Introduction to Fractalsand Fractal Dimension
Christos Faloutsos
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Intro to Fractals - Outline
• Motivation – 3 problems / case studies
• Definition of fractals and power laws
• Solutions to posed problems
• More examples
• Discussion - putting fractals to work!
• Conclusions – practitioner’s guide
• Appendix: gory details - boxcounting plots
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Road end-points of Montgomery county:
•Q : distribution?
• not uniform
• not Gaussian
• ???
• no rules/pattern at all!?
Problem #1: GIS - points
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Problem #2 - spatial d.m.
Galaxies (Sloan Digital Sky Survey w/ B. Nichol)
- ‘spiral’ and ‘elliptical’ galaxies
(stores and households ...)
- patterns?
- attraction/repulsion?
- how many ‘spi’ within r from an ‘ell’?
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Problem #3: traffic
• disk trace (from HP - J. Wilkes); Web traffic
time
#bytes
Poisson
- how many explosions to expect?
- queue length distr.?
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Common answer for all three:
• Fractals / self-similarities / power laws
• Seminal works from Hilbert, Minkowski, Cantor, Mandelbrot, (Hausdorff, Lyapunov, Ken Wilson, …)
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Road map
• Motivation – 3 problems / case studies
• Definition of fractals and power laws
• Solutions to posed problems
• More examples and tools
• Discussion - putting fractals to work!
• Conclusions – practitioner’s guide
• Appendix: gory details - boxcounting plots
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Definitions (cont’d)
• Simple 2-D figure has finite perimeter and finite area– area ≈ perimeter2
• Do all 2-D figures have finite perimeter,
finite area, and area ≈ perimeter2?
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What is a fractal?
= self-similar point set, e.g., Sierpinski triangle:
...zero area;
infinite length!
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Definitions (cont’d)
• Paradox: Infinite perimeter ; Zero area!
• ‘dimensionality’: between 1 and 2
• actually: Log(3)/Log(2) = 1.58…
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Definitions (cont’d)
• “self similar”– pieces look like whole independent of scale
• “dimension”dim = log(# self-similar pieces) / log(scale)
– dim(line) = log(2)/log(2) = 1– dim(square) = log(4)/log(2) = 2– dim(cube) = log(8)/log(2) = 3
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Intrinsic (‘fractal’) dimension
• Q: fractal dimension of a line?
• A: 1 (= log(2)/log(2))
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Intrinsic (‘fractal’) dimension
• Q: dfn for a given set of points?
42
33
24
15
yx
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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^2Fd = slope of (log(nn) vs
log(r) )
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Intrinsic (‘fractal’) dimension
• Algorithm, to estimate it?Notice• avg nn(<=r) is exactly tot#pairs(<=r) / (2*N)
including ‘mirror’ pairs
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Dfn of fd:
ONLY for a perfectly self-similar point set:
=log(n)/log(f) = log(3)/log(2) = 1.58
...zero area;
infinite length!
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Sierpinsky triangle
log( r )
log(#pairs within <=r )
1.58
== ‘correlation integral’
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Observations:
• Euclidean objects have integer fractal dimensions – point: 0– lines and smooth curves: 1– smooth surfaces: 2
• fractal dimension -> roughness of the periphery
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Important properties
• fd = embedding dimension -> uniform pointset
• a point set may have several fd, depending on scale
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Road map
• Motivation – 3 problems / case studies
• Definition of fractals and power laws
• Solutions to posed problems
• More examples and tools
• Discussion - putting fractals to work!
• Conclusions – practitioner’s guide
• Appendix: gory details - boxcounting plots
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Cross-roads of Montgomery county:
•any rules?
Problem #1: GIS points
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Solution #1
A: self-similarity ->• <=> fractals • <=> scale-free• <=> power-laws
(y=x^a, F=C*r^(-2))• avg#neighbors(<= r )
= r^D
log( r )
log(#pairs(within <= r))
1.51
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Solution #1
A: self-similarity• avg#neighbors(<= r )
~ r^(1.51)
log( r )
log(#pairs(within <= r))
1.51
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Examples:MG county
• Montgomery County of MD (road end-points)
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Examples:LB county
• Long Beach county of CA (road end-points)
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Solution#2: spatial d.m.Galaxies ( ‘BOPS’ plot - [sigmod2000])
log(#pairs)
log(r)
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Solution#2: spatial d.m.
log(r)
log(#pairs within <=r )
spi-spi
spi-ell
ell-ell
- 1.8 slope
- plateau!
- repulsion!
