Dynamics of Complex Networks I: Networks II: Percolation

Dynamics of Complex Networks

I: NetworksII: Percolation

Panos Argyrakis

Department of Physics

University of Thessaloniki

Characteristics of networks:

• Structures that are formed by two distinct entities:NodesConnections, synapses, edges

N= 5 nodes, n=4 connections

• They can be simple structures or very complicated (belonging to the class of complex systems)

• How did it all start?

What is Complexity?

Main Entry: 1com·plexFunction: nounEtymology: Late Latin complexus totality, from Latin, embrace, from complectiDate: 16431 : a whole made up of complicated or interrelated parts

non-linear systems chaos fractals

A popular paradigm: Simple systems display complex behavior

3 Body Problem

Earth( ) Jupiter ( ) Sun ( )

Image of the Internet at the AS level

Scale-free P(k) ~ k-γ

Network Structure

Internet and WWW explosive growth

1990 – 1 web site 1996 – 105 web sites Now – 109 web sites

1970 – 10 hosts 1990 – 1.75*105 hosts Now – 1.2*109 hosts

routers

nodes

WWW

edges communicationlines

Internet

nodes

edges

web-sites

hyperlinks (URL)

Internet and WWW are characteristic complex networks

domain (AS)

Connectivity between nodes in the Internet

•How are the different nodes in the Internet connected?•No real top-down approach ever existed•Has been characterized by a large degree of randomness•What is the probability distribution function (PDF) of connectivities of all nodes?•Naïve approach: Normal distribution (Gaussian)•Experimental result?•Faloutsos, Faloutsos, and Faloutsos, 1997

Assuming random connections

0 10 20 30 40 50 60 70 80 90 1000.00

0.02

0.04

0.06

0.08

0.10

P(k)

k

Degree distribution, P(k):Probability that a node has k links (connections) with other nodes

Gaussian distribution

Probability distribution P(k) of the No. of connections(degree distribution)

0 200 400 600 800 1000 1200

0

5000

10000

15000

20000

P(k

)

k

P(k) ~ k-

INTERNET BACKBONE

(Faloutsos, Faloutsos and Faloutsos, 1997)

Nodes: computers, routers Links: physical lines

P(k)

k

k=number of connections of a nodeP(k)= the distribution of k

Assuming random connections

100 101 102 103 10410-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

P(k)

k

Degree distributionin a scale-free network

100 101 102 103 10410-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

P(k)

k

Slope = 2.0

P(k) ~ k -

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

Netwotk with a power law(scale-free network)

P(k) ~ k-

World Wide Web

800 million documents (S. Lawrence, 1999)

ROBOT: collects all URL’s found in a document and follows them recursively

Nodes: WWW documents Links: URL links

R. Albert, H. Jeong, A-L Barabasi, Nature, 401 130 (1999)

WWW It has a power-law

Diameter of WWW:

<d>=0.35 + 2.06 logN

For: Ν=8x108 <d>=18.59

[compare to a square of same size:

edge length=30000]

Communication networksThe Earth is developing an electronic nervous system, a network with diverse nodes and links are

-computers

-routers

-satellites

-phone lines

-TV cables

-EM waves

Communication networks: Many non-identical components with diverse connections between them.

Nodes: scientist (authors) Links: write paper together

(Newman, 2000, H. Jeong et al 2001)

SCIENCE COAUTHORSHIP

SCIENCE CITATION INDEX

( = 3)

Nodes: papers Links: citations

(S. Redner, 1998)

P(k) ~k-

2212

25

1736 PRL papers (1988)

Witten-SanderPRL 1981

ACTOR 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

Sex-webNodes: people (Females; Males)Links: sexual relationships

Liljeros et al. Nature 2001

4781 Swedes; 18-74; 59% response rate.

Yeast protein networkNodes: proteins

Links: physical interactions (binding)

P. Uetz, et al. Nature 403, 623-7 (2000).

What does it mean?Poisson distribution

Exponential Network

Power-law distribution

Scale-free Network

Network Backbone at University of Thessaloniki

The model of Erdös-Rényi (1960)

