finding patterns in large, real networks
DESCRIPTION
Finding patterns in large, real networks. Christos Faloutsos CMU. Thanks to. Deepayan Chakrabarti (CMU/Yahoo) Michalis Faloutsos (UCR) Spiros Papadimitriou (CMU/IBM) George Siganos (UCR). Introduction. Protein Interactions [genomebiology.com]. Internet Map [lumeta.com]. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/1.jpg)
CMU SCS
Finding patterns in large, real networks
Christos Faloutsos
CMU
![Page 2: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/2.jpg)
LLNL 05 C. Faloutsos 2
CMU SCS
Thanks to
• Deepayan Chakrabarti (CMU/Yahoo)
• Michalis Faloutsos (UCR)
• Spiros Papadimitriou (CMU/IBM)
• George Siganos (UCR)
![Page 3: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/3.jpg)
LLNL 05 C. Faloutsos 3
CMU SCS
Introduction
Internet Map [lumeta.com]
Food Web [Martinez ’91]
Protein Interactions [genomebiology.com]
Friendship Network [Moody ’01]
Graphs are everywhere!
![Page 4: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/4.jpg)
LLNL 05 C. Faloutsos 4
CMU SCS
Outline
• Laws– static
– time-evolution laws
– why so many power laws?
• tools/results
• future steps
![Page 5: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/5.jpg)
LLNL 05 C. Faloutsos 5
CMU SCS
Motivating questions
• Q1: What do real graphs look like?• Q2: How to generate realistic graphs?
![Page 6: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/6.jpg)
LLNL 05 C. Faloutsos 6
CMU SCS
Virus propagation
• Q3: How do viruses/rumors propagate?
• Q4: Who is the best person/computer to immunize against a virus?
![Page 7: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/7.jpg)
LLNL 05 C. Faloutsos 7
CMU SCS
Graph clustering & mining
• Q5: which edges/nodes are ‘abnormal’?
• Q6: split a graph in k ‘natural’ communities - but how to determine k?
![Page 8: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/8.jpg)
LLNL 05 C. Faloutsos 8
CMU SCS
Outline
• Laws– static
– time-evolution laws
– why so many power laws?
• tools/results
• future steps
![Page 9: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/9.jpg)
LLNL 05 C. Faloutsos 9
CMU SCS
Topology
Q1: How does the Internet look like? Any rules?
(Looks random – right?)
![Page 10: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/10.jpg)
LLNL 05 C. Faloutsos 10
CMU SCS
Laws – degree distributions
• Q: avg degree is ~2 - what is the most probable degree?
degree
count ??
2
![Page 11: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/11.jpg)
LLNL 05 C. Faloutsos 11
CMU SCS
Laws – degree distributions
• Q: avg degree is ~2 - what is the most probable degree?
degreedegree
count ??
2
count
2
![Page 12: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/12.jpg)
LLNL 05 C. Faloutsos 12
CMU SCS
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
![Page 13: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/13.jpg)
LLNL 05 C. Faloutsos 13
CMU SCS
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
![Page 14: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/14.jpg)
LLNL 05 C. Faloutsos 14
CMU SCS
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
![Page 15: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/15.jpg)
LLNL 05 C. Faloutsos 15
CMU SCS
IV. The Node Neighborhood
• N(h) = # of pairs of nodes within h hops
![Page 16: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/16.jpg)
LLNL 05 C. Faloutsos 16
CMU SCS
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 -> 3h
![Page 17: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/17.jpg)
LLNL 05 C. Faloutsos 17
CMU SCS
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 -> 3h
WE HAVE DUPLICATES!
![Page 18: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/18.jpg)
LLNL 05 C. Faloutsos 18
CMU SCS
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 -> 3h
‘avg’ degree: meaningless!
![Page 19: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/19.jpg)
LLNL 05 C. Faloutsos 19
CMU SCS
IV. Power-law: hopplot H
Pairs of nodes as a function of hops N(h)= hH
H = 4.86
Dec 98Hops
# of Pairs # of Pairs
Hops Router level ’95
H = 2.83
![Page 20: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/20.jpg)
LLNL 05 C. Faloutsos 20
CMU SCS
Observation• Q: Intuition behind ‘hop exponent’?• A: ‘intrinsic=fractal dimensionality’ of the
network
N(h) ~ h1 N(h) ~ h2
...
