Complex networks, such as biological, social, and communication networks, often entail uncertainty, and thus, can be modeled as probabilistic graphs. Similar to the problem of similarity search in standard graphs, a fundamental problem for probabilistic graphs is to efficiently answer k-nearest neighbor queries (k-NN), which is the problem of computing the k closest nodes to some specific node. In this paper we introduce a framework for processing k-NN queries in probabilistic graphs. We propose novel distance functions that extend well-known graph concepts, such as shortest paths. In order to compute them in probabilistic graphs, we design algorithms based on sampling. During k-NN query processing we efficiently prune the search space using novel techniques. Our experiments indicate that our distance functions outperform previously used alternatives in identifying true neighbors in real-world biological data. We also demonstrate that our algorithms scale for graphs with tens of millions of edges.

1. k-Nearest Neighbors
in Uncertain Graphs
Michalis Potamias Francesco Bonchi
Aristides Gionis George Kollios

2. Thesis
• Many complex networks are modeled as
probabilistic (i.e., uncertain) graphs.
• The probabilistic treatment of such graphs leads
to better understanding of real data.
Nearest Neighbors in Uncertain Graphs @ VLDB 2010 2

3. Probabilistic Protein-Protein
Interaction Networks
Possible interactions between
proteins are established
through biological experiments
that entail uncertainty.
The edge probability
represents that uncertainty.
A
0.2 0.6
0.4
B C
0.3 0.7
D
Source: Asthana et al., Genome Research 2004
Nearest Neighbors in Uncertain Graphs @ VLDB 2010 3

4. Probabilistic Protein-Protein
Interaction Networks
• Neighbors of a given node in a standard graph?
– Nodes close in terms of shortest path distance!
A
• How do we define neighbors
0.2 0.6
in probabilistic graphs?
0.4
B C
• How do we define the distance?
0.3 0.7
D
– Treat them as weighted graphs (N06)
– Nodes with high reliability(GR04)
– Most probable path (BI03)
– …shortest paths? (VLDB10)
Nearest Neighbors in Uncertain Graphs @ VLDB 2010 4

5. Probabilistic Protein-Protein
Interaction Networks
• Why is it important to find good neighbors of
proteins in PPI networks?
– Detection of candidate co-complex relationships.
– Actual co-complex relationships can be
established through experiments in the lab.
Nearest Neighbors in Uncertain Graphs @ VLDB 2010 5

6. Outline
• Thesis
• Probabilistic PPI Networks
• Distance Definition
• Sampling Algorithms
• kNN Pruning
• Experiments
Nearest Neighbors in Uncertain Graphs @ VLDB 2010 6

7. Outline
• Thesis
• Probabilistic PPI Networks
• Distance Definition
• Sampling Algorithms
• kNN Pruning
• Experiments
Nearest Neighbors in Uncertain Graphs @ VLDB 2010 7

8. A
0.6
B
0.2
0.4
C
Distance Definition
0.3 0.7
D
A A A A A A A A
B C B C B C B C B C B C B C B C
D D D D D D D D
A A A A A A A A
B C B C B C B C B C B C B C B C
D D D D D D D D
A A A A A A A A
B C B C B C B C B C B C B C B C
D D D D D D D D
A A A A A A A A
B C B C B C B C B C B C B C B C
D D D D D D D D
Nearest Neighbors in Uncertain Graphs @ VLDB 2010 8

9. Distance Definition
the graph
A
0.2 0.6
0.4
B C
0.3 0.7
D
Nearest Neighbors in Uncertain Graphs @ VLDB 2010 9

10. Distance Definition
the graph a world
A A
0.2 0.6 Pr(world ) p( A, B) p( B, D)
0.4 (1 p( B, C )) (1 p(C , D)) (1 p( A, D))
B C B C
0.3 0.7
D D
Nearest Neighbors in Uncertain Graphs @ VLDB 2010 10

11. Distance Definition
the graph a world
A A
0.2 0.6 Pr(world ) p( A, B) p( B, D)
0.4 (1 p( B, C )) (1 p(C , D)) (1 p( A, D))
B C B C
0.3 0.7
D D
PDF .44
.3
.26
1 2 inf
shortest path length d(B,D)
Nearest Neighbors in Uncertain Graphs @ VLDB 2010 12

