Advanced Data Mining Tools: Fractals and Power Laws for Graphs, Streams and Traditional Data

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Advanced Data Mining Tools Fractals and Power
Laws for Graphs, Streams and Traditional Data

• Christos Faloutsos
• Carnegie Mellon University

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THANK YOU!

• Prof. Jiawei Han
• Prof. Kevin Chang

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Overview

• Goals/ motivation find patterns in large
  datasets
• (A) Sensor data
• (B) network/graph data
• Solutions self-similarity and power laws
• Discussion

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Applications of sensors/streams

• Smart house monitoring temperature, humidity
  etc
• Financial, sales, economic series

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Motivation - Applications

• Medical ECGs blood pressure etc monitoring
• Scientific data seismological astronomical
  environment / anti-pollution meteorological

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Motivation - Applications (contd)

• civil/automobile infrastructure
• bridge vibrations Oppenheim02
• road conditions / traffic monitoring

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Motivation - Applications (contd)

• Computer systems
• web servers (buffering, prefetching)
• network traffic monitoring
• ...

http//repository.cs.vt.edu/lbl-conn-7.tar.Z
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Problem definition

• Given one or more sequences
• x1 , x2 , , xt , (y1, y2, , yt, )
• Find
• patterns clusters outliers forecasts

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Problem 1
bytes

• Find patterns, in large datasets

time
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Problem 1
bytes

• Find patterns, in large datasets

time
Poisson indep., ident. distr
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Problem 1
bytes

• Find patterns, in large datasets

time
Poisson indep., ident. distr
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Problem 1
bytes

• Find patterns, in large datasets

time
Poisson indep., ident. distr
Q Then, how to generate such bursty traffic?
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Overview

• Goals/ motivation find patterns in large
  datasets
• (A) Sensor data
• (B) network/graph data
• Solutions self-similarity and power laws
• Discussion

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Problem 2 - network and graph mining

• How does the Internet look like?
• How does the web look like?
• What constitutes a normal social network?
• What is the market value of a customer?
• which gene/species affects the others the most?

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Problem2

• Given a graph

• which node to market-to / defend / immunize
  first?
• Are there un-natural sub-graphs? (eg.,
  criminals rings)?

from Lumeta ISPs 6/1999
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Solutions

• New tools power laws, self-similarity and
  fractals work, where traditional assumptions
  fail
• Lets see the details

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Overview

• Goals/ motivation find patterns in large
  datasets
• (A) Sensor data
• (B) network/graph data
• Solutions self-similarity and power laws
• Discussion

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What is a fractal?

• self-similar point set, e.g., Sierpinski
  triangle

zero area (3/4)inf infinite length! (4/3)inf
...
Q What is its dimensionality??
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What is a fractal?

• self-similar point set, e.g., Sierpinski
  triangle

zero area (3/4)inf infinite length! (4/3)inf
...
Q What is its dimensionality?? A log3 / log2
1.58 (!?!)
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Intrinsic (fractal) dimension

• Q fractal dimension of a (finite set of points
  on a) line?

x y
5 1
4 2
3 3
2 4
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Intrinsic (fractal) dimension

• Q fractal dimension of a line?
• A nn ( lt r ) r1
• (power law yxa)

• Q fd of a plane?
• A nn ( lt r ) r2
• fd slope of (log(nn) vs.. log(r) )

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Sierpinsky triangle
correlation integral CDF of pairwise
distances
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Observations Fractals lt-gt power laws

• Closely related
• fractals ltgt
• self-similarity ltgt
• scale-free ltgt
• power laws ( y xa
• FK r-2)
• (vs ye-ax or yxab)

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Outline

• Problems
• Self-similarity and power laws
• Solutions to posed problems
• Discussion

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Solution 1 traffic

• disk traces self-similar (also Leland94)
• How to generate such traffic?

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Solution 1 traffic

• disk traces (80-20 law) multifractals

bytes
time
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80-20 / multifractals
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80
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80-20 / multifractals
20
80

• p (1-p) in general
• yes, there are dependencies

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Overview

• Goals/ motivation find patterns in large
  datasets
• (A) Sensor data
• (B) network/graph data
• Solutions self-similarity and power laws
• sensor/traffic data
• network/graph data
• Discussion

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Problem 2 - topology

• How does the Internet look like? Any rules?

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Patterns?

• avg degree is, say 3.3
• pick a node at random - what is the degree you
  expect it to have?

count
?
avg 3.3
degree
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Patterns?

