1. CMU SCS Large Graph Mining Christos Faloutsos CMU

2. CMU SCS ICDM-LDMTA 2009C. Faloutsos 2 Thank you! Dr. Yan Liu Dr. Jimeng Sun

3. CMU SCS ICDM-LDMTA 2009C. Faloutsos 3 Outline Introduction – Motivation Problem#1: Patterns in graphs Problem#2: Generators Problem#3: Scalability Conclusions

4. CMU SCS ICDM-LDMTA 2009C. Faloutsos 4 Graphs - why should we care? Internet Map [lumeta.com] Food Web [Martinez ’91] Protein Interactions [genomebiology.com] Friendship Network [Moody ’01]

5. CMU SCS ICDM-LDMTA 2009C. Faloutsos 5 Graphs - why should we care? IR: bi-partite graphs (doc-terms) web: hyper-text graph... and more: D1D1 DNDN T1T1 TMTM...

6. CMU SCS ICDM-LDMTA 2009C. Faloutsos 6 Graphs - why should we care? network of companies & board-of-directors members ‘viral’ marketing web-log (‘blog’) news propagation computer network security: email/IP traffic and anomaly detection....

7. CMU SCS ICDM-LDMTA 2009C. Faloutsos 7 Outline Introduction – Motivation Problem#1: Patterns in graphs –Static graphs –Weighted graphs –Time evolving graphs Problem#2: Generators Problem#3: Scalability Conclusions

8. CMU SCS ICDM-LDMTA 2009C. Faloutsos 8 Problem #1 - network and graph mining How does the Internet look like? How does the web look like? What is ‘normal’/‘abnormal’? which patterns/laws hold?

9. CMU SCS ICDM-LDMTA 2009C. Faloutsos 9 Graph mining Are real graphs random?

10. CMU SCS ICDM-LDMTA 2009C. Faloutsos 10 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 (‘it is amazing what you hear, when you listen’)

11. CMU SCS ICDM-LDMTA 2009C. Faloutsos 11 Solution# S.1 Power law in the degree distribution [SIGCOMM99] log(rank) log(degree) -0.82 internet domains att.com ibm.com

12. CMU SCS ICDM-LDMTA 2009C. Faloutsos 12 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

13. CMU SCS ICDM-LDMTA 2009C. Faloutsos 13 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

14. CMU SCS ICDM-LDMTA 2009C. Faloutsos 14 But: How about graphs from other domains?

15. CMU SCS ICDM-LDMTA 2009C. Faloutsos 15 More settings w/ power laws: citation counts: (citeseer.nj.nec.com 6/2001) log(#citations) log(count) Ullman

16. CMU SCS ICDM-LDMTA 2009C. Faloutsos 16 More power laws: web hit counts [w/ A. Montgomery] Web Site Traffic log(in-degree) log(count) Zipf users sites ``ebay’’

17. CMU SCS ICDM-LDMTA 2009C. Faloutsos 17 epinions.com who-trusts-whom [Richardson + Domingos, KDD 2001] (out) degree count trusts-2000-people user

18. CMU SCS ICDM-LDMTA 2009C. Faloutsos 18 Outline Introduction – Motivation Problem#1: Patterns in graphs –Static graphs degree, diameter, eigen*, triangles cliques –Weighted graphs –Time evolving graphs Problem#2: Generators

19. CMU SCS ICDM-LDMTA 2009C. Faloutsos 19 Solution# S.3: Triangle ‘Laws’ Real social networks have a lot of triangles

20. CMU SCS ICDM-LDMTA 2009C. Faloutsos 20 Triangle ‘Laws’ Real social networks have a lot of triangles –Friends of friends are friends Any patterns?

