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Nonparametric Link Prediction in Dynamic Graphs Purnamrita Sarkar (UC Berkeley) Deepayan Chakrabarti (Facebook) Michael Jordan (UC Berkeley) 1

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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 2

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Link Prediction Alice Bob Charlie Movie recommendation in Netflix Should Netflix suggest this movie to Alice? 3

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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? 4

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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] 5

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Goal Link Prediction incorporating graph dynamics, requiring weak modeling assumptions, allowing fast predictions, and offering consistency guarantees. 6

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Outline Model Estimator Consistency Scalability Experiments 7

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The Link Prediction Problem in Dynamic Graphs G1G1 G2G2 G T+1 …… Y 1 (i,j)=1 Y 2 (i,j)=0 Y T+1 (i,j)=? Y T+1 (i,j) | G 1,G 2, …,G T ~ Bernoulli (g G1,G2,…GT (i,j)) Edge in T+1 Features of previous graphs and this pair of nodes 8

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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 η t = #pairs in G t with these features 1 ≤ cn ≤ 3 3 ≤ deg ≤ 6 1 ≤ ℓℓ ≤ 2 η t + = #pairs in G t with these features, which had an edge in G t+1 high η t + /η t this feature combination is more likely to create a new edge at time t+1 9

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G1G1 G2G2 GTGT …… Y 1 (i,j)=1 Y 2 (i,j)=0 Y T+1 (i,j)=? 1 ≤ cn(i,j) ≤ 3 3 ≤ deg(i,j) ≤ 6 1 ≤ ℓℓ (i,j) ≤ 2 Including graph-based features How do we form these datacubes? Vanilla idea: One datacube for G t →G t+1 aggregated over all pairs (i,j) Does not allow for differently evolving communities 10

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Y T+1 (i,j)=? 1 ≤ cn(i,j) ≤ 3 3 ≤ deg(i,j) ≤ 6 1 ≤ ℓℓ (i,j) ≤ 2 Our Model How do we form these datacubes? Our Model: One datacube for each neighborhood Captures local evolution G1G1 G2G2 GTGT …… Y 1 (i,j)=1 Y 2 (i,j)=0 11

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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) ≤ 3 3 ≤ deg(i,j) ≤ 6 1 ≤ ℓℓ (i,j) ≤ 2 Neighborhood N t (i)= nodes within 2 hops Features extracted from (N t-p,…N t ) 12

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Our Model Datacube d t (i) captures graph evolution in the local neighborhood of a node in the recent past Model: What is g(.)? Y T+1 (i,j) | G 1,G 2, …,G T ~ Bernoulli ( g G1,G2,…GT (i,j)) g(d t (i), s t (i,j) ) Features of the pair Local evolution patterns 13

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Outline Model Estimator Consistency Scalability Experiments 14

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Kernel Estimator for g G1G1 G 2 …… GTGT G T-1 G T-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 { { { { { { { { … { { 15

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Factorize the similarity function Allows computation of g(.) via simple lookups } } } K(, )I{ == } Kernel Estimator for g 16

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Kernel Estimator for g G1G1 G 2 …… GTGT G T-1 G T-2 datacubes t=1 datacubes t=2 datacubes t=3 compute similarities only between data cubes w1w1 w2w2 w3w3 w4w4 η 1, η 1 + η 2, η 2 + η 3, η 3 + η 4, η

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Factorize the similarity function Allows computation of g(.) via simple lookups What is K(, )? } } } K(, )I{ == } Kernel Estimator for g 18

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Similarity between two datacubes Idea 1 For each cell s, take (η 1 + /η 1 – η 2 + /η 2 ) 2 and sum Problem: Magnitude of η is ignored 5/10 and 50/100 are treated equally Consider the distribution η 1, η 1 + η 2, η

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Similarity between two datacubes 0**
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Want to show: Kernel Estimator for g 21

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Outline Model Estimator Consistency Scalability Experiments 22

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Consistency of Estimator Lemma 1: As T→∞, for some R>0, Proof using: As T→∞, 23

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Consistency of Estimator Lemma 2: As T→∞, 24

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Consistency of Estimator Assumption: finite graph Proof sketch: Dynamics are Markovian with finite state space the chain must eventually enter a closed, irreducible communication class geometric ergodicity if class is aperiodic (if not, more complicated…) strong mixing with exponential decay variances decay as o(1/T) 25

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Consistency of Estimator Theorem: Proof Sketch: for some R>0 So 26

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Outline Model Estimator Consistency Scalability Experiments 27

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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] 28

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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 29

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Using LSH Step 1: Map datacubes to bit vectors Use B 2 bits for each bucket For probability mass p the first bits are set to 1 Use B 1 buckets to discretize [0,1] Total M*B1*B2 bits, where M = max number of occupied cells << total number of cells 30

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Using LSH 31

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Fast Search Using LSH

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Outline Model Estimator Consistency Scalability Experiments 33

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Experiments 34

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Setup G1G1 G2G2 GTGT Training data Test data G T+1 35

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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 season different communities in each season Feature vectors evolve smoothly over time evolving community structures 36

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Simulations NonParam is much better than others in the presence of seasonality CN, AA, and Katz implicitly assume smooth evolution 37

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Sensor Network * * 38

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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 39

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