stanford universitysnap.stanford.edu/na09/07-evolution-annot.pdf · 2009. 10. 13. · stanford...
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CS 322: (Social and Information) Network AnalysisJure LeskovecStanford University
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3 pages per group 3 pages per group Should include both Reaction part Reaction part Proposed work part
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Random networkRandom network Scale‐free (power‐law) network(Erdos‐Renyi random graph)
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Preferential attachment Preferential attachment [Price 1965, Albert‐Barabasi 1999]: Nodes arrive in orderNodes arrive in order A new node j creates m out‐links Prob. of linking to a previous node i is g pproportional to its degree di
idijP )(
kk
i
dijP )(
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[Mitzenmacher, ‘03]
Pages are created in order Pages are created in order 1,2,…,n
When node j is created it When node j is created it produces a single link to some earlier node i chosen:earlier node i chosen:
1) With prob. p, j links to i chosen f l duniformly at random
2) With prob. 1-p, j links to i with ) p p, jprob. proportional to di (degree of i)
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How does degree d of node i grow over time? How does degree di of node i grow over time?
dd ii 1dtdq
tp
td ii
1d
d
q=1-p
What is the degree di(t) of node i at time t?
q 1 p
1)(
q
i itptd
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iq6
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Fraction of nodes with degree >d at time t Fraction of nodes with degree >d at time t
dqtidtptd
1
11)(
i dpqtid
iqptd 11)(
Fraction of nodes with degree d?1
kq q
1111111
11
10/13/2009 Jure Leskovec, Stanford CS322: Network Analysis
pqpp
1
7
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We observe how the connectivity (length of the paths) of the network changes as the vertices get removed g[Albert et al. 00; Palmer et al. 01]
Vertices can be removed: Uniformly at random In order of decreasing degreeIn order of decreasing degree
It is important for epidemiology Removal of vertices corresponds to p
vaccination
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Real‐world networks are resilient to random attacks One has to remove all web‐pages of degree > 5 to disconnect the webOne has to remove all web pages of degree > 5 to disconnect the web But this is a very small percentage of web pages
Random network has better resilience to targeted attacks Random networkInternet (Autonomous systems)
h
Preferentialremoval
Random network( y )
path
leng
thM
ean Random
removal
Fraction of removed nodes Fraction of removed nodesJure Leskovec, Stanford CS322: Network Analysis10/13/2009 9
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G : Gnp: not realistic B t i ll di t But gives small diameters
Small world:
Degree distribution of the
Small world: small diameter + local structure
l dgSmall world model But no power‐law degrees
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Preferential Preferential attachment: Power law degree Power law degree distributions But no local But no local clustering
Can we get all ofCan we get all of them?
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Preferential attachment is a network growth Preferential attachment is a network growthmodel
What governs the network growth and What governs the network growth and evolution? P1) Node arrival process: nodes enter the network P1) Node arrival process: nodes enter the network P2) Edge initiation process: each node decides
when to initiate an edgewhen to initiate an edge P3) Edge destination process: determines
destination after a node decides to initiatedestination after a node decides to initiate
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[Leskovec et al. KDD 08]
4 online social networks with 4 online social networks with exact edge arrival sequence For every edge (u,v) we know exactFor every edge (u,v) we know exact time of the appearance tuv
Directly observe mechanisms leadingand so on for millions… Directly observe mechanisms leading
to global network properties
(F)(D)(A)
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(A)(L)
13
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(F) (D)(F) (D)
Flickr: Exponential
Delicious: Linear
(A) (L)
Answers: S b li
LinkedIn: Q d ti
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Sub‐linear Quadratic
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How long do nodes live? How long do nodes live? How often nodes “wake up” to create edges?
How long do nodes live? Node life time is the time bet een the 1st and the Node life‐time is the time between the 1st and the last edge of a node
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Lifetime aLi k dI Lifetime a: time between node’s first d l t d
LinkedInand last edge
Node lifetime is exponential: p(a) = λ exp(‐λa) 10/13/2009 Jure Leskovec, Stanford CS322: Network Analysis 16
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How long do nodes live? How long do nodes live? How often nodes “wake up” to create edges? Edge gap δ(d): time between dth and d+1st edge Edge gap δ(d): time between dth and d+1st edge of a node: Let ti(d) be the creation time of d‐th edge of node iLet ti(d) be the creation time of d th edge of node i δi(d) = ti(d+1) ‐ ti(d) Then δ(d) is a distribution (histogram) of δi(d) over g iall nodes i
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Edge gap δ(d):Edge gap δ(d): inter‐arrival time between
LinkedIndth and d+1stedge
For every dwe get a different plotp
)()(),);(( dg eddp
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As the degree of the node degree increases As the degree of the node degree increases, how α and β change?
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α is const, β linear in d – gaps get smaller with dα is const, β linear in d gaps get smaller with d
ty
Degreed=1
d=3d=2
Proba
bilit
d 1P
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Edge gap20
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What do we know so far? What do we know so far?
Process Our findingProcess Our finding
P1) Node arrival • Node arrival function is given
P2) Edge initiation• Node lifetime is exponential• Edge gaps get smaller as the g g g gdegree increases
Jure Leskovec, Stanford CS322: Network Analysis
P3) Edge destination
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Source node wakes up and creates an edge Source node wakes up and creates an edge How does it find a target node? What is the degree of the target? What is the degree of the target? Do preferential attachment really hold?
How many hops away if the target? How many hops away if the target? Are edges really local?
Wh t i th d f th t t?What is the degree of the target?
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kk)( H f d d ?kkpe )(Network τ
How far do edges go?
Network τGnm 0PAPA 1F 1DD 1A 0.9L 0.6
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Fraction of triad closing edges
Network %Δ
F 66%
g g
F 66%
D 28%
A %A 23%
L 50%
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New triad closing edge (u w) appears next New triad‐closing edge (u,w) appears next We model this as:1 Ch ’ i hb
wv’
1. Choose u’s neighbor v2. Choose v’s neighbor w
( )
uv
3. Connect (u,w) Compute edge prob. under Random‐R d ( )Random: p(u,w) =
S f h ( )Score of a graph = p(u,w)
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Impro ement o er the baseline Improvement over the baseline:Strategy to select v (1st node)
ode)
t w (2
ndno
Select
Strategies to pick a neighbor:Strategies to pick a neighbor: random: uniformly at random deg: proportional to its degree com: prop. to the number of common friends u
wv
Jure Leskovec, Stanford CS322: Network Analysis
last: prop. to time since last activity comlast: prop. to com*last
v
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Th d lif i d d l d Theorem: node lifetimes and edge gaps lead to power law degree distribution
Interesting as temporal behavior predicts structural network propertystructural network property
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Node lifetime: pl(a) = pl( ) Let a node have life‐time a, what will its final degree D be? g
D~
The two exponential functions “cancel out” pand the power law part remains
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Given our model one can take an existing Given our model one can take an existing network continue its evolution
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Take Flickr at time T/2 and then further evolve it continue evolving it using PA and our model.g g
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How do networks evolve at the macroscopic How do networks evolve at the macroscopic level?
Are there global phenomena of network evolution?evolution?
What are analogs of the “small world” and What are analogs of the small world and “power‐law degrees”
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