online dating recommender systems: the split-complex number approach
DESCRIPTION
A typical recommender setting is based on two kinds of relations: similarity between users (or between objects) and the taste of users towards certain objects. In environments such as online dating websites, these two relations are difficult to separate, as the users can be similar to each other, but also have preferences towards other users, i.e., rate other users. In this paper, we present a novel and unified way to model this duality of the relations by using split-complex numbers, a number system related to the complex numbers that is used in mathematics, physics and other fields. We show that this unified representation is capable of modeling both notions of relations between users in a joint expression and apply it for recommending potential partners. In experiments with the Czech dating website Libimseti.cz we show that our modeling approach leads to an improvement over baseline recommendation methods in this scenario.TRANSCRIPT
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
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Online Dating Recommender Systems: The Split-complex Number Approach 2
Friend Recommendation
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Friend/Foe Recommendation
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DISSIMILAR
LIKE
DISLIKE
SIMILAR
Dating Recommendation
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Triangle Closing
Friend Friend
Friend × Friend = Friend
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+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)
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Dating RecommendationLike Like
Like × Like = Similar
Similar Similar
Similar × Similar = Similar
Similar Like
Similar × Like = Like
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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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+j
+1
−1
−j
0Re
Im
LIKE
DISLIKE
SIMILAR
DISSIMILAR
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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
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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
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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
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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
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Computation
B ≡ A
AT
exp(B) ≡ cosh(A)
sinh(A)
sinh(A)
cosh(A)
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(“Do you like me”) – Czech dating site
Evaluation
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Evaluation
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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
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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