• avg degree is, say 3.3
• pick a node at random - what is the degree you
  expect it to have?
• A 1!!

count
avg 3.3
degree
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Patterns?

• avg degree is, say 3.3
• pick a node at random - what is the degree you
  expect it to have?
• A 1!!
• A very skewed distr.
• Corollary the mean is meaningless!
• (and std -gt infinity (!))

count
avg 3.3
degree
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Solution2 Rank exponent R

• A1 Power law in the degree distribution
  SIGCOMM99

internet domains
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Solution2 Eigen Exponent E
Eigenvalue
Exponent slope
E -0.48
May 2001
Rank of decreasing eigenvalue

• A2 power law in the eigenvalues of the adjacency
  matrix

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Solution2 Hop Exponent H

• A3 neighborhood function N(h) number of pairs
  within h hops or less - power law, too!

Hop exp. 1
log(pairs)
internet
Hop exp. 2
hop exponent
log(hops)
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Power laws - discussion

• do they hold, over time?
• do they hold on other graphs/domains?

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Power laws - discussion

• do they hold, over time?
• Yes! for multiple years Siganos
• do they hold on other graphs/domains?
• Yes!
• web sites and links Tomkins, Barabasi
• peer-to-peer graphs (gnutella-style)
• who-trusts-whom (epinions.com)

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Time Evolution rank R
Domain level

• The rank exponent has not changed! Siganos

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The Peer-to-Peer Topology
count
Jovanovic
degree

• Number of immediate peers ( degree), follows a
  power-law

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epinions.com

• who-trusts-whom Richardson Domingos, KDD 2001

count
(out) degree
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Outline

• problems
• Fractals
• Solutions
• Discussion
• what else can they solve?
• how frequent are fractals?

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What else can they solve?

• separability KDD02
• forecasting CIKM02
• dimensionality reduction SBBD00
• non-linear axis scaling KDD02
• disk trace modeling PEVA02
• selectivity of spatial queries PODS94, VLDB95,
  ICDE00
• ...

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Problem 3 - spatial d.m.

• Galaxies (Sloan Digital Sky Survey w/ B. Nichol)

• - spiral and elliptical galaxies
• - patterns? (not Gaussian not uniform)
• attraction/repulsion?
• separability??

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Solution3 spatial d.m.
CORRELATION INTEGRAL!
log(pairs within ltr )
- 1.8 slope - plateau! - repulsion!
ell-ell
spi-spi
spi-ell
log(r)
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Solution3 spatial d.m.
w/ Seeger, Traina, Traina, SIGMOD00
log(pairs within ltr )
- 1.8 slope - plateau! - repulsion!
ell-ell
spi-spi
spi-ell
log(r)
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spatial d.m.
Heuristic on choosing of clusters
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Solution3 spatial d.m.
log(pairs within ltr )
- 1.8 slope - plateau! - repulsion!
ell-ell
spi-spi
spi-ell
log(r)
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Problem4 dim. reduction
skip

• given attributes x1, ... xn
• possibly, non-linearly correlated
• drop the useless ones

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Problem4 dim. reduction
skip

• given attributes x1, ... xn
• possibly, non-linearly correlated
• drop the useless ones
• (Q why?
• A to avoid the dimensionality curse)
• Solution keep on dropping attributes, until the
  f.d. changes! SBBD00

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Outline

• problems
• Fractals
• Solutions
• Discussion
• what else can they solve?
• how frequent are fractals?

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Fractals power laws

• appear in numerous settings
• medical
• geographical / geological
• social
• computer-system related
• ltand many-many more! see Mandelbrotgt

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Fractals Brain scans

• brain-scans

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More fractals

• periphery of malignant tumors 1.5
• benign 1.3
• Burdet

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More fractals

• cardiovascular system 3 (!) lungs 2.9

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Fractals power laws

• appear in numerous settings
• medical
• geographical / geological
• social
• computer-system related

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More fractals

• Coastlines 1.2-1.58

1.1
1
1.3
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(No Transcript)
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GIS points

• Cross-roads of Montgomery county
• any rules?