21. CMU SCS ICDM-LDMTA 2009C. Faloutsos 21 Triangle Law: #S.3 [Tsourakakis ICDM 2008] ASNHEP-TH Epinions X-axis: # of Triangles a node participates in Y-axis: count of such nodes

22. CMU SCS ICDM-LDMTA 2009C. Faloutsos 22 Triangle Law: #S.4 [Tsourakakis ICDM 2008] SNReuters Epinions X-axis: degree Y-axis: mean # triangles Notice: slope ~ degree exponent (insets)

23. CMU SCS ICDM-LDMTA 2009C. Faloutsos 23 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

24. CMU SCS ICDM-LDMTA 2009C. Faloutsos 24 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, we only need the top few eigenvalues! details

25. CMU SCS ICDM-LDMTA 2009C. Faloutsos 25 Triangle Law: Computations [Tsourakakis ICDM 2008] 1000x+ speed-up, high accuracy details

26. CMU SCS ICDM-LDMTA 2009C. Faloutsos 26 Outline Introduction – Motivation Problem#1: Patterns in graphs –Static graphs degree, diameter, eigen*, triangles cliques –Weighted graphs –Time evolving graphs Problem#2: Generators

27. CMU SCS Large Human Communication Networks Patterns and a Utility-Driven Generator Nan Du, Christos Faloutsos, Bai Wang, Leman Akoglu KDD 2009

28. CMU SCS ICDM-LDMTA 2009C. Faloutsos 28 Cliques Clique is a complete subgraph. If a clique can not be contained by any larger clique, it is called the maximal clique. 20 13 4

29. CMU SCS ICDM-LDMTA 2009C. Faloutsos 29 Clique Clique is a complete subgraph. If a clique can not be contained by any larger clique, it is called the maximal clique. 20 13 4

30. CMU SCS ICDM-LDMTA 2009C. Faloutsos 30 Clique Clique is a complete subgraph. If a clique can not be contained by any larger clique, it is called the maximal clique. 20 13 4

31. CMU SCS ICDM-LDMTA 2009C. Faloutsos 31 Clique Clique is a complete subgraph. If a clique can not be contained by any larger clique, it is called the maximal clique. {0,1,2}, {0,1,3}, {1,2,3} {2,3,4}, {0,1,2,3} are cliques; {0,1,2,3} and {2,3,4} are the maximal cliques. 20 13 4

32. CMU SCS ICDM-LDMTA 2009C. Faloutsos 32 S.5: Clique-Degree Power-Law Power law: More friends, even more social circles ! # maximal cliques of node i degree of node i Dataset: who-calls-whom anonymized, Over several time units

33. CMU SCS ICDM-LDMTA 2009C. Faloutsos 33 S.5 Clique-Degree Power-Law Outlier Detection

34. CMU SCS ICDM-LDMTA 2009C. Faloutsos 34 1.5 Clique-Degree Power-Law Outlier Detection

35. CMU SCS ICDM-LDMTA 2009C. Faloutsos 35 Outline Introduction – Motivation Problem#1: Patterns in graphs –Static graphs degree, diameter, eigen*, triangles cliques –Weighted graphs –Time evolving graphs Problem#2: Generators

36. CMU SCS ICDM-LDMTA 2009C. Faloutsos 36 Observation W.1 Question : Nodes in a triangle are topologically equivalent. Will they also give equal number of phone calls to each other ? Max Weight Min Weight Mid Weight

37. CMU SCS ICDM-LDMTA 2009C. Faloutsos 37 Observation W.1:Triangle Weight Law Periods S1 – S3

38. CMU SCS ICDM-LDMTA 2009C. Faloutsos 38 Other 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

39. CMU SCS ICDM-LDMTA 2009C. Faloutsos 39 Observation W.2: fortification Q: How do the weights of nodes relate to degree?

40. CMU SCS Edges (# donors) In-weights ($) ICDM-LDMTA 2009C. Faloutsos 40 Observation W.2: fortification: Snapshot Power Law weight of a node is super-linear on the in-degree with PL exponent ‘iw’: –i.e. 1.01 < iw < 1.26, super-linear Orgs-Candidates e.g. John Kerry, $10M received, from 1K donors More donors, even more $ $10 $5

41. CMU SCS ICDM-LDMTA 2009C. Faloutsos 41 Are there deviations from P.L.? A: yes – but they also correspond to very skewed distributions (‘black/gray swans’)

42. CMU SCS “ Mobile Call Graphs: Beyond Power Law and Lognormal Distributions” K D D ’ 0 8 Mukund Seshadri, Sridhar Machiraju, Ashwin Sridharan, Jean Bolot Christos Faloutsos, Jure Leskovec

