Microscopic Evolution of Social Networks

Jure Leskovec, CMULars Backstrom, CornellRavi Kumar, Yahoo! ResearchAndrew Tomkins, Yahoo! Research

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)

We observe individual edge and node arrivals in large

social networks

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?

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

Structure of our contributions

Experiments and the complete model of network evolution

Process Our finding

P1) 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 lifetime is exponential

Lifetime a: time between node’s first and last edge

LinkedIn

Node lifetime is exponential: p(a) = λ exp(-λa)

)()(),);(( dg eddp

2) How are edges initiated?

Edge gap δ(d): inter-arrival time between dth and d+1st edge

LinkedIn

Degreed=1

d=3d=2

Edge time gap (time between 2 consecutive edges of a node)

Pro

bab

ilit

y

2) How do α & β evolve with degree?

Degree of Preferential Attachment

Edge Locality: closing triangles

Degree of PA & Edge locality

kkpe )(Networ

kτ

Gnm 0

PA 1F 1D 1A 0.9L 0.6

Network

% Δ

F 66%

D 28%

A 23%

L 50%

Fraction of triad closing

edges

How to close triangles?

We consider 25 strategies for choosing node v and then w

Compute likelihood of each strategy

Triad closing strategies

Log-likelihood improvement over the baseline

Strategy to select v (1st node)

Sele

ct

w (

2n

d n

od

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

uw

v

The complete 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 )(

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 1.74

D 2.38 2.30

A 1.90 1.75

L 2.11 2.08

Evolving the networks

Given our model one can take an existing network continue its evolution

Comparison with other models

Take Flickr at time T/2 and then further evolve it continue evolving it using PA and our model.

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

Choosing edge source and destination