1

Nonparametric Link Prediction in Dynamic Graphs

Purnamrita Sarkar (UC Berkeley)Deepayan Chakrabarti (Facebook)Michael Jordan (UC Berkeley)

2

Link Prediction Who is most likely to be interact with a given node?

Friend suggestion in Facebook

Should Facebook suggest Alice

as a friend for Bob?

Bob

Alice

3

Link Prediction

Alice

Bob

Charlie

Movie recommendation in Netflix

Should Netflix suggest this

movie to Alice?

4

Link Prediction Prediction using simple features

degree of a node number of common neighbors last time a link appeared

What if the graph is dynamic?

5

Related Work

Generative models Exp. family random graph models [Hanneke+/’06] Dynamics in latent space [Sarkar+/’05] Extension of mixed membership block models

[Fu+/10] Other approaches

Autoregressive models for links [Huang+/09] Extensions of static features [Tylenda+/09]

6

Goal

Link Prediction incorporating graph dynamics, requiring weak modeling assumptions, allowing fast predictions, and offering consistency guarantees.

7

Outline

Model Estimator Consistency Scalability Experiments

8

The Link Prediction Problem in Dynamic Graphs

G1 G2 GT+1……

Y1 (i,j)=1

Y2 (i,j)=0

YT+1 (i,j)=?

YT+1(i,j) | G1,G2, …,GT ~ Bernoulli (gG1,G2,…GT(i,j))

Edge in T+1 Features of previous graphsand this pair of nodes

9

cn

ℓℓ

deg

Including graph-based features

Example set of features for pair (i,j): cn(i,j) (common neighbors) ℓℓ(i,j) (last time a link was formed) deg(j)

Represent dynamics using “datacubes” of these features. ≈ multi-dimensional histogram on binned feature values

ηt = #pairs in Gt with these features

1 ≤ cn ≤ 33 ≤ deg ≤ 61 ≤ ℓℓ ≤ 2

ηt+ = #pairs in Gt with these

features, which had an edge in Gt+1

high ηt+/ηt this feature

combination is more likely to create a new edge at time t+1

10

G1 G2 GT……

Y1 (i,j)=1 Y2 (i,j)=0 YT+1 (i,j)=?

1 ≤ cn(i,j) ≤ 33 ≤ deg(i,j) ≤ 61 ≤ ℓℓ (i,j) ≤ 2

Including graph-based features

How do we form these datacubes? Vanilla idea: One datacube for Gt→Gt+1

aggregated over all pairs (i,j) Does not allow for differently evolving communities

11

YT+1 (i,j)=?

1 ≤ cn(i,j) ≤ 33 ≤ deg(i,j) ≤ 61 ≤ ℓℓ (i,j) ≤ 2

Our Model

How do we form these datacubes? Our Model: One datacube for each neighborhood

Captures local evolution

G1 G2 GT……

Y1 (i,j)=1 Y2 (i,j)=0

12

Our Model

Number of node pairs- with feature s- in the neighborhood of i- at time t

Number of node pairs- with feature s- in the neighborhood of i- at time t- which got connected at time t+1

Datacube

1 ≤ cn(i,j) ≤ 33 ≤ deg(i,j) ≤ 61 ≤ ℓℓ (i,j) ≤ 2

Neighborhood Nt(i)= nodes within 2 hops

Features extracted from (Nt-p,…Nt)

13

Our Model

Datacube dt(i) captures graph evolution in the local neighborhood of a node in the recent past

Model:

What is g(.)?

YT+1(i,j) | G1,G2, …,GT ~ Bernoulli ( gG1,G2,…GT(i,j))g(dt(i), st(i,j))

Features of the pair

Local evolution patterns

14

Outline

Model Estimator Consistency Scalability Experiments

15

Kernel Estimator for g

G1 G2 …… GTGT-1GT-2

query data-cube at T-1 and feature vector at time T

compute similarities

datacube, feature pair

t=1

{{

{

{

{

{

{

{

…

datacube, feature pair

t=2

{{

{

{

{

{

{

{

…datacube,

feature pair t=3

{{

{

{

{

{

{

{

…{

{

16

Factorize the similarity function Allows computation of g(.) via simple lookups

}} }

K( , )I{ == }

Kernel Estimator for g

17

Kernel Estimator for g

G1 G2 …… GTGT-1GT-2

datacubes t=1

datacubes t=2

datacubes t=3

compute similarities only between data cubes

w1

w2

w3

w4

η1 , η1+

η2 , η2+

η3 , η3+

η4 , η4+

44332211

44332211

wwwwwwww

18

Factorize the similarity function Allows computation of g(.) via simple lookups What is K( , )?