![Page 21: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/21.jpg)
LLNL 05 C. Faloutsos 21
CMU SCS
Hop plots
• More on fractal/intrinsic dimensionalities: very soon
![Page 22: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/22.jpg)
LLNL 05 C. Faloutsos 22
CMU SCS
Any other ‘laws’?
Yes!
![Page 23: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/23.jpg)
LLNL 05 C. Faloutsos 23
CMU SCS
Any other ‘laws’?
Yes!
• Small diameter– six degrees of separation / ‘Kevin Bacon’
– small worlds [Watts and Strogatz]
![Page 24: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/24.jpg)
LLNL 05 C. Faloutsos 24
CMU SCS
Any other ‘laws’?• Bow-tie, for the web [Kumar+ ‘99]
• IN, SCC, OUT, ‘tendrils’
• disconnected components
![Page 25: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/25.jpg)
LLNL 05 C. Faloutsos 25
CMU SCS
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:2n:3
![Page 26: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/26.jpg)
LLNL 05 C. Faloutsos 26
CMU SCS
More Power laws
• Also hold for other web graphs [Barabasi+, ‘99], [Kumar+, ‘99]
• citation graphs (see later)
• and many more
![Page 27: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/27.jpg)
LLNL 05 C. Faloutsos 27
CMU SCS
Outline
• Laws– static
– time-evolution laws
– why so many power laws?
• tools/results
• future steps
![Page 28: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/28.jpg)
LLNL 05 C. Faloutsos 28
CMU SCS
How do graphs evolve?
![Page 29: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/29.jpg)
LLNL 05 C. Faloutsos 29
CMU SCS
Time Evolution: rank R
The rank exponent has not changed! [Siganos+, ‘03]
Domainlevel
#days since Nov. ‘97
![Page 30: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/30.jpg)
LLNL 05 C. Faloutsos 30
CMU SCS
How do graphs evolve?
• degree-exponent seems constant - anything else?
![Page 31: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/31.jpg)
LLNL 05 C. Faloutsos 31
CMU SCS
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?
![Page 32: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/32.jpg)
LLNL 05 C. Faloutsos 32
CMU SCS
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
![Page 33: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/33.jpg)
LLNL 05 C. Faloutsos 33
CMU SCS
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
![Page 34: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/34.jpg)
LLNL 05 C. Faloutsos 34
CMU SCS
Shrinking diameter• Authors &
publications
• 1992– 318 nodes
– 272 edges
• 2002– 60,000 nodes
• 20,000 authors
• 38,000 papers
– 133,000 edges
![Page 35: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/35.jpg)
LLNL 05 C. Faloutsos 35
CMU SCS
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
![Page 36: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/36.jpg)
LLNL 05 C. Faloutsos 36
CMU SCS
Shrinking diameter• Autonomous
systems
• 1997– 3,000 nodes
– 10,000 edges
• 2000– 6,000 nodes
– 26,000 edges
• One graph per day
![Page 37: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/37.jpg)
LLNL 05 C. Faloutsos 37
CMU SCS
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)
![Page 38: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/38.jpg)
LLNL 05 C. Faloutsos 38
CMU SCS
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!
