Predicting Directed Links using Nondiagonal Matrix Decompositions
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Web Science & Technologies University of Koblenz ▪ Landau, Germany Predicting Directed Links using Nondiagonal Matrix Decompositions Jérôme Kunegis & Jörg Fliege Int. Conf. on Data Mining 2012
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Transcript of Predicting Directed Links using Nondiagonal Matrix Decompositions
Web Science & Technologies
University of Koblenz ▪ Landau, Germany
Predicting Directed Links using Nondiagonal Matrix Decompositions
Jérôme Kunegis & Jörg Fliege
Int. Conf. on Data Mining 2012
Jérôme Kunegis & Jörg [email protected]
Predicting Directed Links using Nondiagonal Matrix DecompositionsICDM 2012 2
Trust Prediction
?
Goal: predict trusted edges
Jérôme Kunegis & Jörg [email protected]
Predicting Directed Links using Nondiagonal Matrix DecompositionsICDM 2012 3
Triangle Closing
?
Jérôme Kunegis & Jörg [email protected]
Predicting Directed Links using Nondiagonal Matrix DecompositionsICDM 2012 4
Powers of the Adjacency Matrix
1 2
3
5 6
4
0 1 1 0 1 10 0 0 1 0 00 0 0 0 1 10 0 1 0 0 11 0 0 0 0 00 0 0 1 0 0
A = (A²)14 = 2
(A³)14 = 1
Jérôme Kunegis & Jörg [email protected]
Predicting Directed Links using Nondiagonal Matrix DecompositionsICDM 2012 5
A = U Λ UT
Computing Ak When A is Symmetric
Eigenvalue decomposition:
Ak = (U Λ UT) (U Λ UT) . . . (U Λ UT)
= U Λk UT
Problem: A is asymmetric
Jérôme Kunegis & Jörg [email protected]
Predicting Directed Links using Nondiagonal Matrix DecompositionsICDM 2012 6
Asymmetric Eigenvalue Decomposition
When A is diagonalizable:
Advogato
Problem: A is not
diagonalizable
A = U Λ U−1
Jérôme Kunegis & Jörg [email protected]
Predicting Directed Links using Nondiagonal Matrix DecompositionsICDM 2012 7
Singular Value Decomposition
U Σk VT
= U Σ VT V Σ UT . . . U Σ VT
= A AT . . . A
A = U Λ VT
Problem: This does notequal Ak
Jérôme Kunegis & Jörg [email protected]
Predicting Directed Links using Nondiagonal Matrix DecompositionsICDM 2012 8
DEDICOM
Solution:
“DEDICOM – Decomposition into Directed Components”
A = U X UT
X = Not diagonal
Advogato
Jérôme Kunegis & Jörg [email protected]
Predicting Directed Links using Nondiagonal Matrix DecompositionsICDM 2012 9
Computation of Ak with DEDICOM
Ak = U Xk UT
Ak is easy to compute
Jérôme Kunegis & Jörg [email protected]
Predicting Directed Links using Nondiagonal Matrix DecompositionsICDM 2012 10
Finding a DEDICOM
Singular value decomposition:
U UT ( )A = U Σ VT
Problem: Not computed to full rank
Jérôme Kunegis & Jörg [email protected]
Predicting Directed Links using Nondiagonal Matrix DecompositionsICDM 2012 11
DEDICOM Algorithms
LEFT A = U (Σ VT U) UT
RIGHT A = V (VT U Σ) VT
CLO 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)
Using singular value decomposition A = U Σ VT
Jérôme Kunegis & Jörg [email protected]
Predicting Directed Links using Nondiagonal Matrix DecompositionsICDM 2012 12
Approximation of eA = I + A + ½A² + A⅙ ³ + . . .
DEDICOM algorithms
Approximating A Approximating eA
Advogato trustAdvogato trust
Jérôme Kunegis & Jörg [email protected]
Predicting Directed Links using Nondiagonal Matrix DecompositionsICDM 2012 14
References
Predicting directed links using nondiagonal matrix decompositionsJérôme Kunegis & Jörg FliegeInt. Conf. on Data Mining, 2012
Models for analysis of asymmetrical relationships among n objects or stimuliRichard A. HarshmanContributions to economic analysis 187, 185–204, 1990
A generalization of Takane's algorithm for DEDICOMHenk A. Kliers, Jos M. ten Berge, Yoshio Takane & Jan de LeeuwPsychometrika 55(1), 151–158, 1990
