 C. Faloutsos (CMU) YahooWeb graph (120Gb, 1.4B nodes, 6.6 B edges) effective diameter: surprisingly small. Multi-modality: probably mixture of cores. April '12 ~7 Conjecture:

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

131. CMU SCS C. Faloutsos (CMU) 131 CentralizedHadoop/PEG ASUS Degree Distr. old Pagerank old Diameter/ANF old HERE Conn. Comp old HERE Triangles HERE Visualizationstarted Outline – Algorithms & results April '12

132. CMU SCS Generalized Iterated Matrix Vector Multiplication (GIMV) C. Faloutsos (CMU) 132 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 April '12

133. CMU SCS Generalized Iterated Matrix Vector Multiplication (GIMV) C. Faloutsos (CMU) 133 PageRank proximity (RWR) Diameter Connected components (eigenvectors, Belief Prop. … ) Matrix – vector Multiplication (iterated) April '12 details

134. CMU SCS 134 Example: GIM-V At Work Connected Components – 4 observations: Size Count C. Faloutsos (CMU) April '12

135. CMU SCS 135 Example: GIM-V At Work Connected Components Size Count C. Faloutsos (CMU) April '12 1) 10K x larger than next

136. CMU SCS 136 Example: GIM-V At Work Connected Components Size Count C. Faloutsos (CMU) April '12 2) ~0.7B singleton nodes

137. CMU SCS 137 Example: GIM-V At Work Connected Components Size Count C. Faloutsos (CMU) April '12 3) SLOPE!

138. CMU SCS 138 Example: GIM-V At Work Connected Components Size Count 300-size cmpt X 500. Why? 1100-size cmpt X 65. Why? C. Faloutsos (CMU) April '12 4) Spikes!

139. CMU SCS 139 Example: GIM-V At Work Connected Components Size Count suspicious financial-advice sites (not existing now) C. Faloutsos (CMU) April '12

140. CMU SCS 140 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)April '12

141. CMU SCS C. Faloutsos (CMU) 141 Outline Introduction – Motivation Problem#1: Patterns in graphs Problem#2: Tools Problem#3: Scalability Conclusions April '12

142. CMU SCS C. Faloutsos (CMU) 142 OVERALL CONCLUSIONS – low level: Several new patterns (fortification, triangle-laws, conn. components, etc) New tools: –anomaly detection (OddBall), belief propagation, immunization Scalability: PEGASUS / hadoop April '12

143. CMU SCS C. Faloutsos (CMU) 143 OVERALL CONCLUSIONS – high level BIG DATA: Large datasets reveal patterns/outliers that are invisible otherwise April '12

144. CMU SCS C. Faloutsos (CMU) 144 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) April '12

145. CMU SCS C. Faloutsos (CMU) 145 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: 1316-1324 April '12

146. CMU SCS C. Faloutsos (CMU) 146 References Christos Faloutsos, Tamara G. Kolda, Jimeng Sun: Mining large graphs and streams using matrix and tensor tools. Tutorial, SIGMOD Conference 2007: 1174 April '12

147. CMU SCS C. Faloutsos (CMU) 147 References T. G. Kolda and J. Sun. Scalable Tensor Decompositions for Multi-aspect Data Mining. In: ICDM 2008, pp. 363-372, December 2008. April '12

148. CMU SCS C. Faloutsos (CMU) 148 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 April '12

149. CMU SCS C. Faloutsos (CMU) 149 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, GraphScope: Parameter- free Mining of Large Time-evolving Graphs ACM SIGKDD Conference, San Jose, CA, August 2007 April '12

150. CMU SCS References Jimeng Sun, Dacheng Tao, Christos Faloutsos: Beyond streams and graphs: dynamic tensor analysis. KDD 2006: 374- 383 April '12C. Faloutsos (CMU) 150

151. CMU SCS C. Faloutsos (CMU) 151 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 April '12

152. CMU SCS C. Faloutsos (CMU) 152 References Hanghang Tong, Christos Faloutsos, Brian Gallagher, Tina Eliassi-Rad: Fast best-effort pattern matching in large attributed graphs. KDD 2007: 737-746 April '12

153. CMU SCS C. Faloutsos (CMU) 153 Project info Akoglu, Leman Chau, Polo Kang, U McGlohon, Mary Tong, Hanghang Prakash, Aditya April '12 Thanks to: NSF IIS-0705359, IIS-0534205, CTA-INARC ; Yahoo (M45), LLNL, IBM, SPRINT, Google, INTEL, HP, iLab www.cs.cmu.edu/~pegasus Koutra, Danae