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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.

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Online Dating Recommender Systems: The Split-complex Number Approach

  1. 1. Web Science and Technologies University of Koblenz–Landau, GermanyOnline Dating Recommender Systems: The Split-complex Number Approach Jérôme Kunegis, Gerd Gröner, Thomas Gottron University of Koblenz–Landau RSWeb12 September 9, 2012
  2. 2. Friend Recommendation Jérôme Kunegis et al. Online Dating Recommender Systems: RSWeb12 The Split-complex Number Approach 2
  3. 3. Friend/Foe Recommendation Jérôme Kunegis et al. Online Dating Recommender Systems: RSWeb12 The Split-complex Number Approach 3
  4. 4. Dating Recommendation LIKE DISSIMILAR SIMILAR DISLIKE Jérôme Kunegis et al. Online Dating Recommender Systems: RSWeb12 The Split-complex Number Approach 4
  5. 5. Triangle Closing FriendFriend Friend ‌× Friend = FriendJérôme Kunegis et al. Online Dating Recommender Systems:RSWeb12 The Split-complex Number Approach 5
  6. 6. “The Enemy of My Enemy” Foe Foe Foe ‌× Foe = FriendFriend = +1 +1 × +1 = +1Foe = −1 −1 × +1 = −1 −1 × −1 = +1(Kunegis et al. 1999) Jérôme Kunegis et al. Online Dating Recommender Systems: RSWeb12 The Split-complex Number Approach 6
  7. 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: RSWeb12 The Split-complex Number Approach 7
  8. 8. Split-complex Numbersz = a + bjj × j = +1(a + bj) + (c + dj) = (a + c) + (b + d)j(a + bj) × (c + dj) = (ac + bd) + (ad + bc)jNot a field: (1 + j)(1 − j) = 0Introduced as real tessarines (Cockle 1848) Jérôme Kunegis et al. Online Dating Recommender Systems: RSWeb12 The Split-complex Number Approach 8
  9. 9. Im LIKE +jDISSIMILAR 0 +1 Re −1 SIMILAR −j DISLIKE Jérôme Kunegis et al. Online Dating Recommender Systems: RSWeb12 The Split-complex Number Approach 9
  10. 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 0Auv = 1 when u and v are connected 5 0 0 0 1 0 1Auv = 0 when u and v are not connected 6 0 0 0 0 1 0 Jérôme Kunegis et al. Online Dating Recommender Systems: RSWeb12 The Split-complex Number Approach 10
  11. 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 3exp 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: RSWeb12 The Split-complex Number Approach 11
  12. 12. Split-complex Adjacency Matrix 3 1 2 4 5 6 1 2 3 4 5 6 1 +jBuv = +j when u likes v 2 +j +jBuv = −j when u dislikes v 3 +jBuv = 0 when u and v are not connected B = 4 −j +j 5 −j 6 −jB = jA Jérôme Kunegis et al. Online Dating Recommender Systems: RSWeb12 The Split-complex Number Approach 12
  13. 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: RSWeb12 The Split-complex Number Approach 13
  14. 14. Computation A B≡ T A cosh(A) sinh(A)exp(B) ≡ sinh(A) cosh(A) Jérôme Kunegis et al. Online Dating Recommender Systems: RSWeb12 The Split-complex Number Approach 14
  15. 15. Evaluation (“Do you like me”) – Czech dating siteJérôme Kunegis et al. Online Dating Recommender Systems:RSWeb12 The Split-complex Number Approach 15
  16. 16. EvaluationJérôme Kunegis et al. Online Dating Recommender Systems:RSWeb12 The Split-complex Number Approach 16
  17. 17. Thank YouJérôme Kunegis @kunegisThanks go to Václav Petříček for providing the Libimseti.cz dataset.The research leading to these results has received funding from theEuropean Communitys Seventh Framework Programme under grantagreement n° 257859, ROBUST.konect.uni-koblenz.de/networks/libimseti Jérôme Kunegis et al. Online Dating Recommender Systems: RSWeb12 The Split-complex Number Approach 17
  18. 18. ReferencesJ. Kunegis, G. Gröner, T. Gottron. Online dating recommendersystems: the split-complex number approach. Proc. Workshop onRecommender Systems and the Social Web, 2012.J. Cockle. On certain functions resembling quaternions, and on a newimaginary in algebra. Philos. Mag., 33(3):435–439, 1848.J. Kunegis, A. Lommatzsch, C. Bauckhage. The Slashdot Zoo: mining a socialnetwork with negative edges. Proc. Int. World Wide Web Conf., 741–750,2009.J. Kunegis, D. Fay, C. Bauckhage. Network growth and the spectral evolutionmodel. Proc. Int. Conf. on Information and Knowledge Management, 739–748, 2010. Jérôme Kunegis et al. Online Dating Recommender Systems: RSWeb12 The Split-complex Number Approach 18

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