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spatial d.m.
log(r)
log(#pairs within <=r )
spi-spi
spi-ell
ell-ell
- 1.8 slope
- plateau!
- repulsion!
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spatial d.m.
r1r2
r1
r2
Heuristic on choosing # of clusters
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spatial d.m.
log(r)
log(#pairs within <=r )
spi-spi
spi-ell
ell-ell
- 1.8 slope
- plateau!
- repulsion!
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spatial d.m.
log(r)
log(#pairs within <=r )
spi-spi
spi-ell
ell-ell
- 1.8 slope
- plateau!
-repulsion!!
-duplicates
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Solution #3: traffic
• disk traces: self-similar:
time
#bytes
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Solution #3: traffic
• disk traces (80-20 ‘law’ = ‘multifractal’)
time
#bytes
20% 80%
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Solution#3: traffic
Clarification:
• fractal: a set of points that is self-similar
• multifractal: a probability density function that is self-similar
Many other time-sequences are bursty/clustered: (such as?)
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Tape accesses
time
Tape#1 Tape# N
# tapes needed, to retrieve n records?
(# days down, due to failures / hurricanes / communication noise...)
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Tape accesses
time
Tape#1 Tape# N
# tapes retrieved
# qual. records
50-50 = Poisson
real
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Fast estimation of fd(s):
• How, for the (correlation) fractal dimension?
• A: Box-counting plot:
log( r )
rpi
log(sum(pi ^2))
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Definitions
• pi : the percentage (or count) of points in the i-th cell
• r: the side of the grid
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Fast estimation of fd(s):
• compute sum(pi^2) for another grid side, r’
log( r )
r’
pi’
log(sum(pi ^2))
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Fast estimation of fd(s):• etc; if the resulting plot has a linear part, its
slope is the correlation fractal dimension D2
log( r )
log(sum(pi ^2))
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Hausdorff or box-counting fd:
• Box counting plot: Log( N ( r ) ) vs Log ( r)
• r: grid side
• N (r ): count of non-empty cells
• (Hausdorff) fractal dimension D0:
)log(
))(log(0 r
rND
ƒƒ
−=
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Definitions (cont’d)
• Hausdorff fd:r
log(r)
log(#non-empty cells)
D0
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Observations
• q=0: Hausdorff fractal dimension
• q=2: Correlation fractal dimension (identical to the exponent of the number of neighbors vs radius)
• q=1: Information fractal dimension
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Observations, cont’d
• in general, the Dq’s take similar, but not identical, values.
• except for perfectly self-similar point-sets, where Dq=Dq’ for any q, q’
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Examples:MG county
• Montgomery County of MD (road end-points)
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Examples:LB county
• Long Beach county of CA (road end-points)
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Summary So Far
• many fractal dimensions, with nearby values
• can be computed quickly (O(N) or O(N log(N))
• (code: on the web)
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Dimensionality Reduction with FD
Problem definition: ‘Feature selection’• given N points, with E dimensions• keep the k most ‘informative’ dimensions[Traina+,SBBD’00]
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Dim. reduction - w/ fractals
y
xxx
yy(a) Quarter-circle (c) Spike(b)Line1
0000
1
not informative
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Dim. reduction
Problem definition: ‘Feature selection’• given N points, with E dimensions• keep the k most ‘informative’ dimensionsRe-phrased: spot and drop attributes with
strong (non-)linear correlationsQ: how do we do that?
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Dim. reduction
A: Hint: correlated attributes do not affect the intrinsic/fractal dimension, e.g., if
y = f(x,z,w)we can drop y(hence: ‘partial fd’ (PFD) of a set of
attributes = the fd of the dataset, when projected on those attributes)
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Dim. reduction - w/ fractals
y
xxx
yy(a) Quarter-circle (c) Spike(b)Line1
0000
1
PFD~0
PFD=1
global FD=1
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Dim. reduction - w/ fractals
y
xxx
yy(a) Quarter-circle (c) Spike(b)Line1
0000
1
PFD=1
PFD=1
global FD=1
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Dim. reduction - w/ fractals
y
xxx
yy(a) Quarter-circle (c) Spike(b)Line1
0000
1
PFD~1
PFD~1global FD=1
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Dim. reduction - w/ fractals
• (problem: given N points in E-d, choose k best dimensions)
• Q: Algorithm?
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Dim. reduction - w/ fractals
• Q: Algorithm?
• A: e.g., greedy - forward selection:– keep the attribute with highest partial fd– add the one that causes the highest increase in
pfd– etc., until we are within epsilon from the full
f.d.