- Democratic

- Random

Pál ErdösPál Erdös (1913-1996)

Connections with

probability pp=1/6 N=10

k ~ 1.5 Poisson distribution

Problem:

• construct an Erdos-Renyi Network• use N=1000• use p=0.1666• draw a random number (rn) from a uniform distribution• if rn<0.1666 link exists, otherwise it does not• find the complete distribution of links for all N=1000

nodes• Construct the P(k) vs. k diagram

The Erdos-Renyi model

p=6x10-4

p=10-3

Problem: Scale free networks

• need a random number generator that produces random numbers with a power-law distribution

• P(k)~ k-

• not so simple distribution• check book “Numerical Recipes” by Press et al• construction of network is more cumbersome• can do node-by-node….or• can do link-by-link• many more methods, e.g. thermalization and annealing

of links

Fill node-by-node

3

1

2 1

5 3

212

2

11

1

1

3

21

2

11

k

k-connectivity node, completed with k linksk

k-connectivity node, with missing links

Link-to-link

3

1

2 1

5 3

212

2

11

3

1

2 1

5 3

212

2

11

k

k-connectivity node, completed with k linksk

k-connectivity node, with missing links

Regular network

0 1 2 3 4 5 6 7 80.0

0.2

0.4

0.6

0.8

1.0

P(k)

k

Small world network

The Watts-Strogatz model

C(p) : clustering coeff.

L(p) : average path length(Watts and Strogatz, Nature 393, 440 (1998))

Small world network

Small-world network

Most real networks have similar internal structure:

Scale-free networks

Why?

What does it mean?

Scale-free networks

(1) The number of nodes is not pre-determinedΤhe networks continuously

expand with the addition of new nodesExample: WWW : addition of new pages

Citation : publication of new articles(2) The additions are not

uniformA node that already has a large number of connections is connected with larger probability than another node.Example: WWW : new topics usually go to well-known sites (CNN, YAHOO, NewYork Times, etc) Citation : papers that have a large number of references are more probable to be refered again

Origins SF

Scale-free networks

(1) Growth: At every moment we add a new node with m connections (which is added to the already existing nodes).

(2) Preferential Attachment: The probability Π that a new node will be connected to node i depends on the number ki , the number of connections of this node

A.-L.Barabási, R. Albert, Science 286, 509 (1999)

jj

ii k

kk

)(

P(k) ~k-3

Network categories:

• (1) Random network (Erdos-Renyi)• (2) Network on a regular lattice• (2) Small world network (Strogatz-Watts)• (3) Network with a power-law

Society

Nodes: individuals

Links: social relationship (family/work/friendship/etc.)

S. Milgram (1967)

John Guare

How many (n) connections are needed so that an individual is connected with any other person in the world?N=6 billion people

Result: n~6

Conclusion: We live in a small worldSix Degrees of Separation!!

New books:

Facebook:

•900,000,000 registered users•50,000,000 active users•5,000,000 generate 95% of the traffic

Questions needing answers:•How many people communicate ?•How many connections does one have?•How often does he communicate?•How long does it last?

Real-world phenomena related to communication patterns

• Crowd behavior: strategies to evacuate people and stop panic.

• Search strategies: efficient networks for searching objects and people.

• Traffic flow: optimization of collective flow.• Dynamics of collaboration: human relationship

networks such as collaboration, opinion propagation and email networks.

• Spread of epidemics: efficient immunization strategies.• Patterns in economics and finance: dynamic patterns in

other disciplines, such as Economics and Finance, and Environmental networks.

Where is George?