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GIS

• A self-similarity
• intrinsic dim. 1.51

log(pairs(within lt r))
log( r )
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ExamplesLB county

• Long Beach county of CA (road end-points)

log(pairs)
log(r)
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More power laws areas Korcaks law
Scandinavian lakes Any pattern?
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More power laws areas Korcaks law
log(count( gt area))
Scandinavian lakes area vs complementary
cumulative count (log-log axes)
log(area)
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More power laws Korcak
log(count( gt area))
Japan islands area vs cumulative count (log-log
axes)
log(area)
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More power laws

• Energy of earthquakes (Gutenberg-Richter law)
  simscience.org

Energy released
log(count)
Magnitude log(energy)
day
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Fractals power laws

• appear in numerous settings
• medical
• geographical / geological
• social
• computer-system related

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A famous power law Zipfs law
log(freq)
a

• Bible - rank vs. frequency (log-log)

the
Rank/frequency plot
log(rank)
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TELCO data
count of customers
best customer
of service units
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SALES data store96
count of products
aspirin
units sold
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Olympic medals (Sidney 2000)
log(medals)
log( rank)
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Olympic medals (Athens04)
log(medals)
log( rank)
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Even more power laws

• Income distribution (Paretos law)
• size of firms
• publication counts (Lotkas law)

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Even more power laws

• library science (Lotkas law of publication
  count) and citation counts (citeseer.nj.nec.com
  6/2001)

log(count)
Ullman
log(citations)
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Even more power laws

• web hit counts w/ A. Montgomery

yahoo.com
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Fractals power laws

• appear in numerous settings
• medical
• geographical / geological
• social
• computer-system related

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Power laws, contd

• In- and out-degree distribution of web sites
  Barabasi, IBM-CLEVER

log indegree
from Ravi Kumar, Prabhakar Raghavan, Sridhar
Rajagopalan, Andrew Tomkins
- log(freq)
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Power laws, contd

• In- and out-degree distribution of web sites
  Barabasi, IBM-CLEVER
• length of file transfers Bestavros
• duration of UNIX jobs Harchol-Balter

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Conclusions

• Fascinating problems in Data Mining find
  patterns in
• sensors/streams
• graphs/networks

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Conclusions - contd

• New tools for Data Mining self-similarity
  power laws appear in many cases

Bad news lead to skewed distributions (no
Gaussian, Poisson, uniformity, independence, mean,
variance)
X
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Resources

• Manfred Schroeder Chaos, Fractals and Power
  Laws, 1991
• Jiawei Han and Micheline Kamber Data Mining
  Concepts and Techniques, 2001

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References

• ieeeTN94 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.
• pods94 Christos Faloutsos and Ibrahim Kamel,
  Beyond Uniformity and Independence Analysis of
  R-trees Using the Concept of Fractal Dimension,
  PODS, Minneapolis, MN, May 24-26, 1994, pp. 4-13

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References

• vldb95 Alberto Belussi and Christos Faloutsos,
  Estimating the Selectivity of Spatial Queries
  Using the Correlation' Fractal Dimension Proc.
  of VLDB, p. 299-310, 1995
• vldb96 Christos Faloutsos, Yossi Matias and Avi
  Silberschatz, Modeling Skewed Distributions Using
  Multifractals and the 80-20 Law Conf. on Very
  Large Data Bases (VLDB), Bombay, India, Sept.
  1996.

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References

• vldb96 Christos Faloutsos and Volker Gaede
  Analysis of the Z-Ordering Method Using the
  Hausdorff Fractal Dimension VLD, Bombay, India,
  Sept. 1996
• sigcomm99 Michalis Faloutsos, Petros Faloutsos
  and Christos Faloutsos, What does the Internet
  look like? Empirical Laws of the Internet
  Topology, SIGCOMM 1999

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References

• icde99 Guido Proietti and Christos Faloutsos,
  I/O complexity for range queries on region data
  stored using an R-tree International Conference
  on Data Engineering (ICDE), Sydney, Australia,
  March 23-26, 1999
• sigmod2000 Christos Faloutsos, Bernhard Seeger,
  Agma J. M. Traina and Caetano Traina Jr., Spatial
  Join Selectivity Using Power Laws, SIGMOD 2000

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References

• kdd2001 Agma J. M. Traina, Caetano Traina Jr.,
  Spiros Papadimitriou and Christos Faloutsos
  Tri-plots Scalable Tools for Multidimensional
  Data Mining, KDD 2001, San Francisco, CA.

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Thank you!

• Contact info
• christos _at_ cs.cmu.edu
• www. cs.cmu.edu /christos
• (w/ papers, datasets, code for fractal dimension
  estimation, etc)