The Link Prediction Problem for Social Networks

David Libel-Nowell, MITJohn Klienberg, Cornell

Saswat Mishra sxm111131

Summary

The “Link Prediction Problem”

Given a snapshot of a social network, can we infer which new interactions among its members are likely to occur in the near future?

Based on “proximity” of nodes in a network

Introduction

Natural examples of social networks:

Nodes = people/entitiesEdges = interaction/ collaboration

Nodes Edges

Scientists in a discipline Co-authors of a paper

Employees in a large company

Working on a project

Business Leaders Serve together on a board

Motivation

Understanding how social networks evolve

The link prediction problem Given a snapshot of a social network at time t, we seek

to accurately predict the edges that will be added to the network during the interval (t, t’)

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Why?

To suggest interactions or collaborations that haven’t yet been utilized within an organization

To monitor terrorist networks - to deduce possible interaction between terrorists (without direct evidence)

Used in Facebook and Linked In to suggest friends

Open Question: How does Facebook do it?

(friends of friends, same school, manually…)

Motivation

Co-authorship network for scientists

Scientists who are “close” in the network will have common colleagues & circles – likely to collaborateCaveat: Scientists who have never collaborated might in future - hard to predict

Goal: make that intuitive notion precise; understand which measures of “proximity” lead to accurate predictions

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Goals

Present measures of proximity

Understand relative effectiveness of network proximity measures (adapted from graph theory, CS, social sciences)

Prove that prediction by proximity outperforms random predictions by a factor of 40 to 50

Prove that subtle measures outperform more direct measures

Data and Experimental Setup

Co-authorship network (G) from “author list” of the physics e-Print arXiv (www.arxiv.org)

Took 5 such networks from 5 sections of the print

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Core: set of authors who have at least 3 papers during both training and test

G[1994,1996] = Gcollab = (A,Eold) Enew = new collaborations (edges)

Training interval [1994,1996]Ktraining = 3

Test interval [1997,1999]Ktest = 3

Data

Methods for Link Prediction

Take the input graph during training period Gcollab

Pick a pair of nodes (x, y) Assign a connection weight score(x, y) Make a list in descending order of score

score is a measure of proximity

Any ideas for measures?

Proximity Measures for Link Prediction

Graph distance & Common Neighbors

Graph distance: (Negated) length of shortest path between x and y

Common Neighbors: A and C have 2 common neighbors, more likely to collaborate

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(A, C) -2

(C, D) -2

(A, E) -3

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Jaccard’s coefficient and Adamic / Adar

Jaccard’s coefficient: same as common neighbors, adjusted for degree

Adamic / Adar: weighting rarer neighbors more heavily

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Preferential Attachment

Probability that a new collaboration involves x is proportional to T(x), current neighbors of x

score (x, y) :=

Considering all paths: Katz

Katz: measure that sums over the collection of paths, exponentially damped by length (to count short paths heavily)

β is chosen to be a very small value (for

dampening) A

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DE

Hitting time, PageRank

Hitting time: expected number of steps for a random walk starting at x to reach y

Commute time:

If y has a large stationary probability, Hx,y is small. To counterbalance, we can normalize

PageRank: to cut down on long random walks, walk can return to x with a probablity α at every step y

SimRank

Defined by this recursive definition: two nodes are similar to the extent that they are joined by similar neighbors

Low-rank approximation

Treat the graph as an adjacency matrix

Compute the rank-k matrix Mk (noise-reduction) x is a row, y is a row, score(x, y) = inner product of rows

r(x) and r(y)

-A B C

A 1 0

B 1 1

C 0 1

Unseen bigrams and Clustering

Unseen bigrams: Derived from language modeling Estimating frequency of unseen bigrams – pairs of

words (nodes here) that co-occur in a test corpus but not in the training corpus

Clustering: deleting tenuous edges in Gcollab through a clustering procedure and running predictors on the “cleaned-up” subgraph

Results

The results are presented as:

1. Factor improvement of proposed predictors over Random predictor Graph distance predictor Common neighbors predictor

2. Relative performance vs. the above predictors 3. Common Predictions

Factor Improvement of different measures

Factor Improvement - meta approaches

Relative performance vs. Random Predictions

vs. graph distance predictor, vs. common neighbors predictor

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Common Predictions

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Conclusions

No single clear winner

Many outperform the random predictor => there is useful information in the network topology

Katz + clustering + low-rank approximation perform significantly well

Some simple measures i.e. common neighbors and Adamic/ Adar perform well

Critique

Even the best predictor (Katz on gr-qc) is correct on only 16% of predictions

How good is that?

Treat all collaborations equally. Perhaps, treating recent collaborations as more important than older ones will help?

References

Lada A. Adamic and Eytan Adar. Friends and neighbors on the web. Social Networks, 25(3):211{230, July 2003.

A. L. Barabasi, H. Jeong, Z. N eda, E. Rav asz, A. Schubert, and T. Vicsek. Evolution of the social network of scientist collaboration. Physica A, 311(3{4):590{614, 2002.

Sergey Brin and Lawrence Page. The anatomy of a large-scale hyper textual Web search engine Computer Networks and ISDN Systems, 30(1{7):107{117, 1998.

Rodrigo De Castro and Jerrold W. Grossman. F amous trails to Paul Erdos. Mathematical Intelligencer, 21(3):51{63, 1999.

Question

Question???

Thank You