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# Predicting Directed Links using Nondiagonal Matrix Decompositions

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### Predicting Directed Links using Nondiagonal Matrix Decompositions

1. 1. Web Science & Technologies University of Koblenz ▪ Landau, Germany Predicting Directed Links usingNondiagonal Matrix Decompositions Jérôme Kunegis & Jörg Fliege Int. Conf. on Data Mining 2012
2. 2. Trust Prediction ? Goal: predict trusted edges Jérôme Kunegis & Jörg Fliege Predicting Directed Links using Nondiagonal Matrix Decompositions kunegis@uni-koblenz.de ICDM 2012 2
3. 3. Triangle Closing ? Jérôme Kunegis & Jörg Fliege Predicting Directed Links using Nondiagonal Matrix Decompositions kunegis@uni-koblenz.de ICDM 2012 3
4. 4. Powers of the Adjacency Matrix 1 2 3 4 5 6 0 1 1 0 1 1 0 0 0 1 0 0 0 0 0 0 1 1 (A²)14 = 2 A= 0 0 1 0 0 1 1 0 0 0 0 0 (A³)14 = 1 0 0 0 1 0 0 Jérôme Kunegis & Jörg Fliege Predicting Directed Links using Nondiagonal Matrix Decompositions kunegis@uni-koblenz.de ICDM 2012 4
5. 5. Computing Ak When A is Symmetric Eigenvalue decomposition: A=UΛU T Ak = (U Λ UT) (U Λ UT) . . . (U Λ UT) k T =UΛ U Problem: A is asymmetric Jérôme Kunegis & Jörg Fliege Predicting Directed Links using Nondiagonal Matrix Decompositions kunegis@uni-koblenz.de ICDM 2012 5
6. 6. Asymmetric Eigenvalue DecompositionWhen A is diagonalizable: A=UΛU −1 Problem: A is notdiagonalizable Advogato Jérôme Kunegis & Jörg Fliege Predicting Directed Links using Nondiagonal Matrix Decompositions kunegis@uni-koblenz.de ICDM 2012 6
7. 7. Singular Value Decomposition T A=UΛV k T UΣ V T T T =UΣV VΣU ...UΣV T =AA ...A Problem: This does not equal Ak Jérôme Kunegis & Jörg Fliege Predicting Directed Links using Nondiagonal Matrix Decompositions kunegis@uni-koblenz.de ICDM 2012 7
8. 8. DEDICOM T Solution: A=UXU “DEDICOM – Decomposition into Directed Components” X= Not diagonal Advogato Jérôme Kunegis & Jörg Fliege Predicting Directed Links using Nondiagonal Matrix Decompositions kunegis@uni-koblenz.de ICDM 2012 8
9. 9. Computation of Ak with DEDICOM A =UX Uk k T k A is easy to compute Jérôme Kunegis & Jörg Fliege Predicting Directed Links using Nondiagonal Matrix Decompositions kunegis@uni-koblenz.de ICDM 2012 9
10. 10. Finding a DEDICOM Singular value decomposition: T A = U(Σ V U ) U T Problem: Not computed to full rank Jérôme Kunegis & Jörg Fliege Predicting Directed Links using Nondiagonal Matrix Decompositions kunegis@uni-koblenz.de ICDM 2012 10
11. 11. DEDICOM Algorithms Using singular value decomposition A = U Σ VTLEFT A = U (Σ VT U) UTRIGHT A = V (VT U Σ) VTCLO A = Q X QT U Σ UT + V Σ VT = Q Λ QT (eigenvalue decomp.) X = QT A QITER Iterative algorithm(Harshman 1978) (Kliers et al. 1990) Jérôme Kunegis & Jörg Fliege Predicting Directed Links using Nondiagonal Matrix Decompositions kunegis@uni-koblenz.de ICDM 2012 11
12. 12. Approximation of eA = I + A + ½A² + ⅙A³ + . . . Approximating A Approximating eA Advogato trust Advogato trust DEDICOM algorithms Jérôme Kunegis & Jörg Fliege Predicting Directed Links using Nondiagonal Matrix Decompositions kunegis@uni-koblenz.de ICDM 2012 12
13. 13. Jérôme Kunegiskunegis@uni-koblenz.deJörg Fliegej.ﬂiege@soton.ac.uk Thank You konect.uni-koblenz.de
14. 14. References Predicting directed links using nondiagonal matrix decompositions Jérôme Kunegis & Jörg Fliege Int. Conf. on Data Mining, 2012 Models for analysis of asymmetrical relationships among n objects or stimuli Richard A. Harshman Contributions to economic analysis 187, 185–204, 1990 A generalization of Takanes algorithm for DEDICOM Henk A. Kliers, Jos M. ten Berge, Yoshio Takane & Jan de Leeuw Psychometrika 55(1), 151–158, 1990 Jérôme Kunegis & Jörg Fliege Predicting Directed Links using Nondiagonal Matrix Decompositions kunegis@uni-koblenz.de ICDM 2012 14