1. CMU SCS Large Graph Mining - Patterns, tools and cascade analysis 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 KDD'12 summer school

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

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

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... KDD'12 summer school

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.... KDD'12 summer school

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 KDD'12 summer school

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? KDD'12 summer school

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 KDD'12 summer school

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… KDD'12 summer school

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

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, lets look at the data KDD'12 summer school

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 KDD'12 summer school

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 KDD'12 summer school

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 KDD'12 summer school

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 KDD'12 summer school

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

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 KDD'12 summer school

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

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 KDD'12 summer school C. 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 KDD'12 summer school

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

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? KDD'12 summer school

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) KDD'12 summer school

25. CMU SCS C. Faloutsos (CMU) 25 Triangle Law: #S.3 [Tsourakakis ICDM 2008] ASNHEP-TH Epinions KDD'12 summer school 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 KDD'12 summer school

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 KDD'12 summer school

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 KDD'12 summer school

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

30. CMU SCS Triangle counting for large graphs? Anomalous nodes in Twitter(~ 3 billion edges) [U Kang, Brendan Meeder, +, PAKDD11] 30 KDD'12 summer school 30 C. Faloutsos (CMU)

31. CMU SCS Triangle counting for large graphs? Anomalous nodes in Twitter(~ 3 billion edges) [U Kang, Brendan Meeder, +, PAKDD11] 31 KDD'12 summer school 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 KDD'12 summer school

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

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

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

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

37. CMU SCS EigenSpokes Eigenvectors of adjacency matrix equivalent to singular vectors (symmetric, undirected graph) 37 C. Faloutsos (CMU)KDD'12 summer school 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 KDD'12 summer school 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 KDD'12 summer school

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

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

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

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

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)KDD'12 summer school

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

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 KDD'12 summer school

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 KDD'12 summer school

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

49. CMU SCS C. Faloutsos (CMU) 49 Observation W.1: Fortification More donors, more $ ? $10 $5 KDD'12 summer school 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 KDD'12 summer school

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 … KDD'12 summer school

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

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? KDD'12 summer school

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 KDD'12 summer school

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 KDD'12 summer school

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) KDD'12 summer school

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 KDD'12 summer school

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 KDD'12 summer school

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 … KDD'12 summer school

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 KDD'12 summer school

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

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

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

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

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) [SDM07] KDD'12 summer school

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 KDD'12 summer school

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) KDD'12 summer school

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: Barabasis stack model and like the zero-crossings of a random walk # in links (log) -1.6 days after post (log) KDD'12 summer school

69. CMU SCS KDD'12 summer school C. 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 KDD'12 summer school C. Faloutsos (CMU) 70

71. CMU SCS Probably, power law (?) KDD'12 summer school C. Faloutsos (CMU) 71 ??

72. CMU SCS No Power Law! KDD'12 summer school C. Faloutsos (CMU) 72

73. CMU SCS TLaC: Lazy Contractor The longer a task (phonecall) has taken, The even longer it will take KDD'12 summer school C. 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) KDD'12 summer school C. Faloutsos (CMU)

75. CMU SCS Outliers: KDD'12 summer school C. 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 KDD'12 summer school

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 KDD'12 summer school

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

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)KDD'12 summer school

81. CMU SCS Near-Clique/Star 81 KDD'12 summer school C. Faloutsos (CMU)

82. CMU SCS Near-Clique/Star 82 C. Faloutsos (CMU)KDD'12 summer school

83. CMU SCS Near-Clique/Star 83 C. Faloutsos (CMU)KDD'12 summer school

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

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 KDD'12 summer school

86. CMU SCS KDD'12 summer school C. Faloutsos (CMU) 86 E-bay Fraud detection w/ Polo Chau & Shashank Pandit, CMU [www07]

87. CMU SCS KDD'12 summer school C. Faloutsos (CMU) 87 E-bay Fraud detection

88. CMU SCS KDD'12 summer school C. Faloutsos (CMU) 88 E-bay Fraud detection

89. CMU SCS KDD'12 summer school C. 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?) KDD'12 summer school C. 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 KDD'12 summer school

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 KDD'12 summer school 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 KDD'12 summer school 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 KDD'12 summer school C. 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 KDD'12 summer school

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

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

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

100. CMU SCS Q1: Immunization: ? ? Given a network, k vaccines, and the virus details Which nodes to immunize? KDD'12 summer school 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+, ICDM10] KDD'12 summer school 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) KDD'12 summer school C. 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) KDD'12 summer school C. Faloutsos (CMU) 103 ? ? : attack prob : heal prob depends on connectivity (avg degree? Max degree? variance? Something else?

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

106. CMU SCS KDD'12 summer school C. 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] KDD'12 summer school C. Faloutsos (CMU) 107 ? ?

108. CMU SCS 108 C. Faloutsos (CMU)KDD'12 summer school 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? KDD'12 summer school C. 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+, ICDM10] KDD'12 summer school 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+, ICDM10] KDD'12 summer school 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 KDD'12 summer school

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 KDD'12 summer school 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 KDD'12 summer school 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 KDD'12 summer school 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 KDD'12 summer school

117. CMU SCS KDD'12 summer school C. 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, KDD07] 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 KDD'12 summer school

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, SDM10 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 KDD'12 summer school

120. CMU SCS ???? 19+ [Barabasi+] 120 C. Faloutsos (CMU) Radius Count KDD'12 summer school ~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 KDD'12 summer school ?? ~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 KDD'12 summer school 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 KDD'12 summer school 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 KDD'12 summer school ~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 (?!) KDD'12 summer school

126. CMU SCS Radius Plot of GCC of YahooWeb. 126 C. Faloutsos (CMU)KDD'12 summer school

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. KDD'12 summer school

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. KDD'12 summer school 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. KDD'12 summer school ~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 KDD'12 summer school

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 KDD'12 summer school

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) KDD'12 summer school details

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

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

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

137. CMU SCS 137 Example: GIM-V At Work Connected Components Size Count C. Faloutsos (CMU) KDD'12 summer school 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) KDD'12 summer school 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) KDD'12 summer school

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)KDD'12 summer school

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

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 KDD'12 summer school

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

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) KDD'12 summer school

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 KDD'12 summer school

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 KDD'12 summer school

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. KDD'12 summer school

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 KDD'12 summer school

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 KDD'12 summer school

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

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 KDD'12 summer school

153. CMU SCS C. Faloutsos (CMU) 153 Project info Akoglu, Leman Chau, Polo Kang, U McGlohon, Mary Tong, Hanghang Prakash, Aditya KDD'12 summer school 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