1. CMU SCS Finding patterns in large, real networks Christos Faloutsos CMU

2. CMU SCS LLNL 05C. Faloutsos2 Thanks to Deepayan Chakrabarti (CMU/Yahoo) Michalis Faloutsos (UCR) Spiros Papadimitriou (CMU/IBM) George Siganos (UCR)

3. CMU SCS LLNL 05C. Faloutsos3 Introduction Internet Map [lumeta.com] Food Web [Martinez ’91] Protein Interactions [genomebiology.com] Friendship Network [Moody ’01] Graphs are everywhere!

4. CMU SCS LLNL 05C. Faloutsos4 Outline Laws –static –time-evolution laws –why so many power laws? tools/results future steps

5. CMU SCS LLNL 05C. Faloutsos5 Motivating questions Q1: What do real graphs look like? Q2: How to generate realistic graphs?

6. CMU SCS LLNL 05C. Faloutsos6 Virus propagation Q3: How do viruses/rumors propagate? Q4: Who is the best person/computer to immunize against a virus?

7. CMU SCS LLNL 05C. Faloutsos7 Graph clustering & mining Q5: which edges/nodes are ‘abnormal’? Q6: split a graph in k ‘natural’ communities - but how to determine k?

8. CMU SCS LLNL 05C. Faloutsos8 Outline Laws –static –time-evolution laws –why so many power laws? tools/results future steps

9. CMU SCS LLNL 05C. Faloutsos9 Topology Q1: How does the Internet look like? Any rules? (Looks random – right?)

10. CMU SCS LLNL 05C. Faloutsos10 Laws – degree distributions Q: avg degree is ~2 - what is the most probable degree? degree count ?? 2

11. CMU SCS LLNL 05C. Faloutsos11 Laws – degree distributions Q: avg degree is ~2 - what is the most probable degree? degree count ?? 2 count 2

12. CMU SCS LLNL 05C. Faloutsos12 I.Power-law: outdegree O The plot is linear in log-log scale [FFF’99] freq = degree (-2.15) O = -2.15 Exponent = slope Outdegree Frequency Nov’97 -2.15

13. CMU SCS LLNL 05C. Faloutsos13 II.Power-law: rank R The plot is a line in log-log scale R Rank: nodes in decreasing degree order April’98 log(rank) log(degree) -0.82 att.com ibm.com

14. CMU SCS LLNL 05C. Faloutsos14 III.Power-law: eigen E Eigenvalues in decreasing order (first 20) [Mihail+, 02]: R = 2 * E E = -0.48 Exponent = slope Eigenvalue Rank of decreasing eigenvalue Dec’98

15. CMU SCS LLNL 05C. Faloutsos15 IV. The Node Neighborhood N(h) = # of pairs of nodes within h hops

16. CMU SCS LLNL 05C. Faloutsos16 IV. The Node Neighborhood Q: average degree = 3 - how many neighbors should I expect within 1,2,… h hops? Potential answer: 1 hop -> 3 neighbors 2 hops -> 3 * 3 … h hops -> 3 h

17. CMU SCS LLNL 05C. Faloutsos17 IV. The Node Neighborhood Q: average degree = 3 - how many neighbors should I expect within 1,2,… h hops? Potential answer: 1 hop -> 3 neighbors 2 hops -> 3 * 3 … h hops -> 3 h WE HAVE DUPLICATES!

18. CMU SCS LLNL 05C. Faloutsos18 IV. The Node Neighborhood Q: average degree = 3 - how many neighbors should I expect within 1,2,… h hops? Potential answer: 1 hop -> 3 neighbors 2 hops -> 3 * 3 … h hops -> 3 h ‘avg’ degree: meaningless!

19. CMU SCS LLNL 05C. Faloutsos19 IV. Power-law: hopplot H Pairs of nodes as a function of hops N(h)= h H H = 4.86 Dec 98 Hops # of Pairs Hops Router level ’95 H = 2.83

20. CMU SCS LLNL 05C. Faloutsos20 Observation Q: Intuition behind ‘hop exponent’? A: ‘intrinsic=fractal dimensionality’ of the network N(h) ~ h 1 N(h) ~ h 2...

