Methods to address the link prediction problem on graphs. Lecture at M Sc. Data Science at Sapienza University of Rome, Spring 2016.

1. Link Prediction
Class Data Mining Technology for Business and Society
Program M. Sc. Data Science
University Sapienza University of Rome
Semester Spring 2016
Lecturer Carlos Castillo http://chato.cl/
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
1)Assign a score to every possible link (u,v)
2)Sort links by descending score
3)Predict the top-k links
Or the links with scores above a threshold
4)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. Understanding the Adamic/Adar
heuristic
8 followers
1000
followers
Prolific common friends
Weaker evidence
Less prolific
Stronger evidence
Alice
Bob
Charlie

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

12. Example: score(v5
,v7
)
Exercise, compute:
● Number of common neighbors
● Jaccard coefficient
● Adamic and Adar's
● Preferential attachment

13. 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

14. 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

15. 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

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

17. Graph projections

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

19. 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

20. 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.

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

22. Adamic/Adar + content

23. 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

24. 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

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

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

27. Community prediction

28. 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.

29. 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.

30. Supervised learning

31. Example regression tree

32. 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