12. Distance Definition
• Use well known statistics of the Shortest Path
PDF:
– Median
– Majority (mode)
– ExpectedReliable
• infinity problem
PDF
• Hard! they require .44
d med 2
.3
explicit enumeration .26
d maj inf
of possible worlds:
d exp 1.46
resort to sampling! 1 2 inf
shortest path length d(B,D)
Nearest Neighbors in Uncertain Graphs @ VLDB 2010 14

13. Outline
• Thesis
• Probabilistic PPI Networks
• Distance Definition
• Sampling Algorithms
• kNN Pruning
• Experiments
Nearest Neighbors in Uncertain Graphs @ VLDB 2010 15

14. Sampling Algorithms
1. sample (a small number of) worlds
2. compute sample median (approximation)
3. output result
– Median (Chernoff bound)
– ExpectedReliable (Hoeffding inequality)
– Majority (No bound)
Nearest Neighbors in Uncertain Graphs @ VLDB 2010 16

15. Sampling Algorithms
BIOMINE FLICKR
database of biological entities users from flickr.com. edges have
and uncertain interactions from been created assuming homophily
UHelsinki based on jaccard of flickr groups
1M nodes, 10M edges 77K nodes, 20M edges
Nearest Neighbors in Uncertain Graphs @ VLDB 2010 17

16. Outline
• Thesis
• Probabilistic PPI Networks
• Distance Definition
• Sampling Algorithms
• kNN Pruning
• Experiments
Nearest Neighbors in Uncertain Graphs @ VLDB 2010 18

17. kNN Pruning
• Query: Given a probabilistic graph, and a
source node find the set of k nodes closest to
the source.
• Naïve algorithm:
1. sample worlds
2. run dijkstra traversals and compute a pdf of the sp
distance per node
3. calculate the median distance to all nodes using the
pdf’s
4. compute k-nn
Nearest Neighbors in Uncertain Graphs @ VLDB 2010 19

18. kNN Pruning naive
1nn - median
node: A
sample: 5 worlds
E 0.5
D
0.6
0.8
B 0.3
0.9
A G
0.3
0.7
C
0.4
F
Nearest Neighbors in Uncertain Graphs @ VLDB 2010 20

19. kNN Pruning naive
E
D
1nn - median
B
node: A
A G
sample: 5 worlds
C
F
E 0.5
D
0.6
0.8
B 0.3
0.9
A G
0.3
1 2 3
0.7
C B C D E F G
0.4
F
Nearest Neighbors in Uncertain Graphs @ VLDB 2010 21

20. kNN Pruning naive
E E
D D
1nn - median
B B
node: A
A G A G
sample: 5 worlds
C C
F F
E 0.5
D
0.6
0.8
B 0.3
0.9
A G
0.3
1 2 3
0.7
C B C D E F G
0.4
F
Nearest Neighbors in Uncertain Graphs @ VLDB 2010 22

21. kNN Pruning naive
E E E
D D D
1nn - median
B B B
node: A
A G A G A G
sample: 5 worlds
C C C
F F F
E 0.5
D
0.6
0.8
B 0.3
0.9
A G
0.3
1 1 2 2 3 2 2
0.7
C B C D E F G
0.4
F
Nearest Neighbors in Uncertain Graphs @ VLDB 2010 23

22. kNN Pruning naive
E E E E
D D D D
1nn - median
B B B B
node: A
A G A G A G A G
sample: 5 worlds
C C C C
F F F F
E 0.5
D
0.6
0.8
B 0.3
0.9
A G
0.3
1 1 2 2 3 2 2
0.7
C B C D E F G
0.4
F
Nearest Neighbors in Uncertain Graphs @ VLDB 2010 24

23. kNN Pruning naive
E E E E E
D D D D D
1nn - median
B B B B B
node: A
A G A G A G A G A G
sample: 5 worlds
C C C C C
F F F F F
E 0.5
D
0.6
0.8
B 0.3
0.9
A G
0.3
1 1 2 2 3 2 2
0.7
C B C D E F G
0.4
F
Nearest Neighbors in Uncertain Graphs @ VLDB 2010 25

