Nonparametric Link Prediction in Dynamic Graphs

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Nonparametric Link Prediction in Dynamic Graphs

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

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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

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Link Prediction

Alice

Bob

Charlie

Movie recommendation in Netflix

Should Netflix suggest this

movie to Alice?

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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?

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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]

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Goal

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

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Outline

Model Estimator Consistency Scalability Experiments

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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

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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

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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

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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

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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)

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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

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Outline

Model Estimator Consistency Scalability Experiments

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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

{{

{

{

{

{

{

{

…{

{

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

}} }

K( , )I{ == }

Kernel Estimator for g

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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

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

}}

}

K( , )I{ == }

Kernel Estimator for g

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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+

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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+

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1tη) , K(#1f

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

1tη) , K(

#1h

Want to show: gg

Kernel Estimator for g

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Outline

Model Estimator Consistency Scalability Experiments

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Consistency of Estimator

Lemma 1: As T→∞, for some R>0,

Proof using:

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

As T→∞,

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Consistency of Estimator

Lemma 2: As T→∞,

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

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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)

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Consistency of Estimator

Theorem:

Proof Sketch:

for some R>0

So

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Outline

Model Estimator Consistency Scalability Experiments

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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]

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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

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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

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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

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Fast Search Using LSH

1111111111000000000111111111000

10000101000011100001101010000

10101010000011100001101010000

101010101110111111011010111110

1111111111000000000111111111001

00000001

1111

0011

.

.

.

.

1011

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Outline

Model Estimator Consistency Scalability Experiments

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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

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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

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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

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Simulations

NonParam is much better than others in the presence of seasonality

CN, AA, and Katz implicitly assume smooth evolution

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Sensor Network*

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

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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