![Page 39: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/39.jpg)
LLNL 05 C. Faloutsos 39
CMU SCS
Temporal evolution of graphs
• A: over-doubled - but obeying:
E(t) ~ N(t)a for all t
where 1<a<2
![Page 40: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/40.jpg)
LLNL 05 C. Faloutsos 40
CMU SCS
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
![Page 41: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/41.jpg)
LLNL 05 C. Faloutsos 41
CMU SCS
Densification Power Law
ArXiv: Physics papers
and their citations
1.69
N(t)
E(t)
![Page 42: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/42.jpg)
LLNL 05 C. Faloutsos 42
CMU SCS
Densification Power Law
ArXiv: Physics papers
and their citations
1.69
N(t)
E(t)
‘tree’
1
![Page 43: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/43.jpg)
LLNL 05 C. Faloutsos 43
CMU SCS
Densification Power Law
ArXiv: Physics papers
and their citations
1.69
N(t)
E(t)‘clique’
2
![Page 44: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/44.jpg)
LLNL 05 C. Faloutsos 44
CMU SCS
Densification Power Law
U.S. Patents, citing each other
1.66
N(t)
E(t)
![Page 45: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/45.jpg)
LLNL 05 C. Faloutsos 45
CMU SCS
Densification Power Law
Autonomous Systems
1.18
N(t)
E(t)
![Page 46: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/46.jpg)
LLNL 05 C. Faloutsos 46
CMU SCS
Densification Power Law
ArXiv: authors & papers
1.15
N(t)
E(t)
![Page 47: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/47.jpg)
LLNL 05 C. Faloutsos 47
CMU SCS
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
![Page 48: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/48.jpg)
LLNL 05 C. Faloutsos 48
CMU SCS
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
![Page 49: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/49.jpg)
LLNL 05 C. Faloutsos 49
CMU SCS
Power laws
• Many more power laws, in diverse settings:
![Page 50: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/50.jpg)
LLNL 05 C. Faloutsos 50
CMU SCS
A famous power law: Zipf’s law
• Bible - rank vs frequency (log-log)
log(rank)
log(freq)
“a”
“the”
![Page 51: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/51.jpg)
LLNL 05 C. Faloutsos 51
CMU SCS
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)
![Page 52: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/52.jpg)
LLNL 05 C. Faloutsos 52
CMU SCS
The Peer-to-Peer Topology
• Frequency versus degree
• Number of adjacent peers follows a power-law
[Jovanovic+]
![Page 53: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/53.jpg)
LLNL 05 C. Faloutsos 53
CMU SCS
Web Site Traffic
log(freq)
log(count)Zipf
Click-stream data
u-id’s url’s
log(freq)
log(count)
‘yahoo’
‘super-surfer’
![Page 54: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/54.jpg)
LLNL 05 C. Faloutsos 54
CMU SCS
Lotka’s law
(Lotka’s law of publication count); and citation counts: (citeseer.nj.nec.com 6/2001)
log(#citations)
log(count)
J. Ullman
![Page 55: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/55.jpg)
LLNL 05 C. Faloutsos 55
CMU SCS Swedish sex-web
Nodes: people (Females; Males)Links: sexual relationships
Liljeros et al. Nature 2001
4781 Swedes; 18-74; 59% response rate.
Albert Laszlo Barabasihttp://www.nd.edu/~networks/Publication%20Categories/04%20Talks/2005-norway-3hours.ppt
![Page 56: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/56.jpg)
LLNL 05 C. Faloutsos 56
CMU SCS Swedish sex-web
Nodes: people (Females; Males)Links: sexual relationships
Liljeros et al. Nature 2001
4781 Swedes; 18-74; 59% response rate.
Albert Laszlo Barabasihttp://www.nd.edu/~networks/Publication%20Categories/04%20Talks/2005-norway-3hours.ppt
![Page 57: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/57.jpg)
LLNL 05 C. Faloutsos 57
CMU SCS
Power laws
• Q: Why so many power laws?
• A1: self-similarity
• A2: ‘rich-get-richer’
![Page 58: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/58.jpg)
LLNL 05 C. Faloutsos 58
CMU SCS
Digression: intro to fractals
• Fractals: sets of points that are self similar
![Page 59: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/59.jpg)
LLNL 05 C. Faloutsos 59
CMU SCS
A famous fractal
e.g., Sierpinski triangle:
...zero area;
infinite length!
dimensionality = ??
![Page 60: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/60.jpg)
LLNL 05 C. Faloutsos 60
CMU SCS
A famous fractal
e.g., Sierpinski triangle:
...zero area;
infinite length!
dimensionality = log(3)/log(2) = 1.58
![Page 61: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/61.jpg)
LLNL 05 C. Faloutsos 61
CMU SCS
A famous fractal
equivalent graph:
![Page 62: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/62.jpg)
LLNL 05 C. Faloutsos 62
CMU SCS
Intrinsic (‘fractal’) dimension
How to estimate it?
