1

Complex systemsMade of

many non-identical elements connected by diverse interactions.

NETWORK

New York Times

Slides: thanks to A-L Barabasi

2

(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

4

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))

7

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

8

< 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 > = ??

1

2

3

4

5

6

7

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

9

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

11

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

25

1736 PRL papers (1988)

Witten-SanderPRL 1981

15

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?

18

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

20

Achilles’ Heel of complex network

Internet Protein network

failureattack

R. Albert, H. Jeong, A.L. Barabasi, Nature 406 378 (2000)

21

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

22

Confirm Power Law Structure

23

But Things Are More Complex Than One Might Think …

24

25

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