1. CMU SCS Large Graph Mining – Patterns, Tools and Cascade analysis Christos Faloutsos CMU

2. CMU SCS Thank you! Michael Katehakis Hui Xiong Spiros Papadimitriou Luz Kosar Rutgers, March 2013C. Faloutsos (CMU) 2

3. CMU SCS C. Faloutsos (CMU) 3 Roadmap Introduction – Motivation –Why ‘big data’ –Why (big) graphs? Problem#1: Patterns in graphs Problem#2: Tools Problem#3: Scalability Conclusions Rutgers, March 2013

4. CMU SCS Why ‘big data’ Why? What is the problem definition? Rutgers, March 2013C. Faloutsos (CMU) 4

5. CMU SCS Main message: Big data: often > experts ‘Super Crunchers’ Why Thinking-By-Numbers is the New Way To Be Smart by Ian Ayres, 2008 Google won the machine translation competition 2005 http://www.itl.nist.gov/iad/mig//tests/mt/2005/doc/mt05eval_official_r esults_release_20050801_v3.htmlhttp://www.itl.nist.gov/iad/mig//tests/mt/2005/doc/mt05eval_official_r esults_release_20050801_v3.html Rutgers, March 2013C. Faloutsos (CMU) 5

6. CMU SCS Problem definition – big picture Rutgers, March 2013C. Faloutsos (CMU) 6 Tera/Peta-byte data AnalyticsInsights, outliers

7. CMU SCS Problem definition – big picture Rutgers, March 2013C. Faloutsos (CMU) 7 Tera/Peta-byte data AnalyticsInsights, outliers Main emphasis in this talk

8. CMU SCS C. Faloutsos (CMU) 8 Roadmap Introduction – Motivation –Why ‘big data’ –Why (big) graphs? Problem#1: Patterns in graphs Problem#2: Tools Problem#3: Scalability Conclusions Rutgers, March 2013

9. CMU SCS C. Faloutsos (CMU) 9 Graphs - why should we care? Internet Map [lumeta.com] Food Web [Martinez ’91] >$10B revenue >0.5B users Rutgers, March 2013

10. CMU SCS C. Faloutsos (CMU) 10 Graphs - why should we care? IR: bi-partite graphs (doc-terms) web: hyper-text graph... and more: D1D1 DNDN T1T1 TMTM... Rutgers, March 2013

11. CMU SCS C. Faloutsos (CMU) 11 Graphs - why should we care? web-log (‘blog’) news propagation computer network security: email/IP traffic and anomaly detection ‘viral’ marketing Supplier-supply business chains (-> instabilities).... Subject-verb-object -> graph Many-to-many db relationship -> graph Rutgers, March 2013

12. CMU SCS C. Faloutsos (CMU) 12 Outline Introduction – Motivation Problem#1: Patterns in graphs –Static graphs –Weighted graphs –Time evolving graphs Problem#2: Tools Problem#3: Scalability Conclusions Rutgers, March 2013

13. CMU SCS C. Faloutsos (CMU) 13 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? Rutgers, March 2013

14. CMU SCS C. Faloutsos (CMU) 14 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 Rutgers, March 2013

15. CMU SCS C. Faloutsos (CMU) 15 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… Rutgers, March 2013

16. CMU SCS C. Faloutsos (CMU) 16 Graph mining Are real graphs random? Rutgers, March 2013

17. CMU SCS C. Faloutsos (CMU) 17 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 Rutgers, March 2013

18. CMU SCS C. Faloutsos (CMU) 18 Solution# S.1 Power law in the degree distribution [SIGCOMM99] log(rank) log(degree) internet domains att.com ibm.com Rutgers, March 2013

19. CMU SCS C. Faloutsos (CMU) 19 Solution# S.1 Power law in the degree distribution [SIGCOMM99] log(rank) log(degree) -0.82 internet domains att.com ibm.com Rutgers, March 2013

20. CMU SCS C. Faloutsos (CMU) 20 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 Rutgers, March 2013

21. CMU SCS Many more power laws # 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’ Rutgers, March 2013C. Faloutsos (CMU) 21