![Page 63: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/63.jpg)
LLNL 05 C. Faloutsos 63
CMU SCS
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) )
![Page 64: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/64.jpg)
LLNL 05 C. Faloutsos 64
CMU SCS
Sierpinsky triangle
log( r )
log(#pairs within <=r )
1.58
== ‘correlation integral’
= CDF of pairwise distances
![Page 65: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/65.jpg)
LLNL 05 C. Faloutsos 65
CMU SCS
Sierpinsky triangle
log( r )
log(#pairs within <=r )
1.58
== ‘correlation integral’
= CDF of pairwise distances
![Page 66: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/66.jpg)
LLNL 05 C. Faloutsos 66
CMU SCS
Line
log( r )
log(#pairs within <=r )
1.58
== ‘correlation integral’
= CDF of pairwise distances
1...
![Page 67: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/67.jpg)
LLNL 05 C. Faloutsos 67
CMU SCS
2-d (Plane)
log( r )
log(#pairs within <=r )
1.58
== ‘correlation integral’
= CDF of pairwise distances
2
![Page 68: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/68.jpg)
LLNL 05 C. Faloutsos 68
CMU SCS
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, 1995log(hops)
log(#pairs)
2.8
![Page 69: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/69.jpg)
LLNL 05 C. Faloutsos 69
CMU SCS
Fractals and power laws
They are related concepts:
• fractals <=>
• self-similarity <=>
• scale-free <=>
• power laws ( y= xa )– F = C r(-2)
![Page 70: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/70.jpg)
LLNL 05 C. Faloutsos 70
CMU SCS
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]
![Page 71: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/71.jpg)
LLNL 05 C. Faloutsos 71
CMU SCS
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
![Page 72: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/72.jpg)
LLNL 05 C. Faloutsos 72
CMU SCS
Self-similarity at work
• Q2: How to generate realistic graphs?
• (Q2’) and how to grow it over time?
![Page 73: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/73.jpg)
LLNL 05 C. Faloutsos 73
CMU SCS
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?
![Page 74: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/74.jpg)
LLNL 05 C. Faloutsos 74
CMU SCS
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]
![Page 75: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/75.jpg)
LLNL 05 C. Faloutsos 75
CMU SCS
Kronecker product
![Page 76: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/76.jpg)
LLNL 05 C. Faloutsos 76
CMU SCS
Kronecker product
![Page 77: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/77.jpg)
LLNL 05 C. Faloutsos 77
CMU SCS
Kronecker product
N N*N N**4
![Page 78: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/78.jpg)
LLNL 05 C. Faloutsos 78
CMU SCS
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
![Page 79: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/79.jpg)
LLNL 05 C. Faloutsos 79
CMU SCS
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)
![Page 80: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/80.jpg)
LLNL 05 C. Faloutsos 80
CMU SCS
Kronecker - ArXiv
real
(stochastic)Kronecker
Degree Scree Diameter D.P.L.
(det. Kronecker)
![Page 81: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/81.jpg)
LLNL 05 C. Faloutsos 81
CMU SCS
Kronecker - patents
Degree Scree Diameter D.P.L.
![Page 82: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/82.jpg)
LLNL 05 C. Faloutsos 82
CMU SCS
Kronecker - A.S.
![Page 83: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/83.jpg)
LLNL 05 C. Faloutsos 83
CMU SCS
Outline
• Laws– static
– time-evolution laws
– why so many power laws?
• tools/results– virus propagation
– connection subgraphs
– cross-associations
• future steps
![Page 84: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/84.jpg)
LLNL 05 C. Faloutsos 84
CMU SCS
Virus propagation
• Q3: How do viruses/rumors propagate?
• Q4: Who is the best person/computer to immunize against a virus?
![Page 85: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/85.jpg)
LLNL 05 C. Faloutsos 85
CMU SCS
The model: SIS
• ‘Flu’ like: Susceptible-Infected-Susceptible
• Virus ‘strength’ s= /
Infected
Healthy
NN1
N3
N2
Prob.
