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Small World Networks

Jean VaucherIft6802 - Avril 2005

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Contents

Pertinence of topic Characterization of networks

Regular, Random or Natural Properties of networks

Diameter, clustering coefficient Watt’s network models (alpha & beta) Power Law networks

Clustered networks with short paths

Can these short paths be found ?

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Duncan J. Watts

Six degrees - the science of a connected age, 2003, W.W. Norton.

I read somewhere that everybody on this planet is separated by only six other people. Six degrees of separation between us and everybody on this planet. Six degrees of separation by John Guare

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Networks

Networks are everywhere Internet Neurons is brains Social networks Transportation

Networks have been studied long time Euler (1736): Bridges of Königsberg theory of graphs,

which is now a major (and difficult! – or almost obvious) branch in mathematics

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So what is new?

Global interconnections Internet Power grids Mass travel, mass culture

FAILURES Computer Viruses Power Blackouts Epidemics

Modeling & analysis

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Milgram’s Experiment

Found short chains of acquaintances linking pairs of people in USA who didn’t know each other;

Source person in Nebraska Target person in Massachusetts. Sends message by forwarding to people they knew personally

(who should be closer to target) Average length of the chains that were completed was

between 5 and 6 steps “Six degrees of separation” principle

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Correct question

WHY are there short chains of acquaintances linking together arbitrary pairs of strangers???

Or

Why is this surprising

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Random networks

In a random network, if everybody has 100 friends distributed randomly in the world population, this isn’t strange

In 6 hops, you can reach 1006 people - a million million > 6,000 million (world pop.)

BUT: our social networks tend to be clustered.

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Social networks

Not random But Clustered Most of our friends come from our

geographical or professional neighbourhood.

Our friends tend to have the same friendsBUT In spite of having clustered social networks,

there seem to exist short paths between any random nodes.

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Social network research

Devise various classes of networks

Study their properties

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Network parameters

Network type Regular Random Natural

Size: # of nodes Number of connexions:

average & distribution Selection of neighbours

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STAR TREE

GRID

BUS RING

REGULAR Network Topologies

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Connectivity in Random graphs

Nodes connected by links in a purely random fashion

How large is the largest connected component? (as a fraction of all nodes) Depends on the number of links per

node

(Erdös, Rényi 1959)

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Connecting Nodes

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Random Network (1)

• add random paths

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• paths

• trees

Random Network (2)

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• paths

• trees

• networks

Random Network (3)

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• paths

• trees

• networks …..

Random Network (3+)

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• paths

• trees

• networks

• fully connected

Network Connectivity (4)

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Connectivity of a random graph

1

1

Average number oflinks per node

Fract

ion

of

all

nod

es

in larg

est

com

pon

en

t

0

Dis

con

nect

ed

ph

ase

Con

ect

ed

ph

ase

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Regular or Ordered Network

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Network measures

Connectivity is not main measure. Characteristic Path Length (L) :

the average length of the shortest path connecting each pair of agents (nodes).

Clustering Coefficient (C) is a measure of local interconnection if agent i has ki immediate neighbors, Ci, is the

fraction of the total possible ki*(ki-1) / 2 connections that are realized between i's neighbors. C, is just the average of the Ci's.

Diameter: maximum value of path length

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Regular vs Random Networks

Average number ofconnections/node

Diameter

Number of connectionsneeded to fully connect

few, clustered

Random Regular

fewer, spread

large moderate

many fewer (<2/3)

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Natural networks

Between regular grids and totally random graphs

Need for parametrized models: Regular -> natural -> random

Watts Alpha model ( not intuitive) Beta rewiring model

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Clustering Clustering measures the fraction of neighbors of a node that are

connected themselves Regular Graphs have a high clustering coefficient

but also a high diameter Random Graphs have a low clustering coefficient

but a low diameter Both models do match the properties expected from real networks!

Random Graph (k=4)

Short path length L~logkN

Almost no clustering C~k/n

Regular Graph (k=4)

Long paths L ~ n/(2k)

Highly clustered C~3/4

Base metwork is circle

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Small-World Networks

Random rewiring of regular graph (by Watts and Strogatz) With probability p (or ) rewire each link in a regular graph to a

randomly selected node Resulting graph has properties, both of regular and random

graphs High clustering and short path length

FreeNet has been shown to result in small world graphs

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Example: 4096 node ring

Regular graph:

n nodes, k nearest neighbors

path length ~ n/2k

4096/16 = 256

Random graph:

path length ~ log (n)/log(k)

~ 4

Rewired graph (1% of nodes):

path length ~ random graph

clustering ~ regular graph

Small World Graph

K=4

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Small-worldnetworks

Beta network

Rewiring probability

0 10

1

L

C

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More exactly …. (p = )