22. 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 Rutgers, March 2013

23. CMU SCS C. Faloutsos (CMU) 23 Solution# S.3: Triangle ‘Laws’ Real social networks have a lot of triangles Rutgers, March 2013

24. 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? Rutgers, March 2013

25. CMU SCS C. Faloutsos (CMU) 25 Triangle Law: #S.3 [Tsourakakis ICDM 2008] SNReuters Epinions X-axis: degree Y-axis: mean # triangles n friends -> ~n 1.6 triangles Rutgers, March 2013

26. CMU SCS C. Faloutsos (CMU) 26 Triangle Law: Computations [Tsourakakis ICDM 2008] But: triangles are expensive to compute (3-way join; several approx. algos) – O(d max 2 ) Q: Can we do that quickly? A: details Rutgers, March 2013

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) – O(d max 2 ) 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! - O(E) details Rutgers, March 2013

28. CMU SCS C. Faloutsos (CMU) 28 Triangle Law: Computations [Tsourakakis ICDM 2008] 1000x+ speed-up, >90% accuracy details Rutgers, March 2013

29. CMU SCS Triangle counting for large graphs? Anomalous nodes in Twitter(~ 3 billion edges) [U Kang, Brendan Meeder, +, PAKDD’11] 29 Rutgers, March 2013 29 C. Faloutsos (CMU) ?? ?

30. CMU SCS Triangle counting for large graphs? Anomalous nodes in Twitter(~ 3 billion edges) [U Kang, Brendan Meeder, +, PAKDD’11] 30 Rutgers, March 2013 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 Rutgers, March 2013 31 C. Faloutsos (CMU)

32. CMU SCS Triangle counting for large graphs? Anomalous nodes in Twitter(~ 3 billion edges) [U Kang, Brendan Meeder, +, PAKDD’11] 32 Rutgers, March 2013 32 C. Faloutsos (CMU)

33. CMU SCS C. Faloutsos (CMU) 33 Roadmap Introduction – Motivation Problem#1: Patterns in graphs –Static graphs degree, diameter, eigen, triangles cliques –Weighted graphs –Time evolving graphs Problem#2: Tools Rutgers, March 2013

34. CMU SCS C. Faloutsos (CMU) 34 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 Rutgers, March 2013

35. CMU SCS C. Faloutsos (CMU) 35 Observation W.1: Fortification Q: How do the weights of nodes relate to degree? Rutgers, March 2013

36. CMU SCS C. Faloutsos (CMU) 36 Observation W.1: Fortification More donors, more $ ? $10 $5 Rutgers, March 2013 ‘Reagan’ ‘Clinton’ $7

37. CMU SCS Edges (# donors) In-weights ($) C. Faloutsos (CMU) 37 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 Rutgers, March 2013

38. CMU SCS C. Faloutsos (CMU) 38 Roadmap Introduction – Motivation Problem#1: Patterns in graphs –Static graphs –Weighted graphs –Time evolving graphs Problem#2: Tools … Rutgers, March 2013

39. CMU SCS C. Faloutsos (CMU) 39 Problem: Time evolution with Jure Leskovec (CMU -> Stanford) and Jon Kleinberg (Cornell – sabb. @ CMU) Rutgers, March 2013

40. CMU SCS C. Faloutsos (CMU) 40 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? Rutgers, March 2013

41. CMU SCS C. Faloutsos (CMU) 41 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 Rutgers, March 2013

42. CMU SCS C. Faloutsos (CMU) 42 T.1 Diameter – “Patents” Patent citation network 25 years of data @1999 –2.9 M nodes –16.5 M edges time [years] diameter Rutgers, March 2013

43. CMU SCS C. Faloutsos (CMU) 43 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) Rutgers, March 2013

44. CMU SCS C. Faloutsos (CMU) 44 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’’ Rutgers, March 2013

45. CMU SCS C. Faloutsos (CMU) 45 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 Rutgers, March 2013

46. CMU SCS C. Faloutsos (CMU) 46 Roadmap Introduction – Motivation Problem#1: Patterns in graphs –Static graphs –Weighted graphs –Time evolving graphs Problem#2: Tools … Rutgers, March 2013

47. CMU SCS C. Faloutsos (CMU) 47 T.3 : popularity over time Post popularity drops-off – exponentially? lag: days after post # in links 1 2 3 @t @t + lag Rutgers, March 2013

