Microscopic Evolution of Social Networks

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…


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

(F)(D)(A)(L)


The rest of the talk

Experiments and the complete model of network evolution

Process Our findingP1) Node arrivalP2) Edge initiationP3) Edge destination


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


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)

LinkedIn


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


)()(),);(( 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


The model so far …

What do we know so far?

Process Our finding

P1) Node arrival


P2) Edge initiation • Edge gaps:


dtettp )(


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!


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!


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’


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


The complete model

The complete network evolution model

Process Our finding

P1) Node arrival


P2) Edge initiation • Edge gaps:


•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 1.74D 2.38 2.30A 1.90 1.75L 2.11 2.08

Our theorem accurately predicts degree exponents γ

as observed data


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


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


How do α and β change with node

degree?


)()(),);(( 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

Microscopic Evolution of Social Networks

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