43. CMU SCS ICDM-LDMTA 2009C. Faloutsos 43 Dataset Who-calls-whom (anonymized) Degree distribution, in green Area S1 Time period T1 1 month Count Degree of Mobile-phone call graph.... Observed Data --- Power Law Fit + Observed Data.... LogNormal Fit

44. CMU SCS ICDM-LDMTA 2009C. Faloutsos 44 A poor fit? Power Laws: p(x) ~ x^(-a) Lognormal: log(x) is Normal Count Degree of Mobile-phone call graph.... Observed Data --- Power Law Fit + Observed Data.... LogNormal Fit Area S1 Time period T1 1 month

45. CMU SCS ICDM-LDMTA 2009C. Faloutsos 45 Solution: DPLN Double Pareto Log Normal (Reed, 2003) 4 parameters: [α,β,ν,τ] Linear head and tail. Degree Area S1, Time Period T1 β α “Rich get Richer” BUT Population lifetimes NOT identical count

46. CMU SCS ICDM-LDMTA 2009C. Faloutsos 46 Datasets Anonymized monthly aggregates of call metrics per user –Collected at 4 coverage areas S1,S2,S3,S4 –Collected for two month-long periods T1 and T2, separated by 6 months –Total coverage: >1 million users >10 million calls ~7000 square miles.

47. CMU SCS ICDM-LDMTA 2009C. Faloutsos 47 Degree (No. of Call Partners) Calls Talk Time Total over a month, per user Area S1 Time-period T1 Degree Distribution Metrics

48. CMU SCS ICDM-LDMTA 2009C. Faloutsos 48 Persistence – Across Metric, Time, Space Partners S2, T2 Partners S1, T2 Calls Area S1, Time T1

49. CMU SCS ICDM-LDMTA 2009C. Faloutsos 49 Outlier detection –Many un-answered calls => multiples of 27 sec! Applications Per-user distribution of total talk time (@ T1, S1)

50. CMU SCS ICDM-LDMTA 2009C. Faloutsos 50 Outline Introduction – Motivation Problem#1: Patterns in graphs –Static graphs –Weighted graphs –Time evolving graphs Problem#2: Generators …

51. CMU SCS ICDM-LDMTA 2009C. Faloutsos 51 Problem: Time evolution with Jure Leskovec (CMU -> Stanford) and Jon Kleinberg (Cornell – sabb. @ CMU)

52. CMU SCS ICDM-LDMTA 2009C. Faloutsos 52 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?

53. CMU SCS ICDM-LDMTA 2009C. Faloutsos 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? Diameter shrinks over time

54. CMU SCS ICDM-LDMTA 2009C. Faloutsos 54 T.1 Diameter – “Patents” Patent citation network 25 years of data time [years] diameter

55. CMU SCS ICDM-LDMTA 2009C. Faloutsos 55 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)

56. CMU SCS ICDM-LDMTA 2009C. Faloutsos 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) A: over-doubled! –But obeying the ``Densification Power Law’’

57. CMU SCS ICDM-LDMTA 2009C. Faloutsos 57 T.2 Densification – Patent Citations Citations among patents granted 1999 –2.9 million nodes –16.5 million edges Each year is a datapoint N(t) E(t) 1.66

58. CMU SCS ICDM-LDMTA 2009C. Faloutsos 58 Outline Introduction – Motivation Problem#1: Patterns in graphs –Static graphs –Weighted graphs –Time evolving graphs Problem#2: Generators …

59. CMU SCS ICDM-LDMTA 2009C. Faloutsos 59 More on Time-evolving graphs M. McGlohon, L. Akoglu, and C. Faloutsos Weighted Graphs and Disconnected Components: Patterns and a Generator. SIG-KDD 2008

60. CMU SCS ICDM-LDMTA 2009C. Faloutsos 60 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?

61. CMU SCS ICDM-LDMTA 2009C. Faloutsos 61 Observation T.4: NLCC behavior After the gelling point, the GCC takes off, but NLCC’s remain ~constant (actually, oscillate). IMDB CC size Time-stamp

62. CMU SCS ICDM-LDMTA 2009C. Faloutsos 62 T.5 How do new edges appear? [LBKT’08] Microscopic Evolution of Social Networks Jure Leskovec, Lars Backstrom, Ravi Kumar, Andrew Tomkins. (ACM KDD), 2008.

