Microscopic Evolution of Social NetworksJure Leskovec, CMULars Backstrom, CornellRavi Kumar, Yahoo! ResearchAndrew Tomkins, Yahoo! Research
Leskovec, Backstrom, Kumar & Tomkins: Microscopic Evolution of Social Networks, KDD '08
Introduction
Social networks evolve with additions and deletions of nodes and edges
We talk about the evolution but few have actually directly observed atomic events of network evolution (but only via snapshots)
This talk: We observed individual edge and node arrivals in large social networks
and so on for
millions…
Leskovec, Backstrom, Kumar & Tomkins: Microscopic Evolution of Social Networks, KDD '08
Questions we ask
Test individual edge attachment: Directly observe mechanisms leading to
global network properties▪ E.g., What is really causing power-law degree
distributions? Compare models: via model likelihood
Compare network models by likelihood (and not by summary network statistics)▪ E.g., Is Preferential Attachment best model?
Leskovec, Backstrom, Kumar & Tomkins: Microscopic Evolution of Social Networks, KDD '08
The setting: Edge-by-edge evolution Three processes that govern the evolution
P1) Node arrival process: nodes enter the network P2) Edge initiation process: each node decides when
to initiate an edge P3) Edge destination process: determines destination
after a node decides to initiate
(F)(D)(A)(L)
Leskovec, Backstrom, Kumar & Tomkins: Microscopic Evolution of Social Networks, KDD '08
The rest of the talk
Experiments and the complete model of network evolution
Process Our findingP1) Node arrivalP2) Edge initiationP3) Edge destination
Leskovec, Backstrom, Kumar & Tomkins: Microscopic Evolution of Social Networks, KDD '08
P1) How fast are nodes arriving?(F) (D)
(A) (L)
Flickr: Exponential
Delicious: Linear
Answers: Sub-linear
LinkedIn:
Quadratic
Node arrival process is network
dependent
Leskovec, Backstrom, Kumar & Tomkins: Microscopic Evolution of Social Networks, KDD '08
P1) What is node lifetime?Lifetime a: time between node’s first and last edge
Node lifetime is exponentially distributed:
p(a) = λ exp(-λa)
Leskovec, Backstrom, Kumar & Tomkins: Microscopic Evolution of Social Networks, KDD '08
The model so far …
What do we know so far?
Process Our finding
P1) Node arrival
• Node arrival function is given• Node lifetime is exponential
P2) Edge initiation
P3) Edge destination
Leskovec, Backstrom, Kumar & Tomkins: Microscopic Evolution of Social Networks, KDD '08
)()(),);(( deddp
P2) How do α & β evolve with degree?
Edge gap δ(d): time between dth and d+1st edge of a node
Degreed=1
d=3d=2
Edge time gap (time between 2 consecutive edges of a node)
Prob
abili
ty
Nodes of higher degree start adding edges faster
and faster
Leskovec, Backstrom, Kumar & Tomkins: Microscopic Evolution of Social Networks, KDD '08
The model so far …
What do we know so far?
Process Our finding
P1) Node arrival
• Node arrival function is given• Node lifetime is exponential
P2) Edge initiation • Edge gaps:
P3) Edge destination
dtettp )(
Leskovec, Backstrom, Kumar & Tomkins: Microscopic Evolution of Social Networks, KDD '08
Preferential attachment: Does it hold?
kkpe )(Gnp
PA
(D)
(F)(L)
(A)
Network
τ
Gnp 0PA 1F 1D 1A 0.9L 0.6
We unroll the true network edge arrivals Measure node degrees where edges attach
First direct proof of preferential
attachment!
Leskovec, Backstrom, Kumar & Tomkins: Microscopic Evolution of Social Networks, KDD '08
u wv
PA? Not really. Edges are local! Just before the edge (u,v) is placed how many hops is
between u and v?
Network
% Δ
F 66%D 28%A 23%L 50%
GnpPA
(D)
(F)
(L) (A)
Fraction of triad closing
edges
Real edges are local.Most of them close
triangles!
Leskovec, Backstrom, Kumar & Tomkins: Microscopic Evolution of Social Networks, KDD '08
New triad-closing edge (u,w) appears next We model this as:
1. Choose u’s neighbor v 2. Choose v’s neighbor w3. Connect (u,w)
We consider 25 strategies for choosing v and then w
Can compute likelihood of each strategy Under Random-Random:
p(u,w) = 1/5*1/2+1/5*1
How to close triangles?
uw
v
v’
Leskovec, Backstrom, Kumar & Tomkins: Microscopic Evolution of Social Networks, KDD '08
Triad closing strategies Log-likelihood improvement over the baseline
Strategy to select v (1st node)
Sele
ct w
(2n
d nod
e)
Strategies to pick a neighbor: random: uniformly at random deg: proportional to its degree com: prop. to the number of common friends last: prop. to time since last activity comlast: prop. to com*last
u wv
random-random works
well
Leskovec, Backstrom, Kumar & Tomkins: Microscopic Evolution of Social Networks, KDD '08
The complete model
The complete network evolution model
Process Our finding
P1) Node arrival
• Node arrival function is given• Node lifetime is exponential
P2) Edge initiation • Edge gaps:
P3) Edge destination
•1st edge is created preferentially• Use random-random to close triangles
dtettp )(
Leskovec, Backstrom, Kumar & Tomkins: Microscopic Evolution of Social Networks, KDD '08
Analysis of our model
Theorem: node lifetimes and edge gaps lead to power law degree distribution
Interesting as temporal behavior predicts structural network property
Network
True γ Predicted γ
F 1.73 1.74D 2.38 2.30A 1.90 1.75L 2.11 2.08
Our theorem accurately predicts degree exponents γ
as observed data
Leskovec, Backstrom, Kumar & Tomkins: Microscopic Evolution of Social Networks, KDD '08
Summary and conclusion We observe network evolution at atomic scale We use log-likelihood of edge placements to compare
and infer models Our findings
Preferential attachment holds but it is local Triad closure is fundamental mechanism
We present a 3 process network evolution model P1) Node lifetimes are exponential P2) Edge interarrival time is power law with exp. cutoff P3) Edge destination is chosen by random-random
Gives more realistic evolution that other models
Leskovec, Backstrom, Kumar & Tomkins: Microscopic Evolution of Social Networks, KDD '08
Thanks!
More details and analyses in the paper
Thanks to Yahoo and LinkedIn for providing the data.
http://www.cs.cmu.edu/~jure
2) How are edges initiated?Edge gap δ(d): time between dth and d+1st edge
Edge interarrivals follow power law with exponential cutoff
distribution: )()(),);(( dg eddp
Leskovec, Backstrom, Kumar & Tomkins: Microscopic Evolution of Social Networks, KDD '08
How do α and β change with node
degree?
Leskovec, Backstrom, Kumar & Tomkins: Microscopic Evolution of Social Networks, KDD '08
)()(),);(( dg eddp
2) How do α & β evolve with degree?
This means nodes of higher degree start
adding edges faster and faster
Edge gap time
Prob
abili
ty
Degreed=1
d=3d=2
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