1. Link Prediction Seminar Social Media Mining University UC3M
Date May 2017
Lecturer Carlos Castillo
Sources:
Chapter 10 of Zafarani, Abbasi, and Liu's book on Social Media Mining [slides]
Sarkar, Chakrabarti, Moore: [slides] [slides]

2. Problem definition Given a graph G=(V,E) at time t
Or a series of snapshots of the graph at times ti<=t
Describe the state of the graph at time t'>t
Sometimes, assume V stays the same and E changes

3. Applications
Accelerating formation of connections in professional social networks

4. Applications
Helping find your offline friends online

5. Applications Increase server efficiency through pre-fetching
Determining which links are missing in Wikipedia pages (or other educational resources)
Monitor/control propagation of computer viruses
Fixing corrupted data
You bought five books, one of the titles is lost, can we infer it?
...

6. Basic method Assign a score to every possible link (u,v)
Sort links by descending score
Predict the top-k links
Or the links with scores above a threshold
Profit!

7. Common neighbors
Newman 2001: The probability of scientists collaborating increases with the number of other collaborators they have in common.
Tendency to close triangles, more on this later ...

8. Jaccard similarity
Correct common neighbors by reducing the influence of nodes with many neighbors

9. Adamic/Adar
Count common neighbors but weight down nodes with too many neighbors
The idea is to avoid this

10. Preferential attachment
“Rich-get-richer”
Newman 2001: the probability of two authors collaborating is proportional to the product of their number of collaborators

11. Example: score(v5,v7) Exercise, compute: Number of common neighbors
Jaccard coefficient
Adamic and Adar's
Preferential attachment
Common: 1
Jaccard: 1 / 2 = 0.5
Adamic and Adar: 1/log(3)
Preferential attachment: 1 x 2 = 2

12. Geodesic/shortest path distance
Assumption: social connections are formed by following edges, then finding a new person, then connecting directly
score(u,v) := -(length of shortest path from u to v)
Limit case: triadic closure

13. Katz 1953 or “rooted PageRank”
Score based on weighted counts of paths, with exponential decay on path length. For α < 1
A small α yields predictions which are similar to common neighbors

14. More on random walks Hitting time
Hu,v = expected steps of random walk from u to v
To reduce the influence of well-connected nodes, we can multiply by the probability of a node in stationary state

15. Symmetric hitting time (commute time)
Hitting time is not symmetric, we can symmetrize easily

16. SimRank [Jeh 2002] For directed graphs; follows inlinks
Exercise, compute:
simrank(u,v)
simrank(v,w)
simrank(u,w) u p v q s w r

17. Meta-method / prunning
Compute score(u,v) for all existing edges assuming they do not exist
Remove k% with lower score
Compute score(u,v) in the reduced graph

18. Evaluating link prediction methods
After one of the aforementioned measures is selected, a list of the top most similar pairs of nodes are selected.
These pairs of nodes denote edges predicted to be the most likely to soon appear in the network.
Performance (precision, recall, or accuracy) can be evaluated using the testing graph and by comparing the number of the testing graph’s edges that the link prediction algorithm successfully reveals.
Performance is usually very low, since many edges are created due to reasons not solely available in a social network graph.
So, a common baseline is to compare the performance with random edge predictors and report the factor improvements over random prediction.

19. Performance comparison [Liben-Nowell et al. 2003]
Notes:
Effectiveness in general is very low (challenging problem)

20. Adamic/Adar + content

21. Supervised learning [Hassan et al. 2006]
Input features are all attributes, possibly including node-links as attribute
Predict connected/not-connected learning on a sub-set of the data

22. Example experimental results with supervised learning
Data: co-authorship network in DBLP and BIOBASE
Split into two disjoint ranges of publication years (Ra, Rb)
Example: DBLP, Ra = [1999,2000] Rb=[2001,2004]
Training item is a pair of authors (u,v), both with a paper in Ra, and all their attributes computed in Ra
Ground truth is whether (u,v) co-author during Rb
Positive=yes, Negative=no

23. Example features Content similarity
Keywords in common, conferences in common, ...
Aggregation features
Sum of papers, Sum of neighbors, ...
Topological distance
Shortest-path length, ...

24. Performance results under various learning schemes (same feature set)

25. Community prediction

26. Community membership prediction
Why do users join communities?
What factors affect the community-joining behavior of individuals?
We can observe users who join communities and determine the factors that are common among them
We require a population of users, a community C, and community membership information (i.e., users who are members of C).
To distinguish between users who have already joined the community and those who are now joining it, we need community memberships at two different times t1 and t2, with t2 > t1.
At t2, we determine users such as u who are currently members of the community, but were not members at t1. These new users form the subpopulation that is analyzed for community-joining behavior.

27. Peer influence
Hypothesis: individuals are inclined toward an activity when their friends are engaged in the same activity.
A factor that plays a role in users joining a community is the number of their friends who are already members of the community.

28. Supervised learning

29. Example regression tree

30. Beyond community membership
Communities can be implicit: One can think of individuals buying a product as a community, and people buying the product for the first time as individuals joining the community
Collective Behavior: A group of individuals behaving in a similar way (first defined by sociologist Robert Park)
It can be planned and coordinated, but often is spontaneous and unplanned
Examples:
Individuals standing in line for a new product release
Posting messages online to support a cause or to show support for an individual
Approach can be similar to community membership prediction