}}

}

K( , )I{ == }

Kernel Estimator for g

19

Similarity between two datacubes

Idea 1 For each cell s, take

(η1+/η1 – η2

+/η2)2 and sum

Problem: Magnitude of η is ignored 5/10 and 50/100 are treated

equally

Consider the distribution

η1 , η1+

η2 , η2+

20

Similarity between two datacubes

0 5 10 15 20 25 30 35 40 450

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0 5 10 15 20 25 30 35 40 450

0.02

0.04

0.06

0.08

0.1

0.12

0.14

) , dist(b) , K( 0<b<1

As b0, K( , ) 0 unless dist( , ) =0

Idea 2 For each cell s, compute

posterior distribution of edge creation prob.

dist = total variation distance between distributions summed over all cells

η1 , η1+

η2 , η2+

21

1tη) , K(#1f

) , (f) , (h) , (g

1tη) , K(

#1h

Want to show: gg

Kernel Estimator for g

22

Outline

Model Estimator Consistency Scalability Experiments

23

Consistency of Estimator

Lemma 1: As T→∞, for some R>0,

Proof using:

) , (f) , (h) , (g

As T→∞,

24

Consistency of Estimator

Lemma 2: As T→∞,

) , (f) , (h) , (g

25

Consistency of Estimator

Assumption: finite graph Proof sketch:

Dynamics are Markovian with finite state spacethe chain must eventually enter a closed, irreducible communication classgeometric ergodicity if class is aperiodic(if not, more complicated…)strong mixing with exponential decayvariances decay as o(1/T)

26

Consistency of Estimator

Theorem:

Proof Sketch:

for some R>0

So

27

Outline

Model Estimator Consistency Scalability Experiments

28

Scalability Full solution:

Summing over all n datacubes for all T timesteps Infeasible

Approximate solution: Sum over nearest neighbors of query datacube

How do we find nearest neighbors? Locality Sensitive Hashing (LSH)

[Indyk+/98, Broder+/98]

29

Using LSH

Devise a hashing function for datacubes such that “Similar” datacubes tend to be hashed to the

same bucket “Similar” = small total variation distance

between cells of datacubes

30

0 5 10 15 20 25 30 35 40 450

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Using LSH

Step 1: Map datacubes to bit vectors

Use B2 bits for each bucket For probability mass p the first bits are set to

1Use B1 buckets to discretize [0,1]

Total M*B1*B2 bits, where M = max number of occupied cells << total number of cells

31

Using LSH

Step 1: Map datacubes to bit vectors Total variation distance

L1 distance between distributions Hamming distance between vectors

Step 2: Hash function = k out of MB1B2 bits

32

Fast Search Using LSH

1111111111000000000111111111000

10000101000011100001101010000

10101010000011100001101010000

101010101110111111011010111110

1111111111000000000111111111001

00000001

1111

0011

.

.

.

.

1011

33

Outline

Model Estimator Consistency Scalability Experiments

34

Experiments

Baselines LL: last link (time of last occurrence of a pair)

CN: rank by number of common neighbors in AA: more weight to low-degree common neighbors Katz: accounts for longer paths

CN-all: apply CN to AA-all, Katz-all: similar

ss

35

Setup

Pick random subset S from nodes with degree>0 in GT+1

, predict a ranked list of nodes likely to link to s Report mean AUC (higher is better)

G1 G2 GT

Training data Test dataGT+

1

36

Simulations Social network model of Hoff et al.

Each node has an independently drawn feature vector

Edge(i,j) depends on features of i and j Seasonality effect

Feature importance varies with seasondifferent communities in each season

Feature vectors evolve smoothly over timeevolving community structures

37

Simulations

NonParam is much better than others in the presence of seasonality

CN, AA, and Katz implicitly assume smooth evolution

38

Sensor Network*

* www.select.cs.cmu.edu/data

39

Summary

Link formation is assumed to depend on the neighborhood’s evolution over a time window

Admits a kernel-based estimator Consistency Scalability via LSH

Works particularly well for Seasonal effects differently evolving communities