21. CMU SCS LLNL 05C. Faloutsos21 Hop plots More on fractal/intrinsic dimensionalities: very soon

22. CMU SCS LLNL 05C. Faloutsos22 Any other ‘laws’? Yes!

23. CMU SCS LLNL 05C. Faloutsos23 Any other ‘laws’? Yes! Small diameter –six degrees of separation / ‘Kevin Bacon’ –small worlds [Watts and Strogatz]

24. CMU SCS LLNL 05C. Faloutsos24 Any other ‘laws’? Bow-tie, for the web [Kumar+ ‘99] IN, SCC, OUT, ‘tendrils’ disconnected components

25. CMU SCS LLNL 05C. Faloutsos25 Any other ‘laws’? power-laws in communities (bi-partite cores) [Kumar+, ‘99] 2:3 core (m:n core) Log(m) Log(count) n:1 n:2 n:3

26. CMU SCS LLNL 05C. Faloutsos26 More Power laws Also hold for other web graphs [Barabasi+, ‘99], [Kumar+, ‘99] citation graphs (see later) and many more

27. CMU SCS LLNL 05C. Faloutsos27 Outline Laws –static –time-evolution laws –why so many power laws? tools/results future steps

28. CMU SCS LLNL 05C. Faloutsos28 How do graphs evolve?

29. CMU SCS LLNL 05C. Faloutsos29 Time Evolution: rank R The rank exponent has not changed! [Siganos+, ‘03] Domain level #days since Nov. ‘97

30. CMU SCS LLNL 05C. Faloutsos30 How do graphs evolve? degree-exponent seems constant - anything else?

31. CMU SCS LLNL 05C. Faloutsos31 Evolution of diameter? Prior analysis, on power-law-like graphs, hints that diameter ~ O(log(N)) or diameter ~ O( log(log(N))) i.e.., slowly increasing with network size Q: What is happening, in reality?

32. CMU SCS LLNL 05C. Faloutsos32 Evolution of diameter? Prior analysis, on power-law-like graphs, hints that diameter ~ O(log(N)) or diameter ~ O( log(log(N))) i.e.., slowly increasing with network size Q: What is happening, in reality? A: It shrinks(!!), towards a constant value

33. CMU SCS LLNL 05C. Faloutsos33 Shrinking diameter [Leskovec+05a] Citations among physics papers 11yrs; @ 2003: –29,555 papers –352,807 citations For each month M, create a graph of all citations up to month M

34. CMU SCS LLNL 05C. Faloutsos34 Shrinking diameter Authors & publications 1992 –318 nodes –272 edges 2002 –60,000 nodes 20,000 authors 38,000 papers –133,000 edges

35. CMU SCS LLNL 05C. Faloutsos35 Shrinking diameter Patents & citations 1975 –334,000 nodes –676,000 edges 1999 –2.9 million nodes –16.5 million edges Each year is a datapoint

36. CMU SCS LLNL 05C. Faloutsos36 Shrinking diameter Autonomous systems 1997 –3,000 nodes –10,000 edges 2000 –6,000 nodes –26,000 edges One graph per day

37. CMU SCS LLNL 05C. Faloutsos37 Temporal evolution of graphs N(t) nodes; 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)

38. CMU SCS LLNL 05C. Faloutsos38 Temporal evolution of graphs N(t) nodes; 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!

