Outline Link Prediction Supervised Rank Aggregation based Link Prediction Experiment Conclusion Supervised Rank Aggregation Approach for Link Prediction in Complex Networks Manisha Pujari & Rushed Kanawati LIPN - UMR CNRS 7030 Universit´ e Paris Nord 99 Av. J.B. Clement 93430, Villetaneuse, FRANCE manisha.pujari@lipn.univ-paris13.fr 16 April, 2012 Mining Social Network Dynamics Workshop WWW-2012, Lyon,France M.Pujari & R.Kanawati Supervised Rank Aggregation Approach for Link Prediction 1/22
Outline Link Prediction Supervised Rank Aggregation based Link Prediction Experiment Conclusion Link Prediction 1 Supervised Rank Aggregation based Link Prediction 2 Experiment 3 Conclusion 4 M.Pujari & R.Kanawati Supervised Rank Aggregation Approach for Link Prediction 2/22
Outline Link Prediction Supervised Rank Aggregation based Link Prediction Experiment Conclusion Problem Link Prediction Predicting new links between nodes of a graph. Applications Recommender systems Academic/Professional collaborations Identification of structures of criminal networks Biological networks M.Pujari & R.Kanawati Supervised Rank Aggregation Approach for Link Prediction 3/22
Outline Link Prediction Supervised Rank Aggregation based Link Prediction Experiment Conclusion Link Prediction Approaches Dyadic : Computation of link score for unlinked vertices Structural : Mining rules for evolution of sub-graphs Topology based : Attributes computed for graph Node-feature based : Attributes computed for nodes Hybrid : Combination of the two Temporal : Consider dynamics of the networks Static : Do not consider the dynamics of a network M.Pujari & R.Kanawati Supervised Rank Aggregation Approach for Link Prediction 4/22
Outline Link Prediction Supervised Rank Aggregation based Link Prediction Experiment Conclusion Link Prediction Approaches Dyadic : Computation of link score for unlinked vertices Topology based : Attributes computed for graph Temporal : Consider dynamics of the networks M.Pujari & R.Kanawati Supervised Rank Aggregation Approach for Link Prediction 4/22
Outline Link Prediction Supervised Rank Aggregation based Link Prediction Experiment Conclusion Dyadic Topological Approaches Work of [Liben-Nowell & al.,2007] Prediction on a co-authorship network. For each unlinked node pair ( u , v ) , compute a set of topological attributes [ A 1 , A 2 , ..., A n ]. Rank all ( u , v ) based on attribute values. Considering only top k ranked edges as predicted edges, performance of each attribute is found. Attributes : Neighborhood-based attributes: Jaccard’s coefficient,Common neighbors,Adamic/Adar [Adamic & al.2003], Preferential attachment etc. Shortest path distance,Katz [Katz,1953], Distance-based attributes: Maximum forest algorithm etc. Centrality-based attributes: PageRank, Degree centrality, Clustering coefficient etc. M.Pujari & R.Kanawati Supervised Rank Aggregation Approach for Link Prediction 5/22
Outline Link Prediction Supervised Rank Aggregation based Link Prediction Experiment Conclusion Dyadic Topological Approaches Combining the effect of different topological measures: Application of supervised machine learning algorithms [Benchettara & al.,2010],[Hasan & al., 2006] Examples: ( Node x , Node y ) − → [ a 0 , a 1 , a 2 , ...., a n ] M.Pujari & R.Kanawati Supervised Rank Aggregation Approach for Link Prediction 6/22
Outline Link Prediction Supervised Rank Aggregation based Link Prediction Experiment Conclusion Dyadic Topological Approaches Combining the effect of different topological measures: Application of supervised machine learning algorithms [Benchettara & al.,2010],[Hasan & al., 2006] Examples: ( Node x , Node y ) − → [ a 0 , a 1 , a 2 , ...., a n ] Can we apply rank aggregation methods ? M.Pujari & R.Kanawati Supervised Rank Aggregation Approach for Link Prediction 6/22
Outline Link Prediction Supervised Rank Aggregation based Link Prediction Experiment Conclusion Rank Aggregation (Social choice theory) ⇒ To find an aggregated list with minimum possible disagreement ⇒ Equal weight to all experts Expert 1 = ⇒ L 1 = [ A , B , C , D ] Expert 2 = ⇒ L 2 = [ B , D , A , C ] Expert 3 = ⇒ L 3 = [ C , D , A , B ] Types of input lists ... Full/Complete lists ... Partial lists ... Disjoint lists Expert n = ⇒ L n = [ D , C , A , B ] ——————————————— L aggregate = [? , ? , ? , ?] M.Pujari & R.Kanawati Supervised Rank Aggregation Approach for Link Prediction 7/22