48. CMU SCS C. Faloutsos (CMU) 48 T.3 : popularity over time Post popularity drops-off – exponentially? POWER LAW! Exponent? # in links (log) days after post (log) Rutgers, March 2013

49. CMU SCS C. Faloutsos (CMU) 49 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) Rutgers, March 2013

50. CMU SCS Rutgers, March 2013C. Faloutsos (CMU) 50 -1.5 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)

51. CMU SCS Rutgers, March 2013C. Faloutsos (CMU) 51 -1.5 slope J. G. Oliveira & A.-L. Barabási Human Dynamics: The Correspondence Patterns of Darwin and Einstein. Nature 437, 1251 (2005). [PDF]PDF

52. CMU SCS C. Faloutsos (CMU) 52 Roadmap Introduction – Motivation Problem#1: Patterns in graphs Problem#2: Tools –Belief Propagation –Tensors –Spike analysis Problem#3: Scalability Conclusions Rutgers, March 2013

53. CMU SCS Rutgers, March 2013C. Faloutsos (CMU) 53 E-bay Fraud detection w/ Polo Chau & Shashank Pandit, CMU [www’07]

54. CMU SCS Rutgers, March 2013C. Faloutsos (CMU) 54 E-bay Fraud detection

55. CMU SCS Rutgers, March 2013C. Faloutsos (CMU) 55 E-bay Fraud detection

56. CMU SCS Rutgers, March 2013C. Faloutsos (CMU) 56 E-bay Fraud detection - NetProbe

57. CMU SCS Rutgers, March 2013C. Faloutsos (CMU) 57 E-bay Fraud detection - NetProbe FAH F99% A H49% Compatibility matrix heterophily details

58. CMU SCS Popular press And less desirable attention: E-mail from ‘Belgium police’ (‘copy of your code?’) Open House, 2013C. Faloutsos (CMU) 58

59. CMU SCS Same algo: for malware (‘Polonium’) 50 million machines anonymously reported their executable files 900 million unique files 60 terabytes of data (Jan 2010) Want to label malware and good files 59

60. CMU SCS Polonium: Problem Definition Given –Undirected machine-file bipartite graph 1 billion nodes (machines and files) 37 billion edges Some file labels from Symantec (good or bad) Find Labels for all other files Results : significantly more true positives, same false positives 60

61. CMU SCS C. Faloutsos (CMU) 61 Roadmap Introduction – Motivation Problem#1: Patterns in graphs Problem#2: Tools –Belief Propagation –Tensors –Spike analysis Problem#3: Scalability Conclusions Rutgers, March 2013

62. CMU SCS GigaTensor: Scaling Tensor Analysis Up By 100 Times – Algorithms and Discoveries U Kang Christos Faloutsos KDD’12 Evangelos Papalexakis Abhay Harpale Rutgers, March 2013 62 C. Faloutsos (CMU)

63. CMU SCS Background: Tensor Tensors (=multi-dimensional arrays) are everywhere –Hyperlinks &anchor text [Kolda+,05] URL 1 URL 2 Anchor Text Java C++ C# 1 1 1 1 1 1 1 Rutgers, March 2013 63 C. Faloutsos (CMU)

64. 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 Rutgers, March 2013 64 C. Faloutsos (CMU)

65. CMU SCS Background: Tensor Tensors (=multi-dimensional arrays) are everywhere –Sensor stream (time, location, type) –Predicates (subject, verb, object) in knowledge base Rutgers, March 2013 65 C. Faloutsos (CMU) IP-destination IP-source Time-stamp Anomaly Detection in Computer networks

66. CMU SCS all I learned on tensors: from Rutgers, March 2013C. Faloutsos (CMU) 66 Nikos Sidiropoulos UMN Tamara Kolda, Sandia Labs (tensor toolbox)

67. CMU SCS Problem Definition How to decompose a billion-scale tensor? –Corresponds to SVD in 2D case Rutgers, March 2013 67 C. Faloutsos (CMU)

68. 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 Rutgers, March 2013 68 C. Faloutsos (CMU)