63. CMU SCS ICDM-LDMTA 2009C. Faloutsos 63 Edge gap δ(d): inter-arrival time between d th and d+1 st edge T.5 How do edges appear in time? [LBKT’08]  What is the PDF of  Poisson?

64. CMU SCS ICDM-LDMTA 2009C. Faloutsos 64 Edge gap δ(d): inter-arrival time between d th and d+1 st edge LinkedIn T.5 How do edges appear in time? [LBKT’08]

65. CMU SCS ICDM-LDMTA 2009C. Faloutsos 65 Similarly for Blogs with Mary McGlohon (CMU) Jure Leskovec (CMU->Stanford) Natalie Glance (now at Google) Mat Hurst (now at MSR) [SDM’07]

66. CMU SCS ICDM-LDMTA 2009C. Faloutsos 66 T.6 : popularity over time Post popularity drops-off – exponentially? lag: days after post # in links 1 2 3 @t @t + lag

67. CMU SCS ICDM-LDMTA 2009C. Faloutsos 67 T.6 : popularity over time Post popularity drops-off – exponentially? POWER LAW! Exponent? # in links (log) 1 2 3 days after post (log)

68. CMU SCS ICDM-LDMTA 2009C. Faloutsos 68 T.6 : 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 2 3 -1.6 days after post (log)

69. CMU SCS ICDM-LDMTA 2009C. Faloutsos 69 Outline Introduction – Motivation Problem#1: Patterns in graphs Problem#2: Generators Problem#3: Scalability Conclusions

70. CMU SCS ICDM-LDMTA 2009C. Faloutsos 70 Problem#2: Generation Goal: Generate a realistic sequence of graphs that 1.obey all the patterns including weights on the edges and timestamps on the edges 2.needs no randomness 3.needs no parameters (or as few as possible)

71. CMU SCS ICDM-LDMTA 2009C. Faloutsos 71 PaC generator Why do we need generators? –What-if scenarios –How to attract/retain customers/members –Stress-testing algorithms, when real graphs are unavailable (eg., due to privacy, national/corporate security, etc)

72. CMU SCS ICDM-LDMTA 2009C. Faloutsos 72 (NUMEROUS earlier generators) preferential attachment [Barabasi+] copying model [Kleinberg+] winner does not take all [Giles+] Kronecker graphs... (survey: [Chakrabarti+, ’06])

73. CMU SCS ICDM-LDMTA 2009C. Faloutsos 73 Problem#2: Generation Goal: Generate a realistic sequence of graphs that 1.obey all the patterns including weights on the edges and timestamps on the edges 2.needs no randomness 3.needs no parameters (or as few as possible) Proposed ‘PaC’ model – idea: design ‘agents’, who try to max utility of contacts

74. CMU SCS ICDM-LDMTA 2009C. Faloutsos 74 PaC Model People benefit from calling each other. A Pay and Call game = PaC Model The payoffs are measured as “emotional dollars”. agent Prob p l to stay in the game Friendliness 0<=Fi<=1 capital

75. CMU SCS ICDM-LDMTA 2009C. Faloutsos 75 Outline of Agent Behavior Step 1: decide to stay (P L ) Step 2: if stay, call the most profitable person(s) –Existing friend (‘exploit’) –Stranger (‘explore’) or ask for recommendation (if available) to maximize benefits Exponential lifetime Rich get richer Closing Triangle

76. CMU SCS ICDM-LDMTA 2009C. Faloutsos 76 Validation of PaC Choose the following parameters – Ran 35 simulations 100,000 agents per simulation Variation of the parameters does not change the shape of the distribution

77. CMU SCS ICDM-LDMTA 2009C. Faloutsos 77 Validation of PaC Ran 35 simulations 100,000 agents per simulation Variation of the parameters does not change the shape of the distribution

