Transcript
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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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