1 Complex systems Made of many non-identical elements connected by diverse interactions. NETWORK New...
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Transcript of 1 Complex systems Made of many non-identical elements connected by diverse interactions. NETWORK New...
1
Complex systemsMade of
many non-identical elements connected by diverse interactions.
NETWORK
New York Times
Slides: thanks to A-L Barabasi
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(Internet?) Big Ideas (3)
Structure in complex networks
3Erdös-Rényi model
(1960)
- Democratic
- Random
Pál ErdösPál Erdös (1913-1996)
Connect with probability p
p=1/6 N=10
k ~ 1.5 Poisson distribution
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Small Worlds
Stanley Milgram ’s experiment Small Worlds by Watts/Strogatz (v) = Clustering coefficient of node v
= Percentage of neighbours of v connected to each other
Clustering coefficient:
||V
γ(v)γ Vv
5Cluster CoefficientClustering: My friends will likely know each other!
Probability to be connected C » p
C =# of links between 1,2,…n neighbors
n(n-1)/2
Networks are clustered [large C(p)]
but have a small characteristic path length
[small L(p)].
Network C Crand L N
WWW 0.1078 0.00023 3.1 153127
Internet 0.18-0.3 0.001 3.7-3.763015-6209
Actor 0.79 0.00027 3.65 225226
Coauthorship 0.43 0.00018 5.9 52909
Metabolic 0.32 0.026 2.9 282
Foodweb 0.22 0.06 2.43 134
C. elegance 0.28 0.05 2.65 282
6Watts-Strogatz Model
C(p) : clustering coeff. L(p) : average path length (Watts and Strogatz, Nature 393, 440 (1998))
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k ~ 6
P(k=500) ~ 10-99
NWWW ~ 109
N(k=500)~10-90
What did we expect?
We find:
Pout(k) ~ k-out
P(k=500) ~ 10-6
out= 2.45 in = 2.1
Pin(k) ~ k- in
NWWW ~ 109 N(k=500) ~ 103
J. Kleinberg, et. al, Proceedings of the ICCC (1999)
Web
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< l
>
Finite size scaling: create a network with N nodes with Pin(k) and Pout(k)
< l > = 0.35 + 2.06 log(N)
19 degrees of separation
l15=2 [125]
l17=4 [1346 7]
… < l > = ??
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nd.edu
19 degrees of separation R. Albert et al Nature (99)
based on 800 million webpages [S. Lawrence et al Nature (99)]
A. Broder et al WWW9 (00)IBM
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Power-law Distributions
Gnutella: Node connectivity follows a powerlaw*, i.e. P(k neighbours) k -
* Mapping the Gnutella network: Properties of largescale peer-to-peer systems and implications for system design. M. Ripeanu, A. Iamnitchi, and I. Foster. IEEE Internet Computing Journal 6, 1 (2002), 50-57.
November 2000 March 2001
10What does it mean?Poisson distribution
Exponential Network
Power-law distribution
Scale-free Network
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INTERNET BACKBONE
(Faloutsos, Faloutsos and Faloutsos, 1999)
Nodes: computers, routers Links: physical lines
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13ACTOR CONNECTIVITIES
Nodes: actors Links: cast jointly
N = 212,250 actors k = 28.78
P(k) ~k-
Days of Thunder (1990) Far and Away
(1992) Eyes Wide Shut (1999)
=2.3
14SCIENCE CITATION INDEX
( = 3)
Nodes: papers Links: citations
(S. Redner, 1998)
P(k) ~k-
2212
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1736 PRL papers (1988)
Witten-SanderPRL 1981
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Nodes: scientist (authors) Links: write paper together
(Newman, 2000, H. Jeong et al 2001)
SCIENCE COAUTHORSHIP
16Food Web
Nodes: trophic species Links: trophic interactions
R.J. Williams, N.D. Martinez Nature (2000)R. Sole (cond-mat/0011195)
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Most real world networks have the same internal
structure:
Scale-free networks
Why?
What does it mean?
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SCALE-FREE NETWORKS
(1) The number of nodes (N) is NOT fixed. Networks continuously expand
by the addition of new nodes
Examples: WWW : addition of new documents Citation : publication of new papers
(2) The attachment is NOT uniform.A node is linked with higher probability to a
node that already has a large number of links.
Examples : WWW : new documents link to well known sites (CNN, YAHOO, NewYork Times, etc) Citation : well cited papers are more likely to be cited again
19Scale-free model(1) GROWTH : At every timestep we add a new node with m edges (connected to the nodes already present in the system).
(2) PREFERENTIAL ATTACHMENT : The probability Π that a new node will be connected to node i depends on the connectivity ki of that node
A.-L.Barabási, R. Albert, Science 286, 509 (1999)
jj
ii k
kk
)(
P(k) ~k-3
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Achilles’ Heel of complex network
Internet Protein network
failureattack
R. Albert, H. Jeong, A.L. Barabasi, Nature 406 378 (2000)
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What Does the Web Really Look Like?
Graph Structure in the Web, Broder et al. Analysis of 2 Altavista crawls, each with
over 200M pages and 1.5 billion links
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Confirm Power Law Structure
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But Things Are More Complex Than One Might Think …
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Reading
Emergence of scaling in random networks , Albert-László Barabási, Réka Albert, Science 286 509-512 (1999)
Search in power-law networks, Lada A. Adamic, Rajan M. Lukose, Amit R. Puniyani and Bernardo A. Huberman, Phys. Rev. E, 64 46135 (2001)
Graph structure in the web, Andrei Broder, Ravi Kumar, Farzin Maghoul, Prabhakar Raghavan, Sridhar Rajagopalan, Raymie Stata, Andrew Tomkins, Janet Wiener, Comput. Netw. 33 309
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CMSC 23340-1 (Winter 2005):Course Goals
Primary– Gain deep understanding of fundamental issues
that effect design of large-scale networked systems
– Map primary contemporary research themes
– Gain experience in network research Secondary
– By studying a set of outstanding papers, build knowledge of how to present research
– Learn how to read papers & evaluate ideas
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How the Class Works
Research papers– Prior to each class, we all read and evaluate
two research papers
– During each class, we discuss those papers Project