39. CMU SCS LLNL 05C. Faloutsos39 Temporal evolution of graphs A: over-doubled - but obeying: E(t) ~ N(t) a for all t where 1<a<2

40. CMU SCS LLNL 05C. Faloutsos40 Temporal evolution of graphs A: over-doubled - but obeying: E(t) ~ N(t) a for all t Identically: log(E(t)) / log(N(t)) = constant for all t

41. CMU SCS LLNL 05C. Faloutsos41 Densification Power Law ArXiv: Physics papers and their citations 1.69 N(t) E(t)

42. CMU SCS LLNL 05C. Faloutsos42 Densification Power Law ArXiv: Physics papers and their citations 1.69 N(t) E(t) ‘tree’ 1

43. CMU SCS LLNL 05C. Faloutsos43 Densification Power Law ArXiv: Physics papers and their citations 1.69 N(t) E(t) ‘clique’ 2

44. CMU SCS LLNL 05C. Faloutsos44 Densification Power Law U.S. Patents, citing each other 1.66 N(t) E(t)

45. CMU SCS LLNL 05C. Faloutsos45 Densification Power Law Autonomous Systems 1.18 N(t) E(t)

46. CMU SCS LLNL 05C. Faloutsos46 Densification Power Law ArXiv: authors & papers 1.15 N(t) E(t)

47. CMU SCS LLNL 05C. Faloutsos47 Summary of ‘laws’ Power laws for degree distributions …………... for eigenvalues, bi-partite cores Small & shrinking diameter (‘6 degrees’) ‘Bow-tie’ for web; ‘jelly-fish’ for internet ``Densification Power Law’’, over time

48. CMU SCS LLNL 05C. Faloutsos48 Outline Laws –static –time-evolution laws –why so many power laws? other power laws fractals & self-similarity case study: Kronecker graphs tools/results future steps

49. CMU SCS LLNL 05C. Faloutsos49 Power laws Many more power laws, in diverse settings:

50. CMU SCS LLNL 05C. Faloutsos50 A famous power law: Zipf’s law Bible - rank vs frequency (log-log) log(rank) log(freq) “a” “the”

51. CMU SCS LLNL 05C. Faloutsos51 Power laws, cont’ed length of file transfers [Bestavros+] web hit counts [Huberman] magnitude of earthquakes (Guttenberg- Richter law) sizes of lakes/islands (Korcak’s law) Income distribution (Pareto’s law)

52. CMU SCS LLNL 05C. Faloutsos52 The Peer-to-Peer Topology Frequency versus degree Number of adjacent peers follows a power-law [Jovanovic+]

53. CMU SCS LLNL 05C. Faloutsos53 Web Site Traffic log(freq) log(count) Zipf Click-stream data u-id’s url’s log(freq) log(count) ‘yahoo’ ‘super-surfer’

54. CMU SCS LLNL 05C. Faloutsos54 Lotka’s law (Lotka’s law of publication count); and citation counts: (citeseer.nj.nec.com 6/2001) log(#citations) log(count) J. Ullman

55. CMU SCS LLNL 05C. Faloutsos55 Swedish sex-web Nodes: people (Females; Males) Links: sexual relationships Liljeros et al. Nature 2001 4781 Swedes; 18-74; 59% response rate. Albert Laszlo Barabasi http://www.nd.edu/~networks/ Publication%20Categories/ 04%20Talks/2005-norway- 3hours.ppt

56. CMU SCS LLNL 05C. Faloutsos56 Swedish sex-web Nodes: people (Females; Males) Links: sexual relationships Liljeros et al. Nature 2001 4781 Swedes; 18-74; 59% response rate. Albert Laszlo Barabasi http://www.nd.edu/~networks/ Publication%20Categories/ 04%20Talks/2005-norway- 3hours.ppt

57. CMU SCS LLNL 05C. Faloutsos57 Power laws Q: Why so many power laws? A1: self-similarity A2: ‘rich-get-richer’

58. CMU SCS LLNL 05C. Faloutsos58 Digression: intro to fractals Fractals: sets of points that are self similar

59. CMU SCS LLNL 05C. Faloutsos59 A famous fractal e.g., Sierpinski triangle:... zero area; infinite length! dimensionality = ??