24. kNN Pruning naive
E E E E E
D D D D D
1nn - median
B B B B B
node: A
A G A G A G A G A G
sample: 5 worlds
C C C C C
3 F F F F F
E 0.5
0.6
D 2
0.8
1 B 0.3
0.9
A G
0.3
1 1 2 2 3 2 2
0.7
C B C D E F G
0.4
F
Nearest Neighbors in Uncertain Graphs @ VLDB 2010 26

25. kNN Pruning naive
E E E E E
D D D D D
1nn - median
B B B B B
node: A
A G A G A G A G A G
sample: 5 worlds
C C C C C
3 F F F F F
E 0.5
0.6
D 2
0.8
1 B 0.3
0.9
A G
0.3
1 1 2 2 3 2 2
0.7
C B C D E F G
0.4
F
Nearest Neighbors in Uncertain Graphs @ VLDB 2010 27

26. kNN Pruning
1nn - median
node: A
sample: 5 worlds
E 0.5
D
0.6
0.8
• algorithm
B 0.3
0.9
– sample worlds on the fly
– increase the horizon of each dijkstra one hop at a
A G
time
0.3
0.7 – maintain truncated pdf histograms
C
0.4
F
Nearest Neighbors in Uncertain Graphs @ VLDB 2010 28

27. kNN Pruning
1nn - median
node: A
sample: 5 worlds
E 0.5
D
0.6
0.8
B 0.3
0.9
A G
0.3
0.7
C
0.4
F
Nearest Neighbors in Uncertain Graphs @ VLDB 2010 29

28. kNN Pruning
1nn - median
B
node: A
A
sample: 5 worlds
E 0.5
D
0.6
0.8
B 0.3
0.9
A G
0.3
1
0.7
C B
0.4
F
Nearest Neighbors in Uncertain Graphs @ VLDB 2010 30

29. kNN Pruning
1nn - median
B B
node: A
A
sample: 5 worlds A
E 0.5
D
0.6
0.8
B 0.3
0.9
A G
0.3
1
0.7
C B
0.4
F
Nearest Neighbors in Uncertain Graphs @ VLDB 2010 31

30. kNN Pruning
1nn - median
B B B
node: A
A
sample: 5 worlds A A
C
E 0.5
D
0.6
0.8
B 0.3
0.9
A G
0.3
1 1
0.7
C B C
0.4
F
Nearest Neighbors in Uncertain Graphs @ VLDB 2010 32

31. kNN Pruning
1nn - median
B B B B
node: A
A
sample: 5 worlds A A A
C C
E 0.5
D
0.6
0.8
B 0.3
0.9
A G
0.3
1 1
0.7
C B C
0.4
F
Nearest Neighbors in Uncertain Graphs @ VLDB 2010 33

32. kNN Pruning
1nn - median
B B B B
node: A
A A
sample: 5 worlds A A A
C C
E 0.5
D
0.6
0.8
B 0.3
0.9
A G
0.3
1 1
0.7
C B C
0.4
F
Nearest Neighbors in Uncertain Graphs @ VLDB 2010 34

33. kNN Pruning
1nn - median
B B B B
node: A
A A
sample: 5 worlds A A A
C C
E 0.5
D
0.6
0.8
1 B 0.3
0.9
A G
0.3
1 1
0.7
>1 C B C
0.4
F
Nearest Neighbors in Uncertain Graphs @ VLDB 2010 35

34. kNN Pruning
1nn - median
B B B B
node: A
A A
sample: 5 worlds A A A
C C
E 0.5
0.6
D •B has distance 1
0.8 •C has distance greater than 1
1 B 0.3
•D, E, F, G, … were not discovered (d>1)
0.9
•1NN set is complete with B – no need to cont
A G •just 2 nodes visited (and 2 histograms
0.3
1 1 maintained)
0.7
•worlds were only partially instantiated
>1 C B C •same answer as the naive
0.4
F •with a small cost: dijkstra state needs to be
maintained in memory for all worlds
Nearest Neighbors in Uncertain Graphs @ VLDB 2010 36