Outline Link Prediction Supervised Rank Aggregation based Link Prediction Experiment Conclusion Distance Measure Spearman Footrule Distance: F ( L 1 , L 2 ) = Σ i ∈ n | L 1 ( i ) − L 2 ( i ) | Kendall Tau Distance: K ( L 1 , L 2 ) = | ( i , j ) s . t . L 1 ( i ) < L 2 ( j ) & L 1 ( i ) > L 2 ( j ) | Example: L 1 = [A, B, C, D] and L 2 = [B, D, C, A] F ( L 1 , L 2 ) = | L 1 (A) - L 2 (B) | + | L 1 (B) - L 2 (B) | + | L 1 (C) - L 2 (C) | + | L 1 (D) - L 2 (D) | = 7 K ( L 1 , L 2 ) = 4 M.Pujari & R.Kanawati Supervised Rank Aggregation Approach for Link Prediction 8/22
Outline Link Prediction Supervised Rank Aggregation based Link Prediction Experiment Conclusion Borda’s Method [Borda, 1781] Based on absolute positioning of elements k � B L k ( i ) = { count ( j ) | L k ( j ) < L k ( i )& j ∈ L k } ; B ( i ) = B L t ( i ) (1) t =1 M.Pujari & R.Kanawati Supervised Rank Aggregation Approach for Link Prediction 9/22
Outline Link Prediction Supervised Rank Aggregation based Link Prediction Experiment Conclusion Kemeny Optimal Aggregation [Dwork & al.,2001] Based on relative ranking of elements � SK ( π, L 1 , L 2 , L 3 , ....., L n ) = K ( π, L i ) (2) i ∈ [1 , n ] M.Pujari & R.Kanawati Supervised Rank Aggregation Approach for Link Prediction 10/22
Outline Link Prediction Supervised Rank Aggregation based Link Prediction Experiment Conclusion Supervised Rank Aggregation Combining different rankings to get an aggregation giving different weights to the experts ⇒ Proposed approaches Supervised Borda Supervised local Kemeny w 1 ← Expert 1 = ⇒ L 1 → [ k elements ] w 2 ← Expert 2 = ⇒ L 2 → [ k elements ] w 3 ← Expert 3 = ⇒ L 3 → [ k elements ] ... ... ... w n ← Expert n = ⇒ L n → [ k elements ] M.Pujari & R.Kanawati Supervised Rank Aggregation Approach for Link Prediction 11/22
Outline Link Prediction Supervised Rank Aggregation based Link Prediction Experiment Conclusion Supervised Borda Method Borda score n � B ( i ) = w i ∗ B L t ( i ) ; where t ∈ [1 , k ] (3) t =1 M.Pujari & R.Kanawati Supervised Rank Aggregation Approach for Link Prediction 12/22
Outline Link Prediction Supervised Rank Aggregation based Link Prediction Experiment Conclusion Supervised Local Kemeny Aggregation Steps: L = [ L 1 , L 2 , . . . , L n ], [ w 1 , w 2 , . . . , w n ] , m elements( U ) 1 Initialize m × m matrix M with M ( x , y ) = 0 2 ∀ ( x , y ) ∈ U , Compute 3 score ( x , y ) = � n i =1 ( w i ∗ ( x ≻ y )) where � 0 if L i ( x ) < L i ( y ) x ≻ y = 1 if L i ( x ) > L i ( y ) If score ( x , y ) > 0 . 5 ∗ � n i =1 w i , Insert M ( x , y ) = true and 4 M ( y , x ) = false Initial aggregation R = L 1 5 For x , y ∈ R , Swap ( x , y ) if M ( x , y ) = false 6 R is the final aggregation. 7 M.Pujari & R.Kanawati Supervised Rank Aggregation Approach for Link Prediction 13/22
Outline Link Prediction Supervised Rank Aggregation based Link Prediction Experiment Conclusion Supervised Local Kemeny Aggregation M.Pujari & R.Kanawati Supervised Rank Aggregation Approach for Link Prediction 14/22
Outline Link Prediction Supervised Rank Aggregation based Link Prediction Experiment Conclusion Link Prediction based on Supervised Rank Aggregation Examples : ( Node x , Node y ) − → [ a 0 , a 1 , a 2 , ...., a n ] Steps: Rank learning examples by attribute values 1 Consider only top k examples and compute attribute weight w a i 2 Rank test examples by attribute to get n ranked lists 3 Apply supervised rank aggregation 4 Consider only top k examples of the aggregate list and compute 5 performance. M.Pujari & R.Kanawati Supervised Rank Aggregation Approach for Link Prediction 15/22
Outline Link Prediction Supervised Rank Aggregation based Link Prediction Experiment Conclusion Link Prediction Based on Supervised Rank Aggregation Computation of attribute weights: Maximization of positive precision: W a i = n ∗ Precision a i (4) Minimization of false positive rate: n W a i = (5) FPR a i M.Pujari & R.Kanawati Supervised Rank Aggregation Approach for Link Prediction 16/22
Outline Link Prediction Supervised Rank Aggregation based Link Prediction Experiment Conclusion Experiment DBLP database Training Validation Training examples Test examples Datasets Time Time Positive Total Positive Total Dataset 1 [1970-1975] [1971-1976] 30 1693 41 3471 Dataset 2 [1972-1977] [1973-1978] 87 19332 82 18757 Dataset 3 [1974-1979] [1975-1980] 102 35190 164 60046 Table: DBLP Datasets Performance measure: F = Precision ∗ Recall (6) Precision + Recall M.Pujari & R.Kanawati Supervised Rank Aggregation Approach for Link Prediction 17/22
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