Online Dating Recommender Systems: The Split-complex Number Approach

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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.

Education Technology

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

Friend Recommendation

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

Friend/Foe Recommendation

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

Dating Recommendation

LIKE

DISSIMILAR SIMILAR

DISLIKE

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

Triangle Closing

Friend
Friend

Friend ‌× Friend = Friend

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

“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

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

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

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

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

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

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

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

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

Evaluation

(“Do you like me”) – Czech dating site

Jérôme Kunegis et al. Online Dating Recommender Systems:
RSWeb'12 The Split-complex Number Approach 15

Evaluation

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

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

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

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Online Dating Recommender Systems: The Split-complex Number Approach

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

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