78. CMU SCS ICDM-LDMTA 2009C. Faloutsos 78 Goals of Validation G1: Skewed degree/node weight distribution G2: Snapshot Power-Law G3: Skewed connected components distribution G4: Clique-Degree Power-Law G5: Clique-Participation Law G6: Triangle Weight Law

79. CMU SCS ICDM-LDMTA 2009C. Faloutsos 79 Validation of PaC G1: Skewed Degree / Node Weight Distribution Real Network Synthetic Network

80. CMU SCS ICDM-LDMTA 2009C. Faloutsos 80 Validation of PaC G2: Snapshot Power Law [McGlohon, Akoglu, Faloutsos 08] “ more partners, even more calls” Real NetworkSynthetic Network

81. CMU SCS ICDM-LDMTA 2009C. Faloutsos 81 Validation of PaC G3: Skewed distribution of the connected components Real NetworkSynthetic Network

82. CMU SCS ICDM-LDMTA 2009C. Faloutsos 82 Validation of PaC G4: Clique Degree Power Law Real NetworkSynthetic Network

83. CMU SCS ICDM-LDMTA 2009C. Faloutsos 83 Validation of PaC G5: Clique Participation Law Real NetworkSynthetic Network

84. CMU SCS ICDM-LDMTA 2009C. Faloutsos 84 Validation of PaC G6: Triangle Weight Law Real Network Synthetic Network

85. CMU SCS ICDM-LDMTA 2009C. Faloutsos 85 Validation of PaC G1: Skewed degree/node weight distribution G2: Snapshot Power-Law G3: Skewed connected components distribution G4: Clique-Degree Power-Law G5: Clique-Participation Law G6: Triangle Weight Law

86. CMU SCS ICDM-LDMTA 2009C. Faloutsos 86 Outline Introduction – Motivation Problem#1: Patterns in graphs Problem#2: Generators Problem#3: Scalability Conclusions

87. CMU SCS ICDM-LDMTA 2009C. Faloutsos 87 Scalability How about if graph/tensor does not fit in core? How about handling huge graphs?

88. CMU SCS ICDM-LDMTA 2009C. Faloutsos 88 Scalability How about if graph/tensor does not fit in core? [‘MET’: Kolda, Sun, ICMD’08, best paper award] How about handling huge graphs?

89. CMU SCS ICDM-LDMTA 2009C. Faloutsos 89 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/

90. CMU SCS ICDM-LDMTA 2009C. Faloutsos 90 2’ intro to hadoop master-slave architecture; n-way replication (default n=3) ‘group by’ of SQL (in parallel, fault-tolerant way) e.g, find histogram of word frequency –compute local histograms –then merge into global histogram select course-id, count(*) from ENROLLMENT group by course-id details

91. CMU SCS ICDM-LDMTA 2009C. Faloutsos 91 2’ intro to hadoop master-slave architecture; n-way replication (default n=3) ‘group by’ of SQL (in parallel, fault-tolerant way) e.g, find histogram of word frequency –compute local histograms –then merge into global histogram select course-id, count(*) from ENROLLMENT group by course-id map reduce details

92. CMU SCS ICDM-LDMTA 2009C. Faloutsos 92 User Program Reducer Master Mapper fork assign map assign reduce read local write remote read, sort Output File 0 Output File 1 write Split 0 Split 1 Split 2 Input Data (on HDFS) By default: 3-way replication; Late/dead machines: ignored, transparently (!) details

93. CMU SCS ICDM-LDMTA 2009C. Faloutsos 93 PEGASUS: A Peta-Scale Graph Mining System, ICDM’09 U Kang, Charalampos Tsourakakis, C. Faloutsos

94. CMU SCS ICDM-LDMTA 2009C. Faloutsos 94 the PageRank distribution: power-law Analysis of Real Networks YahooWeb: 1.4B nodes and 6.6B edges, 120Gb (largest public graph analyzed) M45 (Y!): 1.5Pb disk 3Tb RAM 4000 cores among top 500

95. CMU SCS ICDM-LDMTA 2009C. Faloutsos 95 Spikes? Analysis of Real Networks YahooWeb: 1.4 B nodes and 6.6 B edges - Connected components: size count

