JV Small World

7/30/2019 JV Small World

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Small World Networks

Jean Vaucher

Ift6802 - Avril 2005


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Contents

Pertinence of topic

Characterization of networks

Regular, Random or Natural

Properties of networks

Diameter, clustering coefficient

Watts network models (alpha & beta)

Power Lawnetworks Clustered networks with short paths

Can these short paths be found ?


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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 Knigsberg 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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Milgrams Experiment

Found short chains of acquaintances linking pairs ofpeople in USA who didnt know each other; Source person in Nebraska

Target person in Massachusetts.

Sends message by forwarding to people they knewpersonally (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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Social networks

Not random

But Clustered

Most of our friends come from ourgeographical or professionalneighbourhood.

Our friends tend to have the same friends

BUT In spite of having clustered social

networks, there seem to exist short pathsbetween 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 connectedcomponent? (as a fraction of all

nodes)

Depends on the number of links pernode

(Erds, Rnyi 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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Connectivity of a random graph

1

1

Average number oflinks per node

Fractio

nofalln

odes

inlargestcompone

nt

0

Disconnectedph

ase

Con

ectedphase


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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 ihas ki immediate neighbors, Ci, is the

fraction of the total possible ki*(ki-1) / 2connections that are realized between i'sneighbors. 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 of

connections/node

Diameter

Number of connections

needed to fully connect

few, clustered

RandomRegular

fewer, spread

large moderate

many fewer (


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Natural networks

Between regular grids and totallyrandom graphs

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

Watts

Alpha model ( not intuitive)

Beta rewiringmodel


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Small-World Networks

Random rewiring of regular graph (by Watts and Strogatz)

With probabilityp (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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Small-world

networks

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 ofnetwork

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 toother 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 Asimovs I, Robot

novels R. Daneel Olivaw

Elijah Baley

Caves of Steel (Earth)

Solaria


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Two extreme types of social

networks

Cavemans world

people live in isolated communities

probability meeting a random person is high if

you have mutual friends and very low if youdont

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 incommon, they know each other

(Caveman world)

= : A & B dont know each other, no

matter how many common friends theyhave (Solarian world)


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Number of mutual friendsshared by A and B

Like

lihoodthatAmeetsB

Caveman world

Solaria world

=0

=

=1


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Fragmentednetworks

Small-worldnet-

works

Alpha network

Pa

thlengthL

critical

C

lusteringcoefficientC

L drops because we only count

nodes that are connected


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How about realnetworks

All nodes in alpha and beta networks are equal inthe sense that the number of connections eachnodes 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

Barabsi and Albert (1999)

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


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Random Networks

Each node is connected to

a few other nodes.

The number of connections

per 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 Networks

by Albert-Laszlo Barabasi; 2002


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Scale-Free Networks

Each node is connected to

at least one other; most are

connected to only one, while

a few are connected to many.

The number of connections

per node forms a hyperbolic

distribution, with no meaningfulaverage number of connections

per node.


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Random Scale-Free

Scale-free networks are associated with

networks that grow by natural processes

in 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 smallnodes


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Performance

Real power lawnetworks also have

short distances

Existence of central backbone ofhighly 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 bclose 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 toxand is

expressed as

parametershapeis

incomeminimumis

,0,0,/

k

m

mxkmxmxXPk


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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 seriouslyuntil he was 42.

In 1893 he succeeded his mentor, Walras,

as chair of economics at the University ofLausanne.

QuickTime and a

TIFF (Uncompressed) decompressorare needed to s ee this picture.


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Paretos contributions

Pareto optimality. A Pareto-optimal allocation of resources

is achieved when it is not possible tomake anyone better off without makingsomeone 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-freebecauseit 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 existingnode

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 ofcentrality

centers are in networks by design (central control, dictatorship)

by non-design (unnoticed critical resources,informal groups)

or they emerge as a consequence ofcertain 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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Kleinbergs 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 uhas link to the peer vwith probability proportional towhere d(u,v) is the distance between uand 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 slowsdown till finally reaches target itself).

If r > dim we tend to choose more close neighbors (algorithm finds quicklytarget in its 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

Typicallen

gthof

directedse

arch

2

shortpathscannotbe found

no shortpaths


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Influence of r (or)

Given node u if we can partition the remaining peers into setsA1,A2, A3, , AlogN, where Ai, consists of all nodes whose distance

from u is between 2iand 2i+1, i=0..logN-1.

Then given r = dim each long range contact ofu is nearly equally

likely to belong to any of the setsAi

A4

A3

A2A1


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The New Yorker View

When gamma is atits critical value two,the resulting

network has thepeculiar propertythat nodes possessthe same number ofties at all lengthscales (in 2D world)

DHTs (distributed hash tables)


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DHTs (distributed hash tables)

and Kleinberg model

P-Grids

model

Kleinbergs

model

Balanced n-ary search


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More hierarchy

Kleinbergs 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

W tt t l (2002) h i i l


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Watts et al (2002) search in social

networks

Searchablenetworks

H1 10

0

6

Kleinbergcondition

= homophily, thetendency 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 varieswith 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

0

Fraction of random shortcuts

1

Threshold

infectiousness


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Failures in networks

Fault propagation or viruses

Scale-free networks are far more resistantto random failures than ordinary random

networks because of most nodes are leaves

But failure ofhubs can be catastrophicvulnerable or targets of deliberate attacks

which may make scale-free networks morevulnerable 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, Amlie Poulin

Innovation, scientific revolutions

Solar-centric universe

Plate tectonics Is it like the spread of disease ?


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Effect of peers & pundits

Peoples 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 Milgrams 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 infectedneighbors

1

Probabilityofinfection

0

Fraction of neighborschoosing A over B

1

Probabilityofchoosing

optionA

0 Critical

Threshold

Standard disease spreading

model

Social decision making


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Global Cascades

Idea catches on.


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Fin

JV Small World

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