Small world behaviour

C

L

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Effect of short-cuts

Huge effect of just a few short-cuts. First 5 rewirings reduces the path

length by half, regardless of size of network

Further 50% gain requires 50 more short-cuts

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The strength of weak ties

Granovetter (1973): effective social coordination does not arise from densely interlocking strong ties, but derives from the occasional weak ties this is because valuable information

comes from these relations (it is valuable if/because it is not available to other individuals in your immediate network)

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Two ways of constructing

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Alpha model

Watts’ first Model (1999) Inspired by Asimov’s “I, Robot”

novels R. Daneel Olivaw Elijah Baley

Caves of Steel (Earth) Solaria

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Two extreme types of social networks

Caveman’s world people live in isolated communities probability meeting a random person is high if

you have mutual friends and very low if you don’t

Solaria people live isolated from each other but with

supreme communication capabilities your social history is irrelevant to your future

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Alpha network

Alpha () distance parameter

=0 : if A and B have a friend in common, they know each other (Caveman world)

=∞ : A & B don’t know each other, no matter how many common friends they have (Solarian world)

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Number of mutual friendsshared by A and B

Like

lihood

th

at

A m

eets

B

Caveman world

Solaria world

=0

=

=1

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Fragmentednetworks

Small-worldnet- works

Alpha network

Path

len

gth

L

critical

Clu

steri

ng

coeffi

cien

t C

L drops because we only count nodes that are connected

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How about real networks

All nodes in alpha and beta networks are equal in the sense that the number of connections each nodes has is not very far from the average

Watts and Strogatz had used normal distribution

Real world is not like that Sizes of cities, Wealth of individuals in USA, Hubs in

transportation systems Barabási and Albert (1999)

Scale-free networks, whose connectivity is defined by a power-law distribution

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Random Networks

Each node is connected toa few other nodes.

The number of connectionsper node forms a Poisson distribution, with a small average of number of connections per node.

This & three following graphics from:Linked: The New Science of Networksby Albert-Laszlo Barabasi; 2002

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Scale-Free Networks

Each node is connected toat least one other; most areconnected to only one, whilea few are connected to many.

The number of connectionsper node forms a hyperbolic distribution, with no meaningfulaverage number of connectionsper node.

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Random Scale-Free

Scale-free networks are associated with networks that grow by “natural” processesin which the number of nodes increases with time not just the number of connections.

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Power law phenomena

Average & median are far apart Whales and minnows

Average from a few large nodes Median governed by majority of small

nodes

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Performance

Real power law networks also have short distances

Existence of central backbone of highly connected HUBS nodes

Similar phenomena noted in linguistics and economics Zipf Pareto

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Zipf's law - linguistics

Zipf, a Harvard linguistics professor, sought to determine the frequency of use of the 3rd or 8th or 100th most common words in English text.

Zipf's law states that the frequency y is inversely proportional to it's rank r:

Y ~ r -b, with b close to unity.

Zipf Presentations

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The Pareto Income Distribution

The Pareto distribution gives the probability that a person's income is greater than or equal to x and is expressed as

[ ] ( )

parameter shape is

income minimum is

,0,0 ,/

k

m

mxkmxmxXP k ≥>>=≥

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Vilfredo Pareto, 1848-1923

Italian economist Born in Paris Polytechnic Institute in Turin in 1869, Worked for the railroads. Pareto did not study economics seriously

until he was 42. In 1893 he succeeded his mentor, Walras,

as chair of economics at the University of Lausanne.

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

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Pareto’s contributions

Pareto optimality. A Pareto-optimal allocation of resources

is achieved when it is not possible to make anyone better off without making someone else worse off.

Pareto's law of income distribution. In 1906, Italian economist Vilfredo Pareto

created a mathematical formula to describe the unequal distribution of wealth in his country, observing that 20% of the people owned 80% of the wealth.

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0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

10000 60000 110000 160000 210000

x

p(X>=x)

Pareto distribution, m=10000, k=1

0,01

0,1

1

10000 100000 1000000

x

p(X>=x)

log-log plot

Pareto distribution issaid to be scale-free becauseit lacks a characteristic lengthscale

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Building Power-law networks

It is easy to create PL networks

Build network node by node Connect new node to an existing

node Probability of connection proportional

to its number of links The rich get richer The poor get poorer

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Structure and dynamics

The case of centrality centers are in networks

by design (central control, dictatorship) by non-design (unnoticed critical resources,

informal groups) or they emerge as a consequence of

certain events ”he was at the right place at a right time” clapping in unison

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Further applications

Search in networks Short paths are not enough

Epidemics: medical & software Danger of short-cuts Paths + infectiousness

Infection by ideas Fads & Economic Bubbles Individual rationality Peer pressure

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Getting practical: search in networks

A node may be linked to another node via a short path but what does it matter if you cannot find the path?

