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Purnamrita Sarkar (UC Berkeley) Deepayan Chakrabarti (Yahoo! Research) Andrew W. Moore (Google, Inc.) 1

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Which pair of nodes {i,j} should be connected? Alice Bob Charlie Goal: Recommend a movie

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Which pair of nodes {i,j} should be connected? Goal: Suggest friends

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Predict link between nodes Connected by the shortest path With the most common neighbors (length 2 paths) More weight to low-degree common nbrs (Adamic/Adar) 8 followers 1000 followers Prolific common friends Less evidence Less prolific Much more evidence Alice Bob Charlie

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Predict link between nodes Connected by the shortest path With the most common neighbors (length 2 paths) More weight to low-degree common nbrs (Adamic/Adar) With more short paths (e.g. length 3 paths ) exponentially decaying weights to longer paths (Katz measure)

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RandomShortest Path Common Neighbors Adamic/AdarEnsemble of short paths Link prediction accuracy* *Liben-Nowell & Kleinberg, 2003; Brand, 2005; Sarkar & Moore, 2007 How do we justify these observations? Especially if the graph is sparse

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7 Unit volume universe Model: 1.Nodes are uniformly distributed points in a latent space 2.This space has a distance metric 3.Points close to each other are likely to be connected in the graph Logistic distance function (Raftery+/2002)

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8 1 Higher probability of linking radius r determines the steepness The problem of link prediction is to find the nearest neighbor who is not currently linked to the node. Equivalent to inferring distances in the latent space Model: 1.Nodes are uniformly distributed points in a latent space 2.This space has a distance metric 3.Points close to each other are likely to be connected in the graph Logistic distance function (Raftery+/2002)

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RandomShortest Path Common Neighbors Adamic/AdarEnsemble of short paths Link prediction accuracy *Liben-Nowell & Kleinberg, 2003; Brand, 2005; Sarkar & Moore, 2007 Especially if the graph is sparse

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Pr 2 (i,j) = Pr(common neighbor|d ij ) Product of two logistic probabilities, integrated over a volume determined by d ij As Logistic Step function Much easier to analyze! i j

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11 Everyone has same radius r i j # common nbrs gives a bound on distance =Number of common neighbors V(r)=volume of radius r in D dims Unit volume universe

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OPT = node closest to i MAX = node with max common neighbors with i Theorem: w.h.p Link prediction by common neighbors is asymptotically optimal d OPT d MAX d OPT + 2[ /V(1)] 1/D

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Node k has radius r k. i k if d ik r k (Directed graph) r k captures popularity of node k Weighted common neighbors: Predict (i,j) pairs with highest w(r)(r) 13 i k j rkrk m Weight for nodes of radius r # common neighbors of radius r

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Presence of common neighbor is very informative r is close to max radius Absence is very informative Adamic/Adar 1/r Real world graphs generally fall in this range i k j radius

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RandomShortest Path Common Neighbors Adamic/AdarEnsemble of short paths Link prediction accuracy *Liben-Nowell & Kleinberg, 2003; Brand, 2005; Sarkar & Moore, 2007 Especially if the graph is sparse

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Common neighbors = 2 hop paths For longer paths: Bounds are weaker For we need >> to obtain similar bounds justifies the exponentially decaying weight given to longer paths by the Katz measure

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Three key ingredients 1. Closer points are likelier to be linked. Small World Model- Watts, Strogatz, 1998, Kleinberg 2001 2. Triangle inequality holds necessary to extend to - hop paths 3. Points are spread uniformly at random Otherwise properties will depend on location as well as distance

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RandomShortest Path Common Neighbors Adamic/AdarEnsemble of short paths Link prediction accuracy* *Liben-Nowell & Kleinberg, 2003; Brand, 2005; Sarkar & Moore, 2007 The number of paths matters, not the length For large dense graphs, common neighbors are enough Differentiating between different degrees is important In sparse graphs, length 3 or more paths help in prediction.

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20 1 Higher probability of linking Two sources of randomness Point positions: uniform in D dimensional space Linkage probability: logistic with parameters , r , r and D are known radius r determines the steepness The problem of link prediction is to find the nearest neighbor who is not currently linked to the node. Equivalent to inferring distances in the latent space

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1 Factor weak bound for Logistic Can be made tighter, as logistic approaches the step function.

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22 Generative model Link Prediction Heuristics node a Most likely neighbor of node i ? node b Compare A few properties Can justify the empirical observations We also offer some new prediction algorithms

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Combine bounds from different radii But there might not be enough data to obtain individual bounds from each radius New sweep estimator Q r = Fraction of nodes w. radius r, which are common neighbors. Higher Q r smaller d ij w.h.p

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Q r = Fraction of nodes w. radius r, which are common neighbors larger Q r smaller d ij w.h.p T R : = Fraction of nodes w. radius R, which are common neighbors. Smaller T R large d ij w.h.p

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Q r = Fraction of nodes with radius r which are common neighbors T R = Fraction of nodes with radius R which are common neighbors Number of common neighbors of a given radius Large Q r small d ij Small T R large d ij r

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Which pair of nodes {i,j} should be connected? Variant: node i is given Friend suggestion in Facebook Alice Bob Charlie Movie recommendation in Netflix

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27 Nodes are uniformly distributed in a latent space The problem of link prediction is to find the nearest neighbor who is not currently linked to the node. Equivalent to inferring distances in the latent space Raftery et al.s Model: Unit volume universe Points close in this space are more likely to be connected.

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28 1 Higher probability of linking Two sources of randomness Point positions: uniform in D dimensional space Linkage probability: logistic with parameters , r , r and D are known radius r determines the steepness

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i j k 1 ~ Bin[N 1, A(r 1, r 1, d ij )] 2 ~ Bin[N 2, A(r 2, r 2, d ij )] Example graph: N 1 nodes of radius r 1 and N 2 nodes of radius r 2 r 1

Bound d ij as a function of For we need >> to obtain similar bounds justifies the exponentially decaying weight given to longer paths by the Katz measure Also, we can obtain much tighter bounds for long paths if shorter paths are known to exist.