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Dim. reduction - w/ fractals
• (backward elimination: ~ reverse)– drop the attribute with least impact on the p.f.d.– repeat– until we are epsilon below the full f.d.
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Dim. reduction - w/ fractals
• Q: what is the smallest # of attributes we should keep?
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Dim. reduction - w/ fractals
• Q: what is the smallest # of attributes we should keep?
• A: we should keep at least as many as the f.d. (and probably, a few more)
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Dim. reduction - w/ fractals
• Results: E.g., on the ‘currency’ dataset
• (daily exchange rates for USD, HKD, BP, FRF, DEM, JPY - i.e., 6-d vectors, one per day - base currency: CAD)
e.g.:
USD
FRF
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E.g., on the ‘currency’ dataset
correlation integral
1.98
log(r)
log(#pairs(<=r))
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E.g., on the ‘currency’ datasetif unif + indep.
USD HKD FRF DEM BP JY
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E.g., on the eigenface dataset
16-d vectors,one for each of ~1K faces
4.25
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E.g., on the eigenface dataset
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Dim. reduction - w/ fractals
Conclusion:• can do non-linear dim. reduction
y
x
(a) Quarter-circle1
00
1
PFD~1
PFD~1
global FD=1
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More tools
• Zipf’s law
• Korcak’s law / “fat fractals”
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A famous power law: Zipf’s law
• Q: vocabulary word frequency in a document - any pattern?
aaron zoo
freq.
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A famous power law: Zipf’s law
• Bible - rank vs frequency (log-log)
log(rank)
log(freq)
“a”
“the”
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A famous power law: Zipf’s law
• Bible - rank vs frequency (log-log)
• similarly, in many other languages; for customers and sales volume; city populations etc etclog(rank)
log(freq)
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A famous power law: Zipf’s law
•Zipf distr:
freq = 1/ rank
•generalized Zipf:
freq = 1 / (rank)^a
log(rank)
log(freq)
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Olympic medals (Sidney):
rank
log(#medals)
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More power laws: areas – Korcak’s law
Scandinavian lakes
Any pattern?
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More power laws: areas – Korcak’s law
Scandinavian lakes area vs complementary cumulative count (log-log axes)
log(count( >= area))
log(area)
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More power laws: Korcak
Japan islands
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More power laws: Korcak
Japan islands;
area vs cumulative count (log-log axes) log(area)
log(count( >= area))
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(Korcak’s law: Aegean islands)
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Korcak’s law & “fat fractals”
How to generate such regions?
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Korcak’s law & “fat fractals”Q: How to generate such regions?A: recursively, from a single region
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Conclusions
• tool#1: (for points) ‘correlation integral’: (#pairs within <= r) vs (distance r)
• tool#2: (for categorical values) rank-frequency plot (a’la Zipf)
• tool#3: (for numerical values) CCDF: Complementary cumulative distr. function (#of elements with value >= a )
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Fractals - overall conclusions
• Real data often disobey textbook assumptions: not Gaussian, not Poisson, not uniform, not independent)
• self-similar datasets: appear often• powerful tools: correlation integral, rank-
frequency plot, ...• intrinsic/fractal dimension helps:
– find patterns– dimensionality reduction
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NN queries• Q: What about L1, Linf?• A: Same slope, different intercept
log(d)
log(#neighbors)
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Practitioner’s guide:• tool#1: #pairs vs distance, for a set of objects, with a
distance function (slope = intrinsic dimensionality)
log(hops)
log(#pairs)
2.8
log( r )
log(#pairs(within <= r))
1.51
internetMGcounty
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Practitioner’s guide:• tool#2: rank-frequency plot (for categorical attributes)
log(rank)
log(degree)
-0.82
internet domains Biblelog(freq)
log(rank)
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Practitioner’s guide:• tool#3: CCDF, for (skewed) numerical attributes, eg.
areas of islands/lakes, UNIX jobs...)
log(count( >= area))
log(area)
scandinavian lakes
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Resources:
• Software for fractal dimension– http://www.cs.cmu.edu/~christos– [email protected]
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Books
• Strongly recommended intro book:– Manfred Schroeder Fractals, Chaos, Power
Laws: Minutes from an Infinite Paradise W.H. Freeman and Company, 1991
• Classic book on fractals:– B. Mandelbrot Fractal Geometry of Nature,
W.H. Freeman, 1977
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References
•Belussi, A. and C. Faloutsos (Sept. 1995). Estimating the Selectivity of Spatial Queries Using the `Correlation' Fractal Dimension. Proc. of VLDB, Zurich, Switzerland.•Faloutsos, C. and V. Gaede (Sept. 1996). Analysis of the z-ordering Method Using the Hausdorff Fractal Dimension. VLDB, Bombay, India.•Proietti, G. and C. Faloutsos (March 23-26, 1999). I/O complexity for range queries on region data stored using an R-tree. International Conference on Data Engineering (ICDE), Sydney, Australia.