35. kNN Pruning
for 200 worlds and 5NN the speedups were:
247x (BIOMINE), 111x (FLICKR), 269x (DBLP)
BIOMINE FLICKR DBLP
database of biological entities users from flickr.com. edges have authors from dblp. probabilities
and uncertain interactions from been created assuming homophily have been assigned based on
UHelsinki based on jaccard of flickr groups number of coauthored papers
1M nodes, 10M edges 77K nodes, 20M edges 226K nodes, 1.4M edges
Nearest Neighbors in Uncertain Graphs @ VLDB 2010 37

36. kNN Pruning
for 200 worlds and 5NN the speedups were:
247x (BIOMINE), 111x (FLICKR), 269x (DBLP)
BIOMINE FLICKR DBLP
database of biological entities users from flickr.com. edges have authors from dblp. probabilities
and uncertain interactions from been created assuming homophily have been assigned based on
UHelsinki based on jaccard of flickr groups number of coauthored papers
1M nodes, 10M edges 77K nodes, 20M edges 226K nodes, 1.4M edges
Nearest Neighbors in Uncertain Graphs @ VLDB 2010 38

37. Less uncertainty, more pruning
Nearest Neighbors in Uncertain Graphs @ VLDB 2010 39

38. Less uncertainty, more pruning
A A
•boost probabilities of edges by d d
0.2 0.6 1-0.8 1-0.4
giving each edge d chances 0.4
d
1-0.6
B C B C
•d=1: original graph
0.3 0.7 d
•increasing d, p goes to 1 1-0.7 d
1-0.3
D D
Nearest Neighbors in Uncertain Graphs @ VLDB 2010 40

39. Less uncertainty, more pruning
A A
•boost probabilities of edges by d d
0.2 0.6 1-0.8 1-0.4
giving each edge d chances 0.4
d
1-0.6
B C B C
•d=1: original graph
0.3 0.7 d
•increasing d, p goes to 1 1-0.7 d
1-0.3
D D
Nearest Neighbors in Uncertain Graphs @ VLDB 2010 41

40. Less uncertainty, more pruning
A A
•boost probabilities of edges by d d
0.2 0.6 1-0.8 1-0.4
giving each edge d chances 0.4
d
1-0.6
B C B C
•d=1: original graph
0.3 0.7 d
•increasing d, p goes to 1 1-0.7 d
1-0.3
D D
Nearest Neighbors in Uncertain Graphs @ VLDB 2010 42

41. Outline
• Thesis
• Probabilistic PPI Networks
• Distance Definition
• Sampling Algorithms
• kNN Pruning
• Experiments
Nearest Neighbors in Uncertain Graphs @ VLDB 2010 43

42. Experiments
• Dataset
– Probabilistic PPI network
[Krogan et al, Nature 06]
– Protein co-complex
relationships (ground truth)
[Mewes et al, Nuc Acids Res 04]
• Experiment
– Choose a ground truth edge
(A,B)
– Choose a node C s.t. there is
no ground truth edge (A,C)
– Classification task: Distinguish
between the two types of
edges: (A,B) and (A,C)
Nearest Neighbors in Uncertain Graphs @ VLDB 2010 44

43. Experiments
• Dataset
– Probabilistic PPI network
[Krogan et al, Nature 06]
– Protein co-complex
relationships (ground truth)
[Mewes et al, Nuc Acids Res 04]
• Experiment
– Choose a ground truth edge
(A,B)
– Choose a node C s.t. there is
no ground truth edge (A,C)
– Classification task: Distinguish
between the two types of
edges: (A,B) and (A,C)
Nearest Neighbors in Uncertain Graphs @ VLDB 2010 45

44. Conclusion
• Probabilistic graph analysis benefits from
possible-world semantics.
– Extended standard graph concepts to
probabilistic graphs and designed
approximation algorithms to compute them
– Introduced novel pruning algorithms for kNN
in probabilistic graphs
– Confirmed the efficacy of our framework on
real data.
Nearest Neighbors in Uncertain Graphs @ VLDB 2010 46

45. Future Work
• Enrich model
– Node probabilities
– Arbitrary PDFs
• Explore random walks further
Nearest Neighbors in Uncertain Graphs @ VLDB 2010 47

46. Thank you!
?
Nearest Neighbors in Uncertain Graphs @ VLDB 2010 48