Prob. β
Prob.
![Page 86: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/86.jpg)
LLNL 05 C. Faloutsos 86
CMU SCS
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
![Page 87: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/87.jpg)
LLNL 05 C. Faloutsos 87
CMU SCS
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?
![Page 88: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/88.jpg)
LLNL 05 C. Faloutsos 88
CMU SCS
Epidemic threshold
• [Theorem] We have no epidemic, if
β/δ <τ = 1/ λ1,A
![Page 89: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/89.jpg)
LLNL 05 C. Faloutsos 89
CMU SCS
Epidemic threshold
• [Theorem] We have no epidemic, if
β/δ <τ = 1/ λ1,A
largest eigenvalueof adj. matrix A
attack prob.
recovery prob.epidemic threshold
Proof: [Wang+03]
![Page 90: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/90.jpg)
LLNL 05 C. Faloutsos 90
CMU SCS
Experiments (Oregon)
/ > τ (above threshold)
/ = τ (at the threshold)
/ < τ (below threshold)
![Page 91: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/91.jpg)
LLNL 05 C. Faloutsos 91
CMU SCS
Our result:
• Holds for any graph
• includes older results as special cases
![Page 92: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/92.jpg)
LLNL 05 C. Faloutsos 92
CMU SCS
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
![Page 93: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/93.jpg)
LLNL 05 C. Faloutsos 93
CMU SCS
Outline
• Laws– static
– time-evolution laws
– why so many power laws?
• tools/results– virus propagation
– connection subgraphs
– cross-associations
• future steps
![Page 94: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/94.jpg)
LLNL 05 C. Faloutsos 94
CMU SCS
(A-A) Kidman-Diaz
• What are the best paths between ‘Kidman’ and ‘Diaz’?
![Page 95: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/95.jpg)
LLNL 05 C. Faloutsos 95
CMU SCS
(A-A) Kidman-Diaz
Strong, direct link
• What are the best paths between ‘Kidman’ and ‘Diaz’? [KDD’04, w/ McCurley+Tomkins]
![Page 96: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/96.jpg)
LLNL 05 C. Faloutsos 96
CMU SCS
Outline
• Laws– static
– time-evolution laws
– why so many power laws?
• tools/results– virus propagation
– connection subgraphs
– cross-associations
• future steps
![Page 97: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/97.jpg)
LLNL 05 C. Faloutsos 97
CMU SCS
Graph clustering & mining
• Q5: which edges/nodes are ‘abnormal’?
• Q6: split a graph in k ‘natural’ communities - but how to determine k?
![Page 98: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/98.jpg)
LLNL 05 C. Faloutsos 98
CMU SCS
Graph partitioning
• Documents x terms
• Customers x products
• Users x web-sites
![Page 99: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/99.jpg)
LLNL 05 C. Faloutsos 99
CMU SCS
Graph partitioning
• Documents x terms
• Customers x products
• Users x web-sites
• Q: HOW MANY PIECES?
![Page 100: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/100.jpg)
LLNL 05 C. Faloutsos 100
CMU SCS
Graph partitioning
• Documents x terms
• Customers x products
• Users x web-sites
• Q: HOW MANY PIECES?
• A: MDL/ compression
![Page 101: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/101.jpg)
LLNL 05 C. Faloutsos 101
CMU SCS
Cross-associations
1x2 2x2
![Page 102: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/102.jpg)
LLNL 05 C. Faloutsos 102
CMU SCS
Cross-associations
2x3 3x3 3x4
![Page 103: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/103.jpg)
LLNL 05 C. Faloutsos 103
CMU SCS
Cross-associations
![Page 104: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/104.jpg)
LLNL 05 C. Faloutsos 104
CMU SCS
Cross-associations
![Page 105: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/105.jpg)
LLNL 05 C. Faloutsos 105
CMU SCS
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)
![Page 106: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/106.jpg)
LLNL 05 C. Faloutsos 106
CMU SCS
Cross-associations
outlier edgemissing edge
![Page 107: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/107.jpg)
LLNL 05 C. Faloutsos 107
CMU SCS
Outline
• Laws– static
– time-evolution laws
– why so many power laws?
