Fast Reliability Search in Uncertain Graphs Arijit Khan, Francesco Bonchi, Aristides Gionis, Francesco Gullo

Transcription

1 Fast Reliability earch in ncertain Graphs Arijit Khan, Francesco Bonchi, Aristides Gionis, Francesco Gullo ystems Group, ETH Zurich Yahoo Labs, pain Aalto niversity, Finland

2 ncertain Graphs T ocial Network Traffic Network Ad-hoc Mobile Network Protein-interaction Network ncertain Graph 1

3 Motivation Mobile Ad-hoc Network: find the set of sink nodes where a source node can deliver a packet with high probability T Traffic Network: find a set of target locations reachable from a source location with high probability ocial Network: find a set of users who could be influenced with high probability by a target user Packet Delivery Probability in Mobile Ad-hoc Network 2

4 Motivation Mobile Ad-hoc Network: find the set of sink nodes where a source node can deliver a packet with high probability T Traffic Network: find a set of target locations reachable from a source location with high probability ocial Network: find a set of users who could be influenced with high probability by a target user Packet Delivery Probability in Mobile Ad-hoc Network 2

5 Reliability in ncertain Graphs T T ample Edges ncertain Graph Certain Graph (Possible orld) 3

6 Reliability in ncertain Graphs T T ample Edges ncertain Graph Certain Graph (Possible orld) Identity Function 3

7 Reliability earch in ncertain Graphs Given an uncertain graph G, a probability threshold ɳ ϵ (0, 1), and a source node in G, find all nodes in G that are reachable from with probability greater than or equal to threshold ɳ #P - complete 4

8 Related ork Two-terminal reliability All-terminal reliability K-terminal reliability Monte-Carlo (MC) sampling Distance-constraint reliability RHT sampling (Jin et. al., LDB 2011) 5

9 Baseline MC imulation + BF T MC ampling + BF ncertain Graph T Number of amples Certain Graph (Possible orld) 6

10 Can e Be More Efficient? Given a source node and a probability threshold ɳ ϵ (0, 1), can we quickly determine the nodes that are certainly not reachable from with probability greater than or equal to ɳ T Indexing (offline) Filtering + erification (Online) ɳ = ncertain Graph 7

11 RQ-Tree Index T 0.3 ɳ = ncertain Graph out (, *)=0 out (, *)=0.496 out (, *)=0.8 out (, *)=0.8,,,, T ɳ =,,,T, T RQ-Tree Index 7

12 RQ-Tree: Filtering out (, *) =0 out (, *) =0.496,,,,,, T ɳ =,T Max-Flow Min-Cut Based pper Bound: Edge Capacity: out (, *) =0.8 out (, *) =0.8, T RQ-Tree Index c(a) = log (1 p(a)) Compute Max-Flow f from to Outside Cluster C out (, C) = 1 exp(-f) 8

13 RQ-Tree: Filtering out (, *) =0 out (, *) =0.496,,,,,, T ɳ =,T Max-Flow Min-Cut Based pper Bound: Edge Capacity: out (, *) =0.8 out (, *) =0.8, T RQ-Tree Index c(a) = log (1 p(a)) Compute Max-Flow f from to Outside Cluster C out (, C) = 1 exp(-f) Benefits: No false negative (recall = 1) Computation limited only inside cluster C Incremental Max-Flow computation 8

14 RQ-Tree: erification ampling-based erification: MC-ample + BF over the sub-graph formed by the candidate set Pros: high precision, high recall Cons: verification could still be relatively expensive 0.2 Lower-Bound-based erification: Most-Likely-Path Pros: precision = 1, high efficiency Cons: lower recall Pr(--) = * 0.2 = 0.10 Pr(--) = 0.7 * 0.3 = 0.21 Most-Likely-Path: (--) 9

15 RQ-Tree: Online Complexity MC ampling Recursive ampling [LDB 11] RQ-Tree + MC-ampling-based erification [Our Method] O ( m n K ( m n )) RQ-tree + Lower-Bound-based erification [Our Method] O ( m n O ( K ( m n)) O ( n 2 d ) ) K = No of amples m = No of edges n = No of nodes n = No of nodes in the candidate set m = No of edges induced by the candidate nodes d = Diameter of the graph 10

16 RQ-Tree Index Construction 0.1,,,, T T,,,T, T ncertain Graph RQ-Tree Index Hierarchical Clustering: Minimum-cut balanced bi-partition using METI Edge weight: w(a) = log (1 p(a)) 11

17 Experimental Results # Nodes # Edges #Arc Prob: Mean, D, Quartiles DBLP ± 0.11, {0.09, 0.09, 0.18} Flickr ± 0.06, {0.06, 0.07, 0.09} BioMine ± 0.21, {0.12, 0.22, 0.36} Dataset Characteristics 12

18 Accuracy Results RQ-Tree-MC RQ-Tree-LB ɳ=0.4 ɳ= ɳ=0.8 ɳ=0.4 ɳ= ɳ=0.8 DBLP Flickr BioMine Precision RQ-Tree-MC RQ-Tree-LB ɳ=0.4 ɳ= ɳ=0.8 ɳ=0.4 ɳ= ɳ=0.8 DBLP Flickr BioMine Recall 13

19 Efficiency Results RQ-Tree-MC RQ-Tree-LB MC ɳ=0.4 ɳ= ɳ=0.8 ɳ=0.4 ɳ= ɳ=0.8 All ɳ DBLP Flickr BioMine Online query-processing time (sec) 14

20 Pruning Capacity of Filtering Phase Precision of Filtering Phase 15

21 RQ-Tree in Influence Maximization RQ-Tree index in multi-source reliability query and in influence maximization Expected pread (Last.FM) Top-k eed Finding Time (Last.FM) 16

22 Conclusion Indexing method for answering online reliability queries efficiently and effectively. RQ-tree works very well with lower arc probabilities and with higher probability threshold. In future, we shall study reliability search queries when the arc probabilities are not independent. 17

23 Questions?