60. CMU SCS LLNL 05C. Faloutsos60 A famous fractal e.g., Sierpinski triangle:... zero area; infinite length! dimensionality = log(3)/log(2) = 1.58

61. CMU SCS LLNL 05C. Faloutsos61 A famous fractal equivalent graph:

62. CMU SCS LLNL 05C. Faloutsos62 Intrinsic (‘fractal’) dimension How to estimate it?

63. CMU SCS LLNL 05C. Faloutsos63 Intrinsic (‘fractal’) dimension Q: fractal dimension of a line? A: nn ( <= r ) ~ r^1 (‘power law’: y=x^a) Q: fd of a plane? A: nn ( <= r ) ~ r^2 fd== slope of (log(nn) vs log(r) )

64. CMU SCS LLNL 05C. Faloutsos64 Sierpinsky triangle log( r ) log(#pairs within <=r ) 1.58 == ‘correlation integral’ = CDF of pairwise distances

65. CMU SCS LLNL 05C. Faloutsos65 Sierpinsky triangle log( r ) log(#pairs within <=r ) 1.58 == ‘correlation integral’ = CDF of pairwise distances

66. CMU SCS LLNL 05C. Faloutsos66 Line log( r ) log(#pairs within <=r ) 1.58 == ‘correlation integral’ = CDF of pairwise distances 1...

67. CMU SCS LLNL 05C. Faloutsos67 2-d (Plane) log( r ) log(#pairs within <=r ) 1.58 == ‘correlation integral’ = CDF of pairwise distances 2

68. CMU SCS LLNL 05C. Faloutsos68 Recall: Hop Plot Internet routers: how many neighbors within h hops? (= correlation integral!) Reachability function: number of neighbors within r hops, vs r (log- log). Mbone routers, 1995 log(hops) log(#pairs) 2.8

69. CMU SCS LLNL 05C. Faloutsos69 Fractals and power laws They are related concepts: fractals self-similarity scale-free power laws ( y= x a ) –F = C r (-2)

70. CMU SCS LLNL 05C. Faloutsos70 Fractals and power laws Power laws and fractals are closely related And fractals appear in MANY cases –coast-lines: 1.1-1.5 –brain-surface: 2.6 –rain-patches: 1.3 –tree-bark: ~2.1 –stock prices / random walks: 1.5 –... [see Mandelbrot; or Schroeder]

71. CMU SCS LLNL 05C. Faloutsos71 Outline Laws –static –time-evolution laws –why so many power laws? other power laws fractals & self-similarity case study: Kronecker graph generator tools/results future steps

72. CMU SCS LLNL 05C. Faloutsos72 Self-similarity at work Q2: How to generate realistic graphs? (Q2’) and how to grow it over time?

73. CMU SCS LLNL 05C. Faloutsos73 Wish list for a generator: Power-law-tail in- and out-degrees Power-law-tail scree plots shrinking/constant diameter Densification Power Law communities-within-communities Q: how to achieve all of them?

74. CMU SCS LLNL 05C. Faloutsos74 Wish list for a generator: Power-law-tail in- and out-degrees Power-law-tail scree plots shrinking/constant diameter Densification Power Law communities-within-communities Q: how to achieve all of them? A: Kronecker matrix product [Leskovec+05b]

75. CMU SCS LLNL 05C. Faloutsos75 Kronecker product

76. CMU SCS LLNL 05C. Faloutsos76 Kronecker product

77. CMU SCS LLNL 05C. Faloutsos77 Kronecker product N N*N N**4

78. CMU SCS LLNL 05C. Faloutsos78 Properties of Kronecker graphs: Power-law-tail in- and out-degrees Power-law-tail scree plots constant diameter perfect Densification Power Law communities-within-communities

79. CMU SCS LLNL 05C. Faloutsos79 Properties of Kronecker graphs: Power-law-tail in- and out-degrees Power-law-tail scree plots constant diameter perfect Densification Power Law communities-within-communities and we can prove all of the above (first and only generator that does that)

80. CMU SCS LLNL 05C. Faloutsos80 Kronecker - ArXiv real (stochastic) Kronecker DegreeScreeDiameterD.P.L. (det. Kronecker)

81. CMU SCS LLNL 05C. Faloutsos81 Kronecker - patents Degree Scree Diameter D.P.L.

82. CMU SCS LLNL 05C. Faloutsos82 Kronecker - A.S.

83. CMU SCS LLNL 05C. Faloutsos83 Outline Laws –static –time-evolution laws –why so many power laws? tools/results –virus propagation –connection subgraphs –cross-associations future steps

84. CMU SCS LLNL 05C. Faloutsos84 Virus propagation Q3: How do viruses/rumors propagate? Q4: Who is the best person/computer to immunize against a virus?

