k-Nearest Neighbors in Uncertain Graphs (Michalis Potamias, Francesco Bonchi, Aristides Gionis, George Kollios)
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This thesis explores k-nearest neighbors in uncertain graphs, emphasizing the importance of probabilistic models in understanding complex networks, particularly protein-protein interaction networks. It discusses methodologies for defining distance and sampling algorithms to identify reliable neighbors in these probabilistic contexts. The research demonstrates substantial improvements in computational efficiency through advanced pruning techniques, resulting in significant speedups during processing.
![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](https://image.slidesharecdn.com/vldb2010-michalisrevisited-100924102932-phpapp02/85/k-Nearest-Neighbors-in-Uncertain-Graphs-Michalis-Potamias-Francesco-Bonchi-Aristides-Gionis-George-Kollios-42-320.jpg)
![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](https://image.slidesharecdn.com/vldb2010-michalisrevisited-100924102932-phpapp02/85/k-Nearest-Neighbors-in-Uncertain-Graphs-Michalis-Potamias-Francesco-Bonchi-Aristides-Gionis-George-Kollios-43-320.jpg)
k-Nearest Neighbors in Uncertain Graphs (Michalis Potamias, Francesco Bonchi, Aristides Gionis, George Kollios)
- 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