69. CMU SCS Experiments GigaTensor solves 100x larger problem Number of nonzero = I / 50 (J) (I) (K) GigaTensor Tensor Toolbox Out of Memory 100x Rutgers, March 2013 69 C. Faloutsos (CMU)

70. CMU SCS A1: Concept Discovery Concept Discovery in Knowledge Base Rutgers, March 2013 70 C. Faloutsos (CMU)

71. CMU SCS A1: Concept Discovery Rutgers, March 2013 71 C. Faloutsos (CMU)

72. CMU SCS A2: Synonym Discovery Rutgers, March 2013 72 C. Faloutsos (CMU)

73. CMU SCS C. Faloutsos (CMU) 73 Roadmap Introduction – Motivation Problem#1: Patterns in graphs Problem#2: Tools –Belief propagation –Tensors –Spike analysis Problem#3: Scalability -PEGASUS Conclusions Rutgers, March 2013

74. CMU SCS Meme (# of mentions in blogs) –short phrases Sourced from U.S. politics in 2008 74 “you can put lipstick on a pig” “yes we can” Rise and fall patterns in social media C. Faloutsos (CMU)Rutgers, March 2013

75. CMU SCS Rise and fall patterns in social media 75 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)Rutgers, March 2013

76. CMU SCS Rise and fall patterns in social media 76 Answer: YES! We can represent all patterns by single model C. Faloutsos (CMU)Rutgers, March 2013 In Matsubara+ SIGKDD 2012

77. CMU SCS 77 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)Rutgers, March 2013 Infectiveness of a blog-post at age n: -Strength of infection (quality of news) -Decay function Time n=n b +1

78. CMU SCS 78 -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)Rutgers, March 2013 Infectiveness of a blog-post at age n: -Strength of infection (quality of news) -Decay function Time n=n b +1 Main idea - SpikeM

79. CMU SCS SpikeM - with periodicity Full equation of SpikeM 79 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)Rutgers, March 2013

80. CMU SCS Details Analysis – exponential rise and power-raw fall 80 Lin-log Log-log Rise-part SI -> exponential SpikeM -> exponential Rise-part SI -> exponential SpikeM -> exponential C. Faloutsos (CMU)Rutgers, March 2013

81. CMU SCS Details Analysis – exponential rise and power-raw fall 81 Lin-log Log-log Fall-part SI -> exponential SpikeM -> power law Fall-part SI -> exponential SpikeM -> power law C. Faloutsos (CMU)Rutgers, March 2013

82. CMU SCS Tail-part forecasts 82 SpikeM can capture tail part C. Faloutsos (CMU)Rutgers, March 2013

83. CMU SCS “What-if” forecasting 83 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)Rutgers, March 2013 ? ?

84. CMU SCS “What-if” forecasting 84 SpikeM can forecast upcoming spikes (1) First spike (2) Release date (3) Two weeks before release C. Faloutsos (CMU)Rutgers, March 2013

85. CMU SCS C. Faloutsos (CMU) 85 Roadmap Introduction – Motivation Problem#1: Patterns in graphs Problem#2: Tools Problem#3: Scalability –PEGASUS –Diameter –Connected components Conclusions Rutgers, March 2013

86. CMU SCS Rutgers, March 2013C. Faloutsos (CMU) 86 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/

87. CMU SCS C. Faloutsos (CMU) 87 CentralizedHadoop/PEG ASUS Degree Distr. old Pagerank old Diameter/ANF old HERE Conn. Comp old HERE TrianglesdoneHERE Visualizationstarted Roadmap – Algorithms & results Rutgers, March 2013

88. 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) 88 Rutgers, March 2013

89. CMU SCS ???? 19+ [Barabasi+] 89 C. Faloutsos (CMU) Radius Count Rutgers, March 2013 ~1999, ~1M nodes

90. CMU SCS YahooWeb graph (120Gb, 1.4B nodes, 6.6 B edges) Largest publicly available graph ever studied. ???? 19+ [Barabasi+] 90 C. Faloutsos (CMU) Radius Count Rutgers, March 2013 ?? ~1999, ~1M nodes

91. CMU SCS YahooWeb graph (120Gb, 1.4B nodes, 6.6 B edges) Largest publicly available graph ever studied. ???? 19+? [Barabasi+] 91 C. Faloutsos (CMU) Radius Count Rutgers, March 2013 14 (dir.) ~7 (undir.)

