Jure Leskovec, CMU
Lars Backstrom, Cornell
Ravi Kumar, Yahoo! Research
Andrew Tomkins, Yahoo! Research


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)
We observe individual edge and node
arrivals in large social networks

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?

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

Experiments and the complete model of
network evolution
Process
P1) Node arrival
P2) Edge initiation
P3) Edge
destination
Our finding
(F)
(D)
Flickr:
Exponential
(A)
Delicious:
Linear
(L)
Answers:
Sub-linear
LinkedIn:
Quadratic
Leskovec, Backstrom, Kumar & Tomkins: Microscopic Evolution of Social Networks,
KDD '08
LinkedIn

Lifetime a:
time between
node’s first
and last edge
Node lifetime is exponential: p(a) = λ exp(-λa)
LinkedIn
Edge gap δ(d):
inter-arrival
time between
dth and d+1st
edge

pg ( (d );  ,  )   (d ) e
  ( d )
Probability
d=3
d=2
Degree
d=1
Edge time gap (time between 2 consecutive edges of a node)
pe (k )  k
Network
Gnm
PA
F
D
A
L

τ
0
1
1
1
0.9
0.6
Fraction of triad
closing edges
Network
%Δ
F
66%
D
28%
A
23%
L
50%


We consider 25 strategies for choosing node v
and then w
Compute likelihood of each strategy
Log-likelihood improvement over the baseline
Strategy to select v (1st node)
Select w (2nd node)

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
w
v
Process
Our finding
P1) Node arrival
• Node arrival function is given
• Node lifetime is exponential
P2) Edge initiation
• Edge gaps: p(t )  t  e  dt
P3) Edge
destination
•1st edge is created preferentially
• Use random-random to close
triangles


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
1.74
D
2.38
2.30
A
1.90
1.75
L
2.11
2.08

Given our model one can take an existing
network continue its evolution

Take Flickr at time T/2 and then further evolve
it continue evolving it using PA and our model.



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