• tools/results
• other related projects
• future steps
![Page 108: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/108.jpg)
LLNL 05 C. Faloutsos 108
CMU SCS
Motivation
water distribution network
normal operation
May have hundreds of measurements, butit is unlikely they are completely unrelated!
Phase 1 Phase 2 Phase 3
: : : : : :
: : : : : :
chlo
rin
e c
on
cen
trati
on
s
sensorsnear leak
sensorsawayfrom leak
![Page 109: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/109.jpg)
LLNL 05 C. Faloutsos 109
CMU SCS
Phase 1 Phase 2 Phase 3
: : : : : :
: : : : : :
Motivation
water distribution network
normal operation major leak
May have hundreds of measurements, butit is unlikely they are completely unrelated!
chlo
rin
e c
on
cen
trati
on
s
sensorsnear leak
sensorsawayfrom leak
![Page 110: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/110.jpg)
LLNL 05 C. Faloutsos 110
CMU SCS
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
: : : : : :
: : : : : :
chlo
rin
e c
on
cen
trati
on
s
Phase 1
k = 1
![Page 111: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/111.jpg)
LLNL 05 C. Faloutsos 111
CMU SCS
Motivation
We would like to discover a few “hidden(latent) variables” that summarize the key trends
chlo
rin
e c
on
cen
trati
on
s
Phase 1 Phase 1Phase 2 Phase 2
actual measurements(n streams)
k hidden variable(s)
k = 2
: : : : : :
: : : : : :
![Page 112: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/112.jpg)
LLNL 05 C. Faloutsos 112
CMU SCS
Motivation
We would like to discover a few “hidden(latent) variables” that summarize the key trends
chlo
rin
e c
on
cen
trati
on
s
Phase 1 Phase 1Phase 2 Phase 2Phase 3 Phase 3
actual measurements(n streams)
k hidden variable(s)
k = 1
: : : : : :
: : : : : :
![Page 113: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/113.jpg)
LLNL 05 C. Faloutsos 113
CMU SCS
Stream mining
• Solution: SPIRIT [VLDB’05]– incremental, on-line PCA
![Page 114: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/114.jpg)
LLNL 05 C. Faloutsos 114
CMU SCS
Outline
• Laws
• tools/results
• other related projects
• future steps
![Page 115: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/115.jpg)
LLNL 05 C. Faloutsos 115
CMU SCS
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]
![Page 116: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/116.jpg)
LLNL 05 C. Faloutsos 116
CMU SCS
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
![Page 117: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/117.jpg)
LLNL 05 C. Faloutsos 117
CMU SCS
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)
![Page 118: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/118.jpg)
LLNL 05 C. Faloutsos 118
CMU SCS
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
![Page 119: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/119.jpg)
LLNL 05 C. Faloutsos 119
CMU SCS
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
![Page 120: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/120.jpg)
LLNL 05 C. Faloutsos 120
CMU SCS
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.
![Page 121: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/121.jpg)
LLNL 05 C. Faloutsos 121
CMU SCS
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.
![Page 122: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/122.jpg)
LLNL 05 C. Faloutsos 122
CMU SCS
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
![Page 123: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/123.jpg)
LLNL 05 C. Faloutsos 123
CMU SCS
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
![Page 124: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/124.jpg)
LLNL 05 C. Faloutsos 124
CMU SCS
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)
![Page 125: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/125.jpg)
LLNL 05 C. Faloutsos 125
CMU SCS
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)
![Page 126: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/126.jpg)
LLNL 05 C. Faloutsos 126
CMU SCS
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)
![Page 127: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/127.jpg)
LLNL 05 C. Faloutsos 127
CMU SCS
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.
![Page 128: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/128.jpg)
LLNL 05 C. Faloutsos 128
CMU SCS
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)
![Page 129: Finding patterns in large, real networks](https://reader035.vdocuments.site/reader035/viewer/2022062314/56814513550346895db1d602/html5/thumbnails/129.jpg)
LLNL 05 C. Faloutsos 129
CMU SCS
Thank you!
www.cs.cmu.edu/~christos
www.db.cs.cmu.edu