92. CMU SCS YahooWeb graph (120Gb, 1.4B nodes, 6.6 B edges) 7 degrees of separation (!) Diameter: shrunk ???? 19+? [Barabasi+] 92 C. Faloutsos (CMU) Radius Count Rutgers, March 2013 14 (dir.) ~7 (undir.)

93. CMU SCS YahooWeb graph (120Gb, 1.4B nodes, 6.6 B edges) Q: Shape? ???? 93 C. Faloutsos (CMU) Radius Count Rutgers, March 2013 ~7 (undir.)

94. CMU SCS 94 C. Faloutsos (CMU) YahooWeb graph (120Gb, 1.4B nodes, 6.6 B edges) effective diameter: surprisingly small. Multi-modality (?!) Rutgers, March 2013

95. CMU SCS Radius Plot of GCC of YahooWeb. 95 C. Faloutsos (CMU)Rutgers, March 2013

96. CMU SCS 96 C. Faloutsos (CMU) YahooWeb graph (120Gb, 1.4B nodes, 6.6 B edges) effective diameter: surprisingly small. Multi-modality: probably mixture of cores. Rutgers, March 2013

97. CMU SCS 97 C. Faloutsos (CMU) YahooWeb graph (120Gb, 1.4B nodes, 6.6 B edges) effective diameter: surprisingly small. Multi-modality: probably mixture of cores. Rutgers, March 2013 EN ~7 Conjecture: DE BR

98. CMU SCS 98 C. Faloutsos (CMU) YahooWeb graph (120Gb, 1.4B nodes, 6.6 B edges) effective diameter: surprisingly small. Multi-modality: probably mixture of cores. Rutgers, March 2013 ~7 Conjecture:

99. CMU SCS C. Faloutsos (CMU) 99 Roadmap Introduction – Motivation Problem#1: Patterns in graphs Problem#2: Tools Problem#3: Scalability –PEGASUS –Diameter –Connected components Conclusions Rutgers, March 2013

100. CMU SCS Generalized Iterated Matrix Vector Multiplication (GIMV) C. Faloutsos (CMU) 100 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 Rutgers, March 2013

101. CMU SCS Generalized Iterated Matrix Vector Multiplication (GIMV) C. Faloutsos (CMU) 101 PageRank proximity (RWR) Diameter Connected components (eigenvectors, Belief Prop. … ) Matrix – vector Multiplication (iterated) Rutgers, March 2013 details

102. CMU SCS 102 Example: GIM-V At Work Connected Components – 4 observations: Size Count C. Faloutsos (CMU) Rutgers, March 2013

103. CMU SCS 103 Example: GIM-V At Work Connected Components Size Count C. Faloutsos (CMU) Rutgers, March 2013 1) 10K x larger than next

104. CMU SCS 104 Example: GIM-V At Work Connected Components Size Count C. Faloutsos (CMU) Rutgers, March 2013 2) ~0.7B singleton nodes

105. CMU SCS 105 Example: GIM-V At Work Connected Components Size Count C. Faloutsos (CMU) Rutgers, March 2013 3) SLOPE!

106. CMU SCS 106 Example: GIM-V At Work Connected Components Size Count 300-size cmpt X 500. Why? 1100-size cmpt X 65. Why? C. Faloutsos (CMU) Rutgers, March 2013 4) Spikes!

107. CMU SCS 107 Example: GIM-V At Work Connected Components Size Count suspicious financial-advice sites (not existing now) C. Faloutsos (CMU) Rutgers, March 2013

108. CMU SCS 108 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)Rutgers, March 2013

109. CMU SCS C. Faloutsos (CMU) 109 Roadmap Introduction – Motivation Problem#1: Patterns in graphs Problem#2: Tools Problem#3: Scalability Conclusions Rutgers, March 2013

110. CMU SCS C. Faloutsos (CMU) 110 OVERALL CONCLUSIONS – low level: Several new patterns (fortification, triangle-laws, conn. components, etc) New tools: –belief propagation, gigaTensor, etc Scalability: PEGASUS / hadoop Rutgers, March 2013

