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

2. Thanks to Deepayan Chakrabarti (CMU/Yahoo) Michalis Faloutsos (UCR)
Spiros Papadimitriou (CMU/IBM)
George Siganos (UCR)
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3. Introduction Graphs are everywhere!
Protein Interactions [genomebiology.com]
Internet Map [lumeta.com]
Food Web [Martinez ’91]
Graphs are everywhere!
Friendship Network [Moody ’01]
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4. Outline Laws tools/results future steps static time-evolution laws
why so many power laws?
tools/results
future steps
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5. Motivating questions Q1: What do real graphs look like?
Q2: How to generate realistic graphs?
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6. Virus propagation Q3: How do viruses/rumors propagate?
Q4: Who is the best person/computer to immunize against a virus?
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7. Graph clustering & mining
Q5: which edges/nodes are ‘abnormal’?
Q6: split a graph in k ‘natural’ communities - but how to determine k?
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8. Outline Laws tools/results future steps static time-evolution laws
why so many power laws?
tools/results
future steps
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9. Topology Q1: How does the Internet look like? Any rules?
(Looks random – right?)
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10. Laws – degree distributions
Q: avg degree is ~2 - what is the most probable degree?
count ?? 2
degree
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11. Laws – degree distributions
Q: avg degree is ~2 - what is the most probable degree?
degree
count ?? 2
WRONG !
degree
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12. I.Power-law: outdegree O
Frequency
Exponent = slope
O = -2.15
-2.15
Nov’97
Outdegree
The plot is linear in log-log scale [FFF’99]
freq = degree (-2.15)
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13. II.Power-law: rank R The plot is a line in log-log scale att.com
log(rank)
log(degree)
-0.82
att.com
ibm.com
April’98 R
Rank: nodes in decreasing degree order
The plot is a line in log-log scale
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14. III.Power-law: eigen E Eigenvalues in decreasing order (first 20)
Exponent = slope
E = -0.48
Dec’98
Rank of decreasing eigenvalue
Eigenvalues in decreasing order (first 20)
[Mihail+, 02]: R = 2 * E
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15. Any other ‘laws’?
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16. Any other ‘laws’? Yes! Small diameter
six degrees of separation / ‘Kevin Bacon’
small worlds [Watts and Strogatz]
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17. Any other ‘laws’? Bow-tie, for the web [Kumar+ ‘99]
IN, SCC, OUT, ‘tendrils’
disconnected components
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18. Any other ‘laws’?
power-laws in communities (bi-partite cores) [Kumar+, ‘99]
Log(count)
n:1
n:3
n:2
2:3 core
(m:n core)
Log(m)
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19. More Power laws
Also hold for other web graphs [Barabasi+, ‘99], [Kumar+, ‘99]
citation graphs (see later)
and many more
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20. Outline Laws tools/results future steps static time-evolution laws
why so many power laws?
tools/results
future steps
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21. How do graphs evolve?
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22. Time Evolution: rank R #days since Nov. ‘97
log(rank)
log(degree) -
Time Evolution: rank R
Domain
level
#days since Nov. ‘97
The rank exponent has not changed! [Siganos+, ‘03]
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23. How do graphs evolve? degree-exponent seems constant - anything else?
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24. 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?
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25. 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
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26. Shrinking diameter diameter time [Leskovec+05a]
Citations among physics papers
2003:
29,555 papers
352,807 citations
For each month M, create a graph of all citations up to month M
time
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27. Shrinking diameter Authors & publications 1992 2002 318 nodes
272 edges
2002
60,000 nodes
20,000 authors
38,000 papers
133,000 edges
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28. Shrinking diameter Patents & citations 1975 1999
334,000 nodes
676,000 edges
1999
2.9 million nodes
16.5 million edges
Each year is a datapoint
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29. Shrinking diameter Autonomous systems 1997 2000 One graph per day
3,000 nodes
10,000 edges
2000
6,000 nodes
26,000 edges
One graph per day
diameter N
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30. 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)
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31. 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! x
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32. Temporal evolution of graphs
A: over-doubled - but obeying:
E(t) ~ N(t)a for all t
where 1<a<2
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33. Densification Power Law
ArXiv: Physics papers
and their citations
N(t)
E(t)
1.69
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34. Densification Power Law
ArXiv: Physics papers
and their citations
N(t)
E(t) 1
1.69
‘tree’
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35. Densification Power Law
ArXiv: Physics papers
and their citations
‘clique’
N(t)
E(t) 2
1.69
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36. Densification Power Law
U.S. Patents, citing each other
N(t)
E(t)
1.66
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37. Densification Power Law
Autonomous Systems
N(t)
E(t)
1.18
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38. Densification Power Law
ArXiv: authors & papers
N(t)
E(t)
1.15
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39. 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
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40. Outline Laws tools/results future steps static time-evolution laws
why so many power laws?
