1. 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

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. Triangle Closing
?
Jérôme Kunegis & Jörg Fliege Predicting Directed Links using Nondiagonal Matrix Decompositions
kunegis@uni-koblenz.de ICDM 2012 3

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. 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. Asymmetric Eigenvalue Decomposition
When A is diagonalizable: A=UΛU −1
Problem:
A is not
diagonalizable Advogato
Jérôme Kunegis & Jörg Fliege Predicting Directed Links using Nondiagonal Matrix Decompositions
kunegis@uni-koblenz.de ICDM 2012 6

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. 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. 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. 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. DEDICOM Algorithms
Using singular value decomposition A = U Σ VT
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 Q
ITER 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. 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. Jérôme Kunegis
kunegis@uni-koblenz.de
Jörg Fliege
j.fliege@soton.ac.uk
Thank You
konect.uni-koblenz.de

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 Takane's 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