1. The Link Prediction Problem for Social Networks David Libel-Nowell, MIT John Klienberg, Cornell
Saswat Mishra sxm111131

2. 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

3. Introduction Natural examples of social networks: 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
Nodes = people/entities
Edges = interaction/ collaboration

4. ? 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’) ?

5. 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…)

6. Motivation Co-authorship network for scientists
Scientists who are “close” in the network will have common colleagues & circles – likely to collaborate
Caveat: 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 D B A C

7. 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

8. Data and Experimental Setup
Co-authorship network (G) from “author list” of the physics e-Print arXiv (
Took 5 such networks from 5 sections of the print D B B A A C C
Training interval [1994,1996]
Ktraining = 3
Test interval [1997,1999]
Ktest = 3
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)

9. Data

10. 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?
Shortest path, common neighbors

11. Proximity Measures for Link Prediction

12. 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 E D B
(A, C) -2
(C, D)
(A, E) -3 A C E D B A C

13. Jaccard’s coefficient and Adamic / Adar
Jaccard’s coefficient: same as common neighbors, adjusted for degree
Adamic / Adar: weighting rarer neighbors more heavily E D B A C

14. Preferential Attachment
Probability that a new collaboration involves x is proportional to T(x), current neighbors of x
score (x, y) :=

15. 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) E D B A C

16. 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

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

18. 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

19. 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

20. 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

21. Factor Improvement of different measures

22. Factor Improvement - meta approaches

23. Relative performance vs. Random Predictions

24. vs. graph distance predictor, vs. common neighbors predictor

25. Common Predictions a

26. 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

27. 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?

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

29. Question
Question???

30. Thank You