other power laws
fractals & self-similarity
case study: Kronecker graphs
tools/results
future steps
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41. Power laws Many more power laws, in diverse settings: Yahoo 05
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42. A famous power law: Zipf’s law
log(freq)
“a”
Bible - rank vs frequency (log-log)
“the”
log(rank)
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43. 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)
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44. The Peer-to-Peer Topology
[Jovanovic+]
Frequency versus degree
Number of adjacent peers follows a power-law
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45. Click-stream data u-id’s url’s Web Site Traffic log(count) Zipf
‘yahoo’
log(freq)
log(count)
‘super-surfer’
log(freq)
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46. Lotka’s law
(Lotka’s law of publication count); and citation counts: (citeseer.nj.nec.com 6/2001)
log(count)
J. Ullman
log(#citations)
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47. Albert Laszlo Barabasi
Swedish sex-web
Nodes: people (Females; Males)
Links: sexual relationships
Albert Laszlo Barabasi
Publication%20Categories/
04%20Talks/2005-norway-3hours.ppt
4781 Swedes; 18-74;
59% response rate.
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Liljeros et al. Nature 2001

48. Albert Laszlo Barabasi
Swedish sex-web
Nodes: people (Females; Males)
Links: sexual relationships
Albert Laszlo Barabasi
Publication%20Categories/
04%20Talks/2005-norway-3hours.ppt
4781 Swedes; 18-74;
59% response rate.
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Liljeros et al. Nature 2001

49. Power laws Q: Why so many power laws? A1: self-similarity
A2: ‘rich-get-richer’
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50. Digression: intro to fractals
Fractals: sets of points that are self similar
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51. A famous fractal dimensionality = ?? e.g., Sierpinski triangle: ...
zero area;
infinite length!
...
dimensionality = ??
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52. A famous fractal dimensionality = log(3)/log(2) = 1.58
e.g., Sierpinski triangle:
zero area;
infinite length!
...
dimensionality = log(3)/log(2) = 1.58
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53. A famous fractal
equivalent graph:
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54. Intrinsic (‘fractal’) dimension
How to estimate it?
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55. 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) )
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56. Sierpinsky triangle == ‘correlation integral’
= CDF of pairwise distances
log( r )
log(#pairs
within <=r )
1.58
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57. Fractals and power laws
They are related concepts:
fractals <=>
self-similarity <=>
scale-free <=>
power laws ( y= xa )
F = C r(-2)
vs y=e-ax
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58. Fractals and power laws
Power laws and fractals are closely related
And fractals appear in MANY cases
coast-lines:
brain-surface: 2.6
rain-patches: 1.3
tree-bark: ~2.1
stock prices / random walks: 1.5
... [see Mandelbrot; or Schroeder]
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59. Outline Laws tools/results future steps static time-evolution laws
why so many power laws?
other power laws
fractals & self-similarity
case study: Kronecker graph generator
tools/results
future steps
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60. Self-similarity at work
Q2: How to generate a realistic graph?
(Q2’) and how to grow it over time?
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61. 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?
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62. 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]
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63. Kronecker product
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64. Kronecker product
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65. Kronecker product N
N*N
N**4
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66. 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
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67. 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)
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68. Kronecker - patents Scree D.P.L. Degree Diameter Yahoo 05
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69. Outline Laws tools/results future steps static time-evolution laws
why so many power laws?
tools/results
virus propagation
connection subgraphs
cross-associations
future steps
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70. Virus propagation Q3: How do viruses/rumors propagate?
Q4: Who is the best person/computer to immunize against a virus?
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71. The model: SIS ‘Flu’ like: Susceptible-Infected-Susceptible
Virus ‘strength’ s= b/d
Healthy
Prob. d N2
Prob. b
Prob. β N1 N
Infected N3
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72. if strength s = b / d < t
Epidemic threshold t
of a graph: the value of t, such that
if strength s = b / d < t
an epidemic can not happen
Thus,
given a graph
compute its epidemic threshold
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73. Epidemic threshold t What should t depend on?
avg. degree? and/or highest degree?
and/or variance of degree?
and/or third moment of degree?
and/or diameter?
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74. Epidemic threshold β/δ <τ = 1/ λ1,A
[Theorem] We have no epidemic, if
β/δ <τ = 1/ λ1,A
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75. Epidemic threshold β/δ <τ = 1/ λ1,A
[Theorem] We have no epidemic, if
epidemic threshold
recovery prob.