In alpha and beta networks there is no notion of distance, therefore directed searches cannot recognize shortcuts

Kleinberg’s (gamma) networks (2000)

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Kleinberg’s Small-World Model

Embed the graph into an r-dimensional grid (2D in examples) constant number p of short range links (neighborhood) q long range links: choose long-range links such that the probability to have

a long range contact is proportional to 1/dr

Importance of r ! Decentralized (greedy) routing performs best iff. r = dimension of space

(here=2)

r = 2

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Influence of “r” (1)

• Each peer u has link to the peer v with probability proportional to where d(u,v) is the distance between u and v.

• Optimal value: r = dim = dimension of the space• If r < dim we tend to choose more far away neighbors (decentralized

algorithm can quickly approach the neighborhood of target, but then slows down till finally reaches target itself).

• If r > dim we tend to choose more close neighbors (algorithm finds quickly target in it’s neighborhood, but reaches it slowly if it is far away).

• When r = 0 – long range contacts are chosen uniformly. Random graph theory proves that there exist short paths between every pair of vertices, BUT there is no decentralized algorithm capable finding these paths

rvud ),(1

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r (log scale)

p(r)(log scale)

increasing

=0

Typic

al le

ngth

of

dir

ect

ed s

earc

h

2

short paths cannot be found

no short paths

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Influence of “r” (or ) Given node u if we can partition the remaining peers into sets A1,

A2, A3, … , AlogN , where Ai, consists of all nodes whose distance from u is between 2i and 2i+1, i=0..logN-1.

Then given r = dim each long range contact of u is nearly equally likely to belong to any of the sets Ai

A4

A3

A2

A1

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The New Yorker View

When gamma is at its critical value two, the resulting network has the peculiar property that nodes possess the same number of ties at all length scales (in 2D world)

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DHTs (distributed hash tables)and Kleinberg model

P-Grid’s model

Kleinberg’s model

Balanced n-ary search

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More hierarchy

Kleinberg’s model has only one distance measure, geographical (2D)

In human society the social distance is multidimensional

if A is close to B and C is close to B but in different dimension then A and C can be very far from each other ”violation of the triangle inequality” but multidimensionality may enable messages

to be transmitted in networks very efficiently

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Watts et al (2002) search in social networks

Searchablenetworks

H1 10

0

6

Kleinbergcondition

= homophily, the tendency of like toassociate with like

H=number of dimensionsalong which individualsmeasure similarity

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Small Worlds

& Epidemic diseases

Nodes are living entities Link is contact 3 States

Uninfected Infected Recovered (or dead)

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Epidemic diseases

Level of infectiousness needed to start an epidemic varies with presence of shortcuts

In regular grid, disease may die out due to lack of victims In small world, pandemics are facilitated

SRAS Mad cow disease in England

0Fraction of random shortcuts1Threshold infectiousness

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Failures in networks

Fault propagation or viruses Scale-free networks are far more resistant

to random failures than ordinary random networks because of most nodes are leaves

But failure of hubs can be catastrophic vulnerable or targets of deliberate attacks which may make scale-free networks more

vulnerable to deliberate attacks Cascades of failures

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Back to Social Networks

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Spread of ideas

Messages in social networks Fads & fashions

Body piercing, baseball caps Harry Potter, Amélie Poulin

Innovation, scientific revolutions Solar-centric universe Plate tectonics

Is it like the spread of disease ?

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Effect of peers & pundits

People’s decisions are affected by what others do and think Presure to conform ?

Efficient strategy when insufficient knowledge or expertise Ex: picking a restaurant

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Economic models

Selfish agents Individual rationality Markets

Equilibrium ??? Many agents are trend followers Speculation crashes

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Social Experiments

Factors which affect decisions Milgram Asch

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Stanley Milgram (1933-1984)

Controversial social psychologist Yale & Harvard Small world experiment, 1967

6 degrees of separation Obedience to authority - 1963

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Validity of Milgram’s experiment

Global connectivity ? US: Omaha Boston stockbroker Only 96 valid subjects (out of 300)

100 from Boston 100 big investors 96 picked at random in Nebraska

Success? 18 out of 96 Other experiments:

3 out of 60 Worse….

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Conformity

Other presentation

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Threshold models of decisions

Number of infected neighbors

1

Pro

bab

ility

of

infe

ctio

n

0

Fraction of neighborschoosing A over B

1

Pro

bab

ility

of

choosi

ng

op

tion

A

0 CriticalThreshold

Standard disease spreadingmodel

Social decision making

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Global Cascades

Idea catches on….

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Fin