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Road map
• Motivation – 3 problems / case studies
• Definition of fractals and power laws
• Solutions to posed problems
• More tools and examples
• Discussion - putting fractals to work!
• Conclusions – practitioner’s guide
• Appendix: gory details - boxcounting plots
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NN queries
• Q: in NN queries, what is the effect of the shape of the query region? [Belussi+95]
rLinf
L2
L1
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NN queries• Q: in NN queries, what is the effect of the
shape of the query region?• that is, for L2, and self-similar data:
log(d)
log(#pairs-within(<=d))
r L2
D2
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NN queries• Q: What about L1, Linf?
log(d)
log(#pairs-within(<=d))
r L2
D2
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NN queries• Q: What about L1, Linf?• A: Same slope, different intercept
log(d)
log(#pairs-within(<=d))
r L2
D2
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NN queries• Q: what about the intercept? Ie., what can
we say about N2 and Ninf
r L2
N2 neighbors
rLinf
Ninf neighbors
volume: V2 volume: Vinf
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NN queries• Consider sphere with volume Vinf and r’
radius
r L2
N2 neighbors
rLinf
Ninf neighbors
volume: V2 volume: Vinf
r’
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NN queries• Consider sphere with volume Vinf and r’
radius• (r/r’)^E = V2 / Vinf
• (r/r’)^D2 = N2 / N2’• N2’ = Ninf (since shape does not matter)• and finally:
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NN queries ( N2 / Ninf ) ^ 1/D2 = (V2 / Vinf) ^ 1/E
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NN queriesConclusions: for self-similar datasets• Avg # neighbors: grows like (distance)^D2 ,
regardless of query shape (circle, diamond, square, e.t.c. )
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Internet topology
• Internet routers: how many neighbors within h hops?
Reachability function: number of neighbors within r hops, vs r (log-log).
Mbone routers, 1995log(hops)
log(#pairs)
2.8
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More power laws on the Internet
degree vs rank, for Internet domains (log-log) [sigcomm99]
log(rank)
log(degree)
-0.82
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More power laws - internet
• pdf of degrees: (slope: 2.2 )
Log(count)
Log(degree)
-2.2
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Even more power laws on the Internet
Scree plot for Internet domains (log-log) [sigcomm99]
log(i)
log( i-th eigenvalue)
0.47
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More apps: Brain scans
• Oct-trees; brain-scans
octree levels
Log(#octants)
2.63 = fd
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More apps: Medical images
[Burdett et al, SPIE ‘93]:
• benign tumors: fd ~ 2.37
• malignant: fd ~ 2.56
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More fractals:
• cardiovascular system: 3 (!)
• stock prices (LYCOS) - random walks: 1.5
• Coastlines: 1.2-1.58 (Norway!)
1 year 2 years
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More power laws
• duration of UNIX jobs [Harchol-Balter]
• Energy of earthquakes (Gutenberg-Richter law) [simscience.org]
log(freq)
magnitudeday
amplitude
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Even more power laws:
• publication counts (Lotka’s law)• Distribution of UNIX file sizes• Income distribution (Pareto’s law)
• web hit counts [Huberman]
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Power laws, cont’ed
• In- and out-degree distribution of web sites [Barabasi], [IBM-CLEVER]
• length of file transfers [Bestavros+]
• Click-stream data (w/ A. Montgomery (CMU-GSIA) + MediaMetrix)
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Resources:
• Software for fractal dimension– http://www.cs.cmu.edu/~christos– [email protected]
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Books
• Strongly recommended intro book:– Manfred Schroeder Fractals, Chaos, Power
Laws: Minutes from an Infinite Paradise W.H. Freeman and Company, 1991
• Classic book on fractals:– B. Mandelbrot Fractal Geometry of Nature,
W.H. Freeman, 1977
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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
• [PODS94] Faloutsos, C. and I. Kamel (May 24-26, 1994). Beyond Uniformity and Independence: Analysis of R-trees Using the Concept of Fractal Dimension. Proc. ACM SIGACT-SIGMOD-SIGART PODS, Minneapolis, MN.• [Traina+, SBBD’00] Traina, C., A. Traina, et al. (2000). Fast feature selection using the fractal dimension. XV Brazilian Symposium on Databases (SBBD), Paraiba, Brazil.