85. CMU SCS LLNL 05C. Faloutsos85 The model: SIS ‘Flu’ like: Susceptible-Infected-Susceptible Virus ‘strength’ s=  /  Infected Healthy NN1 N3 N2 Prob.  Prob. β Prob. 

86. CMU SCS LLNL 05C. Faloutsos86 Epidemic threshold  of a graph: the value of , such that if strength s =  /  <  an epidemic can not happen Thus, given a graph compute its epidemic threshold

87. CMU SCS LLNL 05C. Faloutsos87 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?

88. CMU SCS LLNL 05C. Faloutsos88 Epidemic threshold [Theorem] We have no epidemic, if β/δ <τ = 1/ λ 1,A

89. CMU SCS LLNL 05C. Faloutsos89 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]

90. CMU SCS LLNL 05C. Faloutsos90 Experiments (Oregon)  /  > τ (above threshold)  /  = τ (at the threshold)  /  < τ (below threshold)

91. CMU SCS LLNL 05C. Faloutsos91 Our result: Holds for any graph includes older results as special cases

92. CMU SCS LLNL 05C. Faloutsos92 Virus propagation Q3: How do viruses/rumors propagate? Q4: Who is the best person/computer to immunize against a virus? A4: the one that causes the highest decrease in 1

93. CMU SCS LLNL 05C. Faloutsos93 Outline Laws –static –time-evolution laws –why so many power laws? tools/results –virus propagation –connection subgraphs –cross-associations future steps

94. CMU SCS LLNL 05C. Faloutsos94 (A-A) Kidman-Diaz What are the best paths between ‘Kidman’ and ‘Diaz’?

95. CMU SCS LLNL 05C. Faloutsos95 (A-A) Kidman-Diaz Strong, direct link What are the best paths between ‘Kidman’ and ‘Diaz’? [KDD’04, w/ McCurley+Tomkins]

96. CMU SCS LLNL 05C. Faloutsos96 Outline Laws –static –time-evolution laws –why so many power laws? tools/results –virus propagation –connection subgraphs –cross-associations future steps

97. CMU SCS LLNL 05C. Faloutsos97 Graph clustering & mining Q5: which edges/nodes are ‘abnormal’? Q6: split a graph in k ‘natural’ communities - but how to determine k?

98. CMU SCS LLNL 05C. Faloutsos98 Graph partitioning Documents x terms Customers x products Users x web-sites

99. CMU SCS LLNL 05C. Faloutsos99 Graph partitioning Documents x terms Customers x products Users x web-sites Q: HOW MANY PIECES?

100. CMU SCS LLNL 05C. Faloutsos100 Graph partitioning Documents x terms Customers x products Users x web-sites Q: HOW MANY PIECES? A: MDL/ compression

101. CMU SCS LLNL 05C. Faloutsos101 Cross-associations 1x2 2x2

102. CMU SCS LLNL 05C. Faloutsos102 Cross-associations 2x3 3x3 3x4

103. CMU SCS LLNL 05C. Faloutsos103 Cross-associations

104. CMU SCS LLNL 05C. Faloutsos104 Cross-associations

105. CMU SCS LLNL 05C. Faloutsos105 Graph clustering & mining Q5: which edges/nodes are ‘abnormal’? Q6: split a graph in k ‘natural’ communities - but how to determine k? A6: choose the k that leads to best overall compression (= MDL = Minimum Description Language)