111. CMU SCS C. Faloutsos (CMU) 111 OVERALL CONCLUSIONS – high level BIG DATA: Large datasets reveal patterns/outliers that are invisible otherwise Rutgers, March 2013

112. CMU SCS ML & Stats. Comp. Systems Theory & Algo. Biology Econ. Social Science Physics 112 Graph Analytics Rutgers, March 2013C. Faloutsos (CMU)

113. CMU SCS C. Faloutsos (CMU) 113 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) Rutgers, March 2013

114. CMU SCS C. Faloutsos (CMU) 114 References D. Chakrabarti, C. Faloutsos: Graph Mining – Laws, Tools and Case Studies, Morgan Claypool 2012 http://www.morganclaypool.com/doi/abs/10.2200/S004 49ED1V01Y201209DMK006 Rutgers, March 2013

115. CMU SCS C. Faloutsos (CMU) 115 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 Rutgers, March 2013

116. CMU SCS C. Faloutsos (CMU) 116 References Christos Faloutsos, Tamara G. Kolda, Jimeng Sun: Mining large graphs and streams using matrix and tensor tools. Tutorial, SIGMOD Conference 2007: 1174 Rutgers, March 2013

117. CMU SCS C. Faloutsos (CMU) 117 References T. G. Kolda and J. Sun. Scalable Tensor Decompositions for Multi-aspect Data Mining. In: ICDM 2008, pp. 363-372, December 2008. Rutgers, March 2013

118. CMU SCS C. Faloutsos (CMU) 118 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 Rutgers, March 2013

119. CMU SCS References Yasuko Matsubara, Yasushi Sakurai, B. Aditya Prakash, Lei Li, Christos Faloutsos, "Rise and Fall Patterns of Information Diffusion: Model and Implications", KDD’12, pp. 6-14, Beijing, China, August 2012 Rutgers, March 2013C. Faloutsos (CMU) 119

120. CMU SCS C. Faloutsos (CMU) 120 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 Rutgers, March 2013

121. CMU SCS References Jimeng Sun, Dacheng Tao, Christos Faloutsos: Beyond streams and graphs: dynamic tensor analysis. KDD 2006: 374- 383 Rutgers, March 2013C. Faloutsos (CMU) 121

122. CMU SCS C. Faloutsos (CMU) 122 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 Rutgers, March 2013

123. CMU SCS C. Faloutsos (CMU) 123 References Hanghang Tong, Christos Faloutsos, Brian Gallagher, Tina Eliassi-Rad: Fast best- effort pattern matching in large attributed graphs. KDD 2007: 737-746 (Best paper award, CIKM'12) Hanghang Tong, B. Aditya Prakash, Tina Eliassi-Rad, Michalis Faloutsos and Christos Faloutsos Gelling, and Melting, Large Graphs by Edge Manipulation, Maui, Hawaii, USA, Oct. 2012. Gelling, and Melting, Large Graphs by Edge Manipulation Rutgers, March 2013

124. CMU SCS C. Faloutsos (CMU) 124 References Hanghang Tong, Spiros Papadimitriou, Christos Faloutsos, Philip S. Yu, Tina Eliassi-Rad: Gateway finder in large graphs: problem definitions and fast solutions. Inf. Retr. 15(3-4): 391-411 (2012) Rutgers, March 2013

125. CMU SCS C. Faloutsos (CMU) 125 Project info & ‘thanks’ Rutgers, March 2013 Thanks to: NSF IIS-0705359, IIS-0534205, CTA-INARC ; Yahoo (M45), LLNL, IBM, SPRINT, Google, INTEL, HP, iLab www.cs.cmu.edu/~pegasus

126. CMU SCS C. Faloutsos (CMU) 126 Cast Akoglu, Leman Chau, Polo Kang, U McGlohon, Mary Tong, Hanghang Prakash, Aditya Rutgers, March 2013 Koutra, Danae Beutel, Alex Papalexakis, Vagelis

127. CMU SCS Take-home message Rutgers, March 2013C. Faloutsos (CMU) 127 Tera/Peta-byte data AnalyticsInsights, outliers Big data reveal insights that would be invisible otherwise (even to experts)