Scale-Free Network Model: Introduction - Lecture Slides | CS 249, Study notes of Computer Science

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‘Scale-Free’ NetworkModel: Introduction

CS 249B: Science of NetworksWeek 08: Monday, 03/20/08Daniel BilarWellesley CollegeSpring 2008

Goals next couple of lectures
^ Quick review of models so far^ 

Power law review  Still have to account for PL node distribution (seemingly) ubiquitous in realworld phenomena BIG QUESTION:

How simple can models be?

For instance: Do we have to account for ‘

domain specificity

Do I need to know the purpose, the makeup, the users and the functionaldescription of the network to make testable predictions or can I simply‘throw the dice’ probabilistically and accurately model real-life phenomena?  We’ll see papers that do not take domain into account, some that do takedomain into account, and some that contrast both with real data and decide… Introduction to Scale-Free Network

Barabasi-Albert

(BA)

Model

^ Mechanism : “Growth and Preferential Attachment”

Review: Erdos-Renyi (1959)^
^ For >1 Crand

~^ 1/N

(if the

average degree isheld constant)  ≈ pN  Short

average path length

l

^ High

average clustering

C

^ Power law

distributions on degree

ki

p^ = 0.0 ;

k^ = 0 p^ = 0.09 ;

k^ = 1 p^ = 1.0 ;

k^ ≈ N-

Review: Watts-Strogatz (1998)^ W-S is not badbut stillP(k) ~ Poisson(K)^ ^ Short

average path length

l

^ High

average clustering

C

^ Power law

distributions on degree

ki

Recap: Power laws
^ A

Power Law

is a function f(x) where the

value y is

proportional to some power

of

the input x :

y = f(x) = x

-α

^ linear scale

^ log-logscale

CDF of degree distribution for sixnetworks

Example: Sexual network, power laws and Preferential Attachment

^ Swedish study: Lewin,B., editor. 1996. Sex inSweden. NationalInstitute of PublicHealth, Stockholm
^ Lewin claimed thedistribution of thenumber of lifetimepartners fit a power-lawcurve indicatingpreferentialattachment, when firstscrutinized
^ Analyzed in the contextof STI spread: Doherty(2005) ”Determinantsand Consequences ofSexual Networks asThey Affect the Spreadof Sexually TransmittedInfections”

W-S mentions spread briefly ..we’ll return to this question ofdiffusion and other questionsof attack and error tolerances

History of SF network analysis
^ “BA model” (1999) is reborn “Price Model” (1965)^ 

Historian-of-science Derek de la Solla Price studied citation networksand described what we would call today a SF network  Built on ideas of Herbert Simon (1955) but applied them to networkgrowth^
^

Simon wanted to explain Pareto distributions
His model could be termed “The Rich get richer”
Someone who is rich has more opportunities to develop additional wealth,while a poor person has trouble getting out of poverty

^ “Cumulative advantage” (Price) became “preferential attachment”(Barabasi)
^ Also known as the “Matthew effect”^ “

For to every one that hath shall be given ..”(Matthew 25:29)

Basic BA-model
^ Very simple algorithm to implement^ 

start with an initial set of m

fully connected nodes 0

^ e.g. m

^ now add new vertices one by one, each one with exactly

m^ edges

^ each new edge connects to an existing vertex in proportion to thenumber of edges that vertex already has

→^ preferential

attachment  easiest if you keep track of edge endpoints in one large array andselect an element from this array at random^
^ the probability of selecting any one vertex will be proportional to thenumber of times it appears in the array – which corresponds to itsdegree

Generating BA graphs^
^ To start, each vertex has anequal number of edges (2)

^ the probability of choosingany vertex is 1/3
We add a new vertex, and itwill have

m^ edges, here take m=2^ ^ draw 2 random elementsfrom the array – supposethey are 2 and 3
Now the probabilities ofselecting 1,2,3,or 4 are1/5, 3/10, 3/10, 1/5
Add a new vertex, draw avertex for it to connect fromthe array^ ^ etc.

Properties: Path length

l

^ Average path lengthsmaller than evenrandom graphs
^ Indicates “that theheterogeneous scale-freetopology is more efficientin bringing the nodesclose than thehomogeneous topology ofrandom graphs” ^ l ~ log(N)/loglog(N)

Properties: Clustering Coeff

C

^ Clustering coefficient

C^ of the

scale-free network is about fivetimes higher than that of therandom graph^ ^ Factor slowly increases with thenumber of nodes
C^ decreases with the networksize, following approximatelypower law C ~ N

Note:
^ ER: C ~ N

^ WS:

C = ¾

(independent of

N)

Analysis
^ We see BA model has two features^ 

Growth^
^ New nodes are being added  Preferential attachment^
^ New nodes attach preferentially to high degreenodes
^ Question^ 

Are both necessary to reproduce SF graphs?

Examining the BA model
^ Barabasi investigated necessity of features toproduce SF graphs
^ Varied original model^ 

Model A: Growth but no PA  Model B: PA but no growth

A^

Found

both

features necessary: Model A produces P(k) ~ exp(-βk)Model B leads after N

2 time steps to

fully connected graph