96. CMU SCS ICDM-LDMTA 2009C. Faloutsos 96 Spikes: due to replications of pages. Analysis of Real Networks YahooWeb: 1.4 B nodes and 6.6 B edges - Connected components: size count

97. CMU SCS ICDM-LDMTA 2009C. Faloutsos 97 Evolution of connected components of LinkedIn: stable, *after* the gelling point (@2004) LinkedIn: 7.5M nodes and 58M edges details

98. CMU SCS ICDM-LDMTA 2009C. Faloutsos 98 OVERALL CONCLUSIONS – low level: Several new patterns (fortification, triangle- laws, etc) New generator: ‘PaC’ (agent based) Scalability / cloud computing -> PetaBytes

99. CMU SCS ICDM-LDMTA 2009C. Faloutsos 99 OVERALL CONCLUSIONS – high level good news: unprecedented opportunities (huge datasets, available on-line) better news: cross-disciplinary work: –DB/sys -> scalability –ML/stat -> tools –Econ, Soc -> experience; what questions to ask –phys, math -> phase transitions, tensors, eigenvalues –…

100. CMU SCS ICDM-LDMTA 2009C. Faloutsos 100 References Hanghang Tong, Christos Faloutsos, and Jia-Yu Pan Fast Random Walk with Restart and Its Applications ICDM 2006, Hong Kong.Fast Random Walk with Restart and Its Applications Hanghang Tong, Christos Faloutsos Center-Piece Subgraphs: Problem Definition and Fast Solutions, KDD 2006, Philadelphia, PACenter-Piece Subgraphs: Problem Definition and Fast Solutions, T. G. Kolda and J. Sun. Scalable Tensor Decompositions for Multi-aspect Data Mining. In: ICDM 2008, pp. 363-372, December 2008.

101. CMU SCS ICDM-LDMTA 2009C. Faloutsos 101 References Jure Leskovec, Jon Kleinberg and Christos Faloutsos Graphs over Time: Densification Laws, Shrinking Diameters and Possible Explanations KDD 2005, Chicago, IL. ("Best Research Paper" award).Graphs over Time: Densification Laws, Shrinking Diameters and Possible Explanations Jure Leskovec, Deepayan Chakrabarti, Jon Kleinberg, Christos Faloutsos Realistic, Mathematically Tractable Graph Generation and Evolution, Using Kronecker Multiplication (ECML/PKDD 2005), Porto, Portugal, 2005.Realistic, Mathematically Tractable Graph Generation and Evolution, Using Kronecker MultiplicationECML/PKDD 2005

102. CMU SCS ICDM-LDMTA 2009C. Faloutsos 102 References Jure Leskovec and Christos Faloutsos, Scalable Modeling of Real Graphs using Kronecker Multiplication, ICML 2007, Corvallis, OR, USA Shashank Pandit, Duen Horng (Polo) Chau, Samuel Wang and Christos Faloutsos NetProbe: A Fast and Scalable System for Fraud Detection in Online Auction Networks WWW 2007, Banff, Alberta, Canada, May 8-12, 2007.NetProbe: A Fast and Scalable System for Fraud Detection in Online Auction Networks Jimeng Sun, Dacheng Tao, Christos Faloutsos Beyond Streams and Graphs: Dynamic Tensor Analysis, KDD 2006, Philadelphia, PA Beyond Streams and Graphs: Dynamic Tensor Analysis,

103. CMU SCS ICDM-LDMTA 2009C. Faloutsos 103 References Jimeng Sun, Yinglian Xie, Hui Zhang, Christos Faloutsos. Less is More: Compact Matrix Decomposition for Large Sparse Graphs, SDM, Minneapolis, Minnesota, Apr 2007. [pdf]pdf 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

104. CMU SCS ICDM-LDMTA 2009C. Faloutsos 104 Contact info: www. cs.cmu.edu /~christos Thanks again to the organizers: Jimeng Sun, Yan Liu Funding sources: NSF IIS-0705359, IIS-0534205, DBI-0640543 LLNL, PITA, IBM, INTEL, HP