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.

1. Web Science and Technologies
University of Koblenz–Landau, Germany
Online Dating Recommender Systems:
The Split-complex Number Approach
Jérôme Kunegis, Gerd Gröner, Thomas Gottron
University of Koblenz–Landau
RSWeb'12
September 9, 2012

2. Friend Recommendation
Jérôme Kunegis et al. Online Dating Recommender Systems:
RSWeb'12 The Split-complex Number Approach 2

3. Friend/Foe Recommendation
Jérôme Kunegis et al. Online Dating Recommender Systems:
RSWeb'12 The Split-complex Number Approach 3

4. Dating Recommendation
LIKE
DISSIMILAR SIMILAR
DISLIKE
Jérôme Kunegis et al. Online Dating Recommender Systems:
RSWeb'12 The Split-complex Number Approach 4

5. Triangle Closing
Friend
Friend
Friend ‌× Friend = Friend
Jérôme Kunegis et al. Online Dating Recommender Systems:
RSWeb'12 The Split-complex Number Approach 5

6. “The Enemy of My Enemy”
Foe
Foe
Foe ‌× Foe = Friend
Friend = +1 +1 × +1 = +1
Foe = −1 −1 × +1 = −1
−1 × −1 = +1
(Kunegis et al. 1999)
Jérôme Kunegis et al. Online Dating Recommender Systems:
RSWeb'12 The Split-complex Number Approach 6

7. Dating Recommendation
Like
Like
Like ‌× Like = Similar
r Simila
Simila r
Similar ‌× Similar = Similar
r Like
Simila
Similar ‌× Like = Like
Jérôme Kunegis et al. Online Dating Recommender Systems:
RSWeb'12 The Split-complex Number Approach 7

8. Split-complex Numbers
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)
Jérôme Kunegis et al. Online Dating Recommender Systems:
RSWeb'12 The Split-complex Number Approach 8

9. Im
LIKE
+j
DISSIMILAR 0 +1
Re
−1 SIMILAR
−j
DISLIKE
Jérôme Kunegis et al. Online Dating Recommender Systems:
RSWeb'12 The Split-complex Number Approach 9

10. Adjacency Matrix
3
1 2 4 5 6
1 2 3 4 5 6
1 0 1 0 0 0 0
2 1 0 1 1 0 0
3 0 1 0 1 0 0
A=
4 0 1 1 0 1 0
Auv = 1 when u and v are connected 5 0 0 0 1 0 1
Auv = 0 when u and v are not connected 6 0 0 0 0 1 0
Jérôme Kunegis et al. Online Dating Recommender Systems:
RSWeb'12 The Split-complex Number Approach 10

11. Recommender Functions
3
0.98 0.76 0.22
1 2 4 5 6 7
exp(A) = I + A + 1/2 A2 + 1/6 A3 + . . .
1 2 3 4 5 6 7
0 1 0 0 0 0 0 1 . 66 1 . 72 0 .93 0 . 98 0 . 28 0 . 06 0 . 01 1
1 0 1 1 0 0 0 1. 72 3 . 57 2. 70 2 . 93 1. 04 0. 29 0 . 06 2
0 1 0 1 0 0 0 0 . 93 2 .70 2. 86 2. 71 0 . 99 0 . 28 0 . 06 3
exp 0 1 1 0 1 0 0 = 0 . 98 2 . 93 2 .71 3 . 63 1. 95 0 . 76 0 . 22 4
0 0 0 1 0 1 0 0 . 28 1. 04 0 . 99 1 . 95 2 .35 1. 59 0 . 64 5
0 0 0 0 1 0 1 0 . 06 0 . 29 0 .28 0 . 76 1 .59 2. 23 1 .38 6
0 0 0 0 0 1 0 0. 01 0 . 06 0. 06 0 . 22 0 .64 1 . 38 1 .59 7
Jérôme Kunegis et al. Online Dating Recommender Systems:
RSWeb'12 The Split-complex Number Approach 11

12. Split-complex Adjacency Matrix
3
1 2 4 5 6
1 2 3 4 5 6
1 +j
Buv = +j when u likes v
2 +j +j
Buv = −j when u dislikes v
3 +j
Buv = 0 when u and v are not connected B = 4 −j +j
5 −j
6 −j
B = jA
Jérôme Kunegis et al. Online Dating Recommender Systems:
RSWeb'12 The Split-complex Number Approach 12

13. Split-complex Numbers as 2×2 Matrices
a b
a + bj ≡
b a
a b c d a+c b+d
+ =
b a d c b+d a+c
a b c d ac+bd ad+bc
× =
b a d c ad+bc ac+bd
Jérôme Kunegis et al. Online Dating Recommender Systems:
RSWeb'12 The Split-complex Number Approach 13

14. Computation
A
B≡ T
A
cosh(A) sinh(A)
exp(B) ≡
sinh(A) cosh(A)
Jérôme Kunegis et al. Online Dating Recommender Systems:
RSWeb'12 The Split-complex Number Approach 14

15. Evaluation
(“Do you like me”) – Czech dating site
Jérôme Kunegis et al. Online Dating Recommender Systems:
RSWeb'12 The Split-complex Number Approach 15

16. Evaluation
Jérôme Kunegis et al. Online Dating Recommender Systems:
RSWeb'12 The Split-complex Number Approach 16

17. Thank You
Jérôme 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. Online Dating Recommender Systems:
RSWeb'12 The Split-complex Number Approach 17

18. References
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.
Jérôme Kunegis et al. Online Dating Recommender Systems:
RSWeb'12 The Split-complex Number Approach 18