106. CMU SCS LLNL 05C. Faloutsos106 Cross-associations outlier edge missing edge

107. CMU SCS LLNL 05C. Faloutsos107 Outline Laws –static –time-evolution laws –why so many power laws? tools/results other related projects future steps

108. CMU SCS LLNL 05C. Faloutsos108 Motivation water distribution network normal operation May have hundreds of measurements, but it is unlikely they are completely unrelated! Phase 1Phase 2Phase 3 : : : chlorine concentrations sensors near leak sensors away from leak

109. CMU SCS LLNL 05C. Faloutsos109 Phase 1Phase 2Phase 3 : : : Motivation water distribution network normal operationmajor leak May have hundreds of measurements, but it is unlikely they are completely unrelated! chlorine concentrations sensors near leak sensors away from leak

110. CMU SCS LLNL 05C. Faloutsos110 Motivation actual measurements (n streams) k hidden variable(s) We would like to discover a few “hidden (latent) variables” that summarize the key trends Phase 1 : : : chlorine concentrations Phase 1 k = 1

111. CMU SCS LLNL 05C. Faloutsos111 Motivation We would like to discover a few “hidden (latent) variables” that summarize the key trends chlorine concentrations Phase 1 Phase 2 actual measurements (n streams) k hidden variable(s) k = 2 : : :

112. CMU SCS LLNL 05C. Faloutsos112 Motivation We would like to discover a few “hidden (latent) variables” that summarize the key trends chlorine concentrations Phase 1 Phase 2 Phase 3 actual measurements (n streams) k hidden variable(s) k = 1 : : :

113. CMU SCS LLNL 05C. Faloutsos113 Stream mining Solution: SPIRIT [VLDB’05] –incremental, on-line PCA

114. CMU SCS LLNL 05C. Faloutsos114 Outline Laws tools/results other related projects future steps

115. CMU SCS LLNL 05C. Faloutsos115 Future steps connection sub-graphs on multi-graphs [Tong] traffic matrix, evolving over time [Sun+] influence propagation on blogs [Leskovec+] automatic web site classification [Leskovec]

116. CMU SCS LLNL 05C. Faloutsos116 Conclusions Power laws & self similarity –excellent tools, for real datasets, –where Gaussian, uniform etc assumptions fail Additional tools –Kronecker, for graph generators –eigenvalues for virus propagation –MDL/compression for graph partitioning

117. CMU SCS LLNL 05C. Faloutsos117 References [Aiello+, '00] William Aiello, Fan R. K. Chung, Linyuan Lu: A random graph model for massive graphs. STOC 2000: 171-180 [Albert+] Reka Albert, Hawoong Jeong, and Albert-Laszlo Barabasi: Diameter of the World Wide Web, Nature 401 130-131 (1999) [Barabasi, '03] Albert-Laszlo Barabasi Linked: How Everything Is Connected to Everything Else and What It Means (Plume, 2003)

118. CMU SCS LLNL 05C. Faloutsos118 References, cont’d [Barabasi+, '99] Albert-Laszlo Barabasi and Reka Albert. Emergence of scaling in random networks. Science, 286:509--512, 1999 [Broder+, '00] Andrei Broder, Ravi Kumar, Farzin Maghoul, Prabhakar Raghavan, Sridhar Rajagopalan, Raymie Stata, Andrew Tomkins, and Janet Wiener. Graph structure in the web, WWW, 2000

119. CMU SCS LLNL 05C. Faloutsos119 References, cont’d [Chakrabarti+, ‘04] RMAT: A recursive graph generator, D. Chakrabarti, Y. Zhan, C. Faloutsos, SIAM-DM 2004 [Dill+, '01] Stephen Dill, Ravi Kumar, Kevin S. McCurley, Sridhar Rajagopalan, D. Sivakumar, Andrew Tomkins: Self- similarity in the Web. VLDB 2001: 69-78

120. CMU SCS LLNL 05C. Faloutsos120 References, cont’d [Fabrikant+, '02] A. Fabrikant, E. Koutsoupias, and C.H. Papadimitriou. Heuristically Optimized Trade-offs: A New Paradigm for Power Laws in the Internet. ICALP, Malaga, Spain, July 2002 [FFF, 99] M. Faloutsos, P. Faloutsos, and C. Faloutsos, "On power-law relationships of the Internet topology," in SIGCOMM, 1999.

