SNA Lecture2CGrowthModels

8/13/2019 SNA Lecture2CGrowthModels

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SNA 2C: Growth &Preferential Attachment

Models

Lada Adamic


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Online Question & Answer Forums


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Uneven participation

100

101

102

103

10-4

10-3

10-2

10-1

100

degree (k)

cumulativeprob

ability

!= 1.87 fit, R2= 0.9730

number of peopleone received

replies from

number ofpeople one

replied to

!answerpeoplemayreply tothousands ofothers

!questionpeoplearealso unevenin thenumber of

repliers totheir posts,but to alesser extent


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Real-world degree distributions

!Sexual networks

!Great variationin contact

numbers


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Power-law distribution

! linear scale ! log-logscale

! high skew (asymmetry)! straight line on a log-log plot

1 2 5 10 20 500.

00005

0.00500

0.

50000

x

P(x)

0 20 40 60 80 100

0.

0

0.

2

0.

4

0.

6

x

P(x)


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Poisson distribution

0 20 40 60 80 100

0.

00

0.

04

0

.08

0.

12

x

P(x)

1 2 5 10 20 501e-64

1e-36

1e-08

x

P(x)

! linear scale ! log-logscale

! little skew (asymmetry)! curved on a log-log plot


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Power law distribution

!Straight line on a log-log plot

!Exponentiate both sides to get thatp(k), theprobability of observing an node of degree

k is given by

p(k)=Ck!!

ln(p(k))= c!!ln(k)

normalizationconstant (probabilities over

all kmust sum to 1)

power law exponent !


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Quiz Q:

! As the exponent !increases, thedownward slope of the line on a log-log

plot

!stays the same!becomes milder!becomes steeper


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2 ingredients in generating power-lawnetworks

!nodes appear over time (growth)


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2 ingredients in generating power-lawnetworks

!nodes prefer to attach to nodes with manyconnections (preferential attachment, cumulative

advantage)


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Ingredient # 1: growth over time

!nodes appear one by one, each selecting mother nodes at random to connect to

m = 2


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random network growth

! one node is born at each time tick! at time t there aretnodes! change in degree kiof node i(born at time i, with 0 < i < t)

t

m

dt

tdki

=

)(

there are mnew edgesbeing added per unit time

(with 1 new node)

the medges are being

distributed amongtnodes


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a node in a randomly grown network

!how many new edges does a node accumulatesince it's birth at time iuntil time t?

!integrate from ito t

t

m

dt

tdki=

)(

)log()(i

tmmtki +=

to get

born with medges


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age and degree

on average

if

)()( tktk ji >

ji

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