online dating recommender systems: the split-complex number approach
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Web Science and Technologies
University of Koblenz–Landau, Germany
Online Dating Recommender Systems: The Split-complex Number Approach
Jérôme Kunegis, Gerd Gröner, Thomas GottronUniversity of Koblenz–Landau
RSWeb'12
September 9, 2012
Jérôme Kunegis et al.RSWeb'12
Online Dating Recommender Systems: The Split-complex Number Approach 2
Friend Recommendation
Jérôme Kunegis et al.RSWeb'12
Online Dating Recommender Systems: The Split-complex Number Approach 3
Friend/Foe Recommendation
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Online Dating Recommender Systems: The Split-complex Number Approach 4
DISSIMILAR
LIKE
DISLIKE
SIMILAR
Dating Recommendation
Jérôme Kunegis et al.RSWeb'12
Online Dating Recommender Systems: The Split-complex Number Approach 5
Triangle Closing
Friend Friend
Friend × Friend = Friend
Jérôme Kunegis et al.RSWeb'12
Online Dating Recommender Systems: The Split-complex Number Approach 6
+1 × +1 = +1−1 × +1 = −1−1 × −1 = +1
“The Enemy of My Enemy”
Foe Foe
Foe × Foe = Friend
Friend = +1Foe = −1
(Kunegis et al. 1999)
Jérôme Kunegis et al.RSWeb'12
Online Dating Recommender Systems: The Split-complex Number Approach 7
Dating RecommendationLike Like
Like × Like = Similar
Similar Similar
Similar × Similar = Similar
Similar Like
Similar × Like = Like
Jérôme Kunegis et al.RSWeb'12
Online Dating Recommender Systems: The Split-complex Number Approach 8
z = a + bj
j × j = +1
(a + bj) + (c + dj) = (a + c) + (b + d)j(a + bj) × (c + dj) = (ac + bd) + (ad + bc)j
Not a field: (1 + j)(1 − j) = 0
Introduced as real tessarines (Cockle 1848)
Split-complex Numbers
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Online Dating Recommender Systems: The Split-complex Number Approach 9
+j
+1
−1
−j
0Re
Im
LIKE
DISLIKE
SIMILAR
DISSIMILAR
Jérôme Kunegis et al.RSWeb'12
Online Dating Recommender Systems: The Split-complex Number Approach 10
Adjacency Matrix
0 1 0 0 0 01 0 1 1 0 00 1 0 1 0 00 1 1 0 1 00 0 0 1 0 10 0 0 0 1 0
Auv
= 1 when u and v are connected
Auv
= 0 when u and v are not connected
1 2 4 5 6
3
A =
1
2
3
4
5
6
1 2 3 4 5 6
Jérôme Kunegis et al.RSWeb'12
Online Dating Recommender Systems: The Split-complex Number Approach 11
Recommender Functions
exp(A) = I + A + 1/2 A2 + 1/6 A3 + . . .
1 2 4 5
3
1 2 3 4 5 61234566
0 1 0 0 0 0 01 0 1 1 0 0 00 1 0 1 0 0 00 1 1 0 1 0 00 0 0 1 0 1 00 0 0 0 1 0 10 0 0 0 0 1 0
exp =
1 .66 1 .72 0 .93 0 .98 0 .28 0 . 06 0 .011.72 3 .57 2.70 2 .93 1. 04 0. 29 0 .060 .93 2 .70 2.86 2.71 0 .99 0 . 28 0 .060 .98 2 .93 2 .71 3 .63 1. 95 0 . 76 0 .220 .28 1.04 0 .99 1 .95 2 .35 1. 59 0 .640 .06 0 .29 0 .28 0 .76 1 .59 2. 23 1 .380.01 0 .06 0.06 0 .22 0 .64 1 . 38 1 .59
76
7
7
0.98 0.76 0.22
Jérôme Kunegis et al.RSWeb'12
Online Dating Recommender Systems: The Split-complex Number Approach 12
Split-complex Adjacency Matrix
Buv
= +j when u likes v
Buv
= −j when u dislikes v
Buv
= 0 when u and v are not connected
B = jA
1 2 4 5 6
3
B =
1
2
3
4
5
6
1 2 3 4 5 6
+j
+j +j
+j
+j−j
−j
−j
Jérôme Kunegis et al.RSWeb'12
Online Dating Recommender Systems: The Split-complex Number Approach 13
Split-complex Numbers as 2×2 Matrices
a + bj ≡ a
b a
b
a
b a
b c
d c
d a+c
b+d a+c
b+d+ =
a
b a
b c
d c
d ac+bd
ad+bc ac+bd
ad+bc× =
Jérôme Kunegis et al.RSWeb'12
Online Dating Recommender Systems: The Split-complex Number Approach 14
Computation
B ≡ A
AT
exp(B) ≡ cosh(A)
sinh(A)
sinh(A)
cosh(A)
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Online Dating Recommender Systems: The Split-complex Number Approach 15
(“Do you like me”) – Czech dating site
Evaluation
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Online Dating Recommender Systems: The Split-complex Number Approach 16
Evaluation
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Online Dating Recommender Systems: The Split-complex Number Approach 17
Thank YouJérôme Kunegis @kunegis
Thanks go to Václav Petříček for providing the Libimseti.cz dataset. The research leading to these results has received funding from the European Community's Seventh Framework Programme under grant agreement n° 257859, ROBUST.
konect.uni-koblenz.de/networks/libimseti
Jérôme Kunegis et al.RSWeb'12
Online Dating Recommender Systems: The Split-complex Number Approach 18
J. Kunegis, G. Gröner, T. Gottron. Online dating recommender systems: the split-complex number approach. Proc. Workshop on Recommender Systems and the Social Web, 2012.
J. Cockle. On certain functions resembling quaternions, and on a new imaginary in algebra. Philos. Mag., 33(3):435–439, 1848.
J. Kunegis, A. Lommatzsch, C. Bauckhage. The Slashdot Zoo: mining a social network with negative edges. Proc. Int. World Wide Web Conf., 741–750, 2009.
J. Kunegis, D. Fay, C. Bauckhage. Network growth and the spectral evolution model. Proc. Int. Conf. on Information and Knowledge Management, 739–748, 2010.
References
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