121. CMU SCS LLNL 05C. Faloutsos121 References, cont’d [Leskovec+05a] 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) [Leskovec+05b] Jure Leskovec, Deepayan Chakrabarti, Jon Kleinberg and Christos Faloutsos, Realistic, Mathematically Tractable Graph Generation and Evolution, Using Kronecker Multiplication, ECML/PKDD 2005, Porto, Portugal.

122. CMU SCS LLNL 05C. Faloutsos122 References, cont’d [Jovanovic+, '01] M. Jovanovic, F.S. Annexstein, and K.A. Berman. Modeling Peer-to-Peer Network Topologies through "Small-World" Models and Power Laws. In TELFOR, Belgrade, Yugoslavia, November, 2001 [Kumar+ '99] Ravi Kumar, Prabhakar Raghavan, Sridhar Rajagopalan, Andrew Tomkins: Extracting Large-Scale Knowledge Bases from the Web. VLDB 1999: 639-650

123. CMU SCS LLNL 05C. Faloutsos123 References, cont’d [Leland+, '94] W. E. Leland, M.S. Taqqu, W. Willinger, D.V. Wilson, On the Self-Similar Nature of Ethernet Traffic, IEEE Transactions on Networking, 2, 1, pp 1-15, Feb. 1994. [Mihail+, '02] Milena Mihail, Christos H. Papadimitriou: On the Eigenvalue Power Law. RANDOM 2002: 254-262

124. CMU SCS LLNL 05C. Faloutsos124 References, cont’d [Milgram '67] Stanley Milgram: The Small World Problem, Psychology Today 1(1), 60-67 (1967) [Montgomery+, ‘01] Alan L. Montgomery, Christos Faloutsos: Identifying Web Browsing Trends and Patterns. IEEE Computer 34(7): 94-95 (2001)

125. CMU SCS LLNL 05C. Faloutsos125 References, cont’d [Palmer+, ‘01] Chris Palmer, Georgos Siganos, Michalis Faloutsos, Christos Faloutsos and Phil Gibbons The connectivity and fault-tolerance of the Internet topology (NRDM 2001), Santa Barbara, CA, May 25, 2001 [Pennock+, '02] David M. Pennock, Gary William Flake, Steve Lawrence, Eric J. Glover, C. Lee Giles: Winners don't take all: Characterizing the competition for links on the web Proc. Natl. Acad. Sci. USA 99(8): 5207-5211 (2002)

126. CMU SCS LLNL 05C. Faloutsos126 References, cont’d [Schroeder, ‘91] Manfred Schroeder Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise W H Freeman & Co., 1991 (excellent book on fractals)

127. CMU SCS LLNL 05C. Faloutsos127 References, cont’d [Siganos+, '03] G. Siganos, M. Faloutsos, P. Faloutsos, C. Faloutsos Power-Laws and the AS-level Internet Topology, Transactions on Networking, August 2003. [Wang+03] Yang Wang, Deepayan Chakrabarti, Chenxi Wang and Christos Faloutsos: Epidemic Spreading in Real Networks: an Eigenvalue Viewpoint, SRDS 2003, Florence, Italy.

128. CMU SCS LLNL 05C. Faloutsos128 Reference [Watts+ Strogatz, '98] D. J. Watts and S. H. Strogatz Collective dynamics of 'small-world' networks, Nature, 393:440-442 (1998) [Watts, '03] Duncan J. Watts Six Degrees: The Science of a Connected Age W.W. Norton & Company; (February 2003)

129. CMU SCS LLNL 05C. Faloutsos129 Thank you! www.cs.cmu.edu/~christos www.db.cs.cmu.edu