β/δ <τ = 1/ λ1,A
largest eigenvalue
of adj. matrix A
attack prob.
Proof: [Wang+03]
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76. Experiments (Oregon) b/d > τ (above threshold)
b/d = τ (at the threshold)
b/d < τ (below threshold)
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77. Our result: Holds for any graph
includes older results as special cases
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78. 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 l1
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79. Outline Laws tools/results future steps static time-evolution laws
why so many power laws?
tools/results
virus propagation
connection subgraphs
cross-associations
future steps
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80. (A-A) Kidman-Diaz What are the best paths between ‘Kidman’ and ‘Diaz’?
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81. (A-A) Kidman-Diaz
What are the best paths between ‘Kidman’ and ‘Diaz’? [KDD’04, w/ McCurley+Tomkins]
Strong, direct link
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82. Outline Laws tools/results future steps static time-evolution laws
why so many power laws?
tools/results
virus propagation
connection subgraphs
cross-associations
future steps
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83. Graph clustering & mining
Q5: which edges/nodes are ‘abnormal’?
Q6: split a graph in k ‘natural’ communities - but how to determine k?
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84. Graph partitioning Documents x terms Customers x products
Users x web-sites
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85. Graph partitioning Documents x terms Customers x products
Users x web-sites
Q: HOW MANY PIECES?
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86. Graph partitioning Documents x terms Customers x products
Users x web-sites
Q: HOW MANY PIECES?
A: MDL/ compression
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87. Cross-associations
2x2
1x2
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88. Cross-associations
3x4
3x3
2x3
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89. Cross-associations
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90. Cross-associations
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91. 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)
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92. Cross-associations missing edge outlier edge Yahoo 05
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93. Outline Laws tools/results other related projects future steps static
time-evolution laws
why so many power laws?
tools/results
other related projects
future steps
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94. Stream mining : : : : : : chlorine concentrations
Phase 1
Phase 2
Phase 3
: : : : : :
chlorine concentrations
This is basically our main contribution: “wake up, you *can* do dimensionality reduction incrementally, in real-time”.
water distribution network
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95. Stream mining k = 1 actual measurements k hidden variable(s)
Phase 1
: : : : : :
chlorine concentrations
Phase 1
k = 1
actual measurements
(n streams)
k hidden variable(s)
We would like to discover a few “hidden
(latent) variables” that summarize the key trends
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96. Stream mining Solution: SPIRIT [VLDB’05] incremental, on-line PCA
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97. Outline Laws tools/results other related projects future steps
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98. 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]
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99. Conclusions Power laws & self similarity Additional tools
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
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100. References
[Aiello+, '00] William Aiello, Fan R. K. Chung, Linyuan Lu: A random graph model for massive graphs. STOC 2000:
[Albert+] Reka Albert, Hawoong Jeong, and Albert-Laszlo Barabasi: Diameter of the World Wide Web, Nature (1999)
[Barabasi, '03] Albert-Laszlo Barabasi Linked: How Everything Is Connected to Everything Else and What It Means (Plume, 2003)
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101. References, cont’d
[Barabasi+, '99] Albert-Laszlo Barabasi and Reka Albert. Emergence of scaling in random networks. Science, 286: , 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
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102. 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
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103. 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.
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104. 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.
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105. 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:
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106. 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
[Mihail+, '02] Milena Mihail, Christos H. Papadimitriou: On the Eigenvalue Power Law. RANDOM 2002:
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107. References, cont’d
[Milgram '67] Stanley Milgram: The Small World Problem, Psychology Today 1(1), (1967)
[Montgomery+, ‘01] Alan L. Montgomery, Christos Faloutsos: Identifying Web Browsing Trends and Patterns. IEEE Computer 34(7): (2001)
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108. 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
[Papadimitriou+,’05] Spiros Papadimitriou, Jimeng Sun and Christos Faloutsos Streaming Pattern Discovery in Multiple Time-Series VLDB 2005, Trondheim, Norway
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109. References, cont’d
[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): (2002)
[Schroeder, ‘91] Manfred Schroeder Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise W H Freeman & Co., 1991 (excellent book on fractals)
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110. 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.
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111. Reference
[Watts+ Strogatz, '98] D. J. Watts and S. H. Strogatz Collective dynamics of 'small-world' networks, Nature, 393: (1998)
[Watts, '03] Duncan J. Watts Six Degrees: The Science of a Connected Age W.W. Norton & Company; (February 2003)
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112. Thank you! www.cs.cmu.edu/~christos www.db.cs.cmu.edu Yahoo 05
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