FP5 data (1998-2002)• 84267 partners in 16558 contracts• 27219 unique partners• 147 countries

FP6 data (2002-2006)• 69237 partners in 8861 contracts

• 19984 unique partners

• 154 countries

24982 partners in the largest cluster (27219 total)

FP5

FP6 Minimum Spanning Tree (countries)

15 EU members

25 EU members

ResearchInnovation

Stock price changes

Financial Time Series

9/25/2001 3/25/2002 9/25/2002 3/25/2003 9/25/2003 3/25/2004 9/25/2004 3/25/20051.61.82.02.22.42.62.83.03.23.43.6

Price

in E

uro

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

Logarith

mic

Retu

rns

Theoretical Economics is dominated by pure mathematics:

• Lemma/theorem style is required

• Little effort to compare theoretical predictions to “experimental data” - say, price record from real stock markets

• Bulk of papers are inaccessible and of no interest to “experimentalists” - practitioners of the field

George Soros on theoretical economics

“Existing theories about the behavior of stock prices are remarkably inadequate. They are of so little value to the practitioner that I am not even fully familiar with them. The fact that I could get by without them speaks for itself.”

G. Soros, “Alchemy of Finance” 1994

Quick experiment: free data fromwww.nyse.com/marketinfo/nysestatistics.html

In a gaussian world the probability of the October 1987 crash would be 10-135!

10%

-10%

Filtering of the correlation matrix•Minimum Spanning Tree of the 100 most capitalized US stocks in 1998 (R.N.Mantegna).

•From n(n-1)/2 connections only n-1 survive.

Spreading phenomena

• Percolation• SIR (Susceptible-Infected-Removed)• SIS (Susceptible-Infected-Susceptible)• SIRS (Susceptible-Infected-Refractory-

Susceptible)• Applications: Forest fires, epidemics, rumor

spreading, virus spreading, etc.

The SIR model

• SIR (Susceptible, Ιnfected, Removed), • q=probability of infection • Initially all nodes are susceptible (S)• Then, a random node is infected (I)• The virus is spread in the network, all I nodes become R• This process continues until the virus either

– has been spread in the entire network, or – has been totally eliminated

• M=infected mass• Duration

SIR model

S

S

S S

S S

SSS

S

SS

I

I

I I

I I

III

I

II

R

R

R R

R R

RRR

R

RR

RIS Susceptible Infected Recovered (or Removed)

Immunization

• Complete immunization of populations is not feasible (pc can not be made significantly lower than 1)

• Try to find an efficient method of immunization• Should we seek for alternative goals?

e.g. try to isolate via immunization the largest possible portion of the population instead– New strategies may be needed

Acquaintance immunization

• Suggested by Cohen, Havlin, ben-Avraham (PRL, 2003)• Strategy: Randomly choose a node and immunize a random

neighbor of this node.

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

M

Infection probability q

Comparison of the 3 network types

Comparison of different network types

Lattice

Small-world network

Scale-free network, =2.0



CASE 2

Robustness of networksComplex systems maintain their basic functions even under errors and failures (cell mutations; Internet router breakdowns)

node failure

fc

0 1Fraction of removed nodes, f

1

S

Distribution of infected mass

Duration of epidemics

Attack tolerance (introduction)

• The stability of the networks under failure or attack is very important.

• In general, the integrity is destroyed after a critical percentage pc of the nodes has been removed (no giant cluster).

• Scale-free networks are extremely robust under random failure (pc→1), but very vulnerable under targeted attacks (pc→0).

• We studied different attack strategies by removing nodes based on their connectivity k, according to a power law ka.

• The parameter a determines the degree of information one has about the network structure.

• The existence of a spanning cluster is based on the criterion

22

k

k

DISEASE SPREADING:The dynamical consequences of a scale-free network are drastically different than those on lattice and small-world

networks.

There is no threshold in the infected mass as a function of the infection transmission probability.

The starting point of the disease is important, since it determines whether the disease will spread or die out.

The spreading is rapid and manifests the small diameter of the network.

ATTACK TOLERANCE:The tolerance of a network depends on its connectivity.

The random node removal is the marginal case where the critical threshold moves from pc=0 to pc=1.

A small bias in the probability of selecting nodes either retains or destroys the compactness of the cluster.

SIR model:• The network with a power-law structure is a more

realistic representation than all the other network categories

• It does NOT show critical behavior

Summary- Conclusions

• Networks is a new area in sciences which started from Physics, but pertains to ALL sciences today

• It shows rich dynamical behavior• It has many-many applications in everyday life• Will influence directly the way we live and act

Dynamics of Complex Networks I: Networks II: Percolation

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