Web Science & Technologies               University of Koblenz ▪ Landau, GermanyOn the Spectral Evolution   of Large Netwo...
Networks                 …are everywhere                                                                                  ...
Social Network               Person                Friendship       Jérôme Kunegis       kunegis@uni-koblenz.de   3 / 29
Trust Network Trust         Jérôme Kunegis         kunegis@uni-koblenz.de   4 / 29
Signed Social Network    Friend                                           Foe         Jérôme Kunegis         kunegis@uni-k...
Interaction Network        Jérôme Kunegis        kunegis@uni-koblenz.de   6 / 29
Recommender Systems                                           :-(  me       Predict who I will add as friend nextFacebooks...
Outline1.   Algebraic Link Prediction2.   Spectral Transformations3.   Learning Link PredictionTake into account the whole...
Adjacency Matrix                                 3            1             2              4        5        6            ...
Baseline: Friend of a Friend Model    Count the number of ways a person can be found as    the friend of a friend    Matri...
Friend of a Friend of a Friend                                 3        1                2               4            5   ...
Eigenvalue Decomposition                           A = UΛUTwhere  U are the eigenvectors                   U TU = I  Λ are...
Computing A3Use the eigenvalue decomposition A = UΛUT        A3 = UΛUT UΛUT UΛUT = UΛ3UTExploit U and Λ:  U TU = I       ...
Matrix Exponential                                     3        0.98                                                      ...
Spectral Transformations   A2 = UΛ2UT                     Friend of a friend   A = UΛ UT     3            3               ...
Outline  1.      Algebraic Link Prediction  2.      Spectral Transformations  3.      Learning Link Prediction            ...
Looking at Real Facebook DataDataset:     Facebook New Orleans             (Viswanath et al. 2009)63,731 persons1,545,686 ...
Evolution of Eigenvalues                       Constant eigenvalue     (Λt)ii                                             ...
Eigenvector EvolutionCosine similarity between (Ut)•i and (Ut+x)•i                      Constant                          ...
Eigenvector PermutationTime split:            old edges A = UΛUT                       new edges B = VDVTEigenvectors perm...
Outline1.     Algebraic Link Prediction2.     Spectral Transformations3.     Learning Link Prediction     a) Learning by E...
a) Learning by Extrapolation                         CIKM 2010 Extrapolate the growth of the spectrum                     ...
b) Learning by Curve Fitting                                    f       A                                        B      UΛ...
Curve Fitting (UTBU)ii                                           Λii        Jérôme Kunegis        kunegis@uni-koblenz.de  ...
Polynomial Curve FittingFit a polynomial             a + bx + cx2 + dx3 + ex4         Jérôme Kunegis         kunegis@uni-k...
Other Curves                                                   ICML 2009                                          Friend o...
Evaluation Methodology                              All edges E          Training set Ea ∪ Eb                          ˙  ...
Experiments Precision of link prediction (1 = perfect)                                 All datasets available at konect.un...
Conclusion●    Observation     ●       Eigenvalue change, eigenvectors are constant●    Why?     ●       Graph kernels, tr...
Selected PublicationsThe Slashdot Zoo: Mining a social network with negative edgesJ. Kunegis, A. Lommatzsch and C. Bauckha...
ReferencesB. Viswanath, A. Mislove, M. Cha, K. P. Gummadi, Onthe evolution of user interaction in Facebook. In Proc.Worksh...
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On the Spectral Evolution of Large Networks (PhD Thesis by Jérôme Kunegis)

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In this thesis, I study the spectral characteristics of large dynamic networks and formulate the spectral evolution model. The spectral
evolution model applies to networks that evolve over time, and describes their spectral decompositions such as the eigenvalue and singular value
decomposition.
The spectral evolution model states that over time, the eigenvalues of a
network change while its eigenvectors stay approximately constant.

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On the Spectral Evolution of Large Networks (PhD Thesis by Jérôme Kunegis)

  1. 1. Web Science & Technologies University of Koblenz ▪ Landau, GermanyOn the Spectral Evolution of Large Networks Jérôme Kunegis
  2. 2. Networks …are everywhere ip r sh tho Au ip dsh Fr ien t Trus o n ic ati n mu eCo m e nc c ti on c urr ra c -oc I nte Co Jérôme Kunegis kunegis@uni-koblenz.de 2 / 29
  3. 3. Social Network Person Friendship Jérôme Kunegis kunegis@uni-koblenz.de 3 / 29
  4. 4. Trust Network Trust Jérôme Kunegis kunegis@uni-koblenz.de 4 / 29
  5. 5. Signed Social Network Friend Foe Jérôme Kunegis kunegis@uni-koblenz.de 5 / 29
  6. 6. Interaction Network Jérôme Kunegis kunegis@uni-koblenz.de 6 / 29
  7. 7. Recommender Systems :-( me Predict who I will add as friend nextFacebooks algorithm: find friends-of-friends → Problem: Rest of the network is ignored! Jérôme Kunegis kunegis@uni-koblenz.de 7 / 29
  8. 8. Outline1. Algebraic Link Prediction2. Spectral Transformations3. Learning Link PredictionTake into account the whole network Jérôme Kunegis kunegis@uni-koblenz.de 8 / 29
  9. 9. Adjacency Matrix 3 1 2 4 5 6 1 2 3 4 5 6Aij = 1 when i and j are connected 1 0 1 0 0 0 0Aij = 0 when i and j are not connected 2 1 0 1 1 0 0 3 0 1 0 1 0 0 A= 4 0 1 1 0 1 0A is square and symmetric 5 0 0 0 1 0 1 6 0 0 0 0 1 0 Jérôme Kunegis kunegis@uni-koblenz.de 9 / 29
  10. 10. Baseline: Friend of a Friend Model Count the number of ways a person can be found as the friend of a friend Matrix product AA = A2 2 3 0 1 0 0 0 0 1 0 1 1 0 0 1 0 1 1 0 0 0 3 1 1 1 0A 2 = 0 0 1 1 0 1 1 0 0 1 0 0 = 1 1 1 1 2 1 1 3 1 0 0 1 0 0 0 1 0 1 0 1 1 0 2 0 1 2 4 0 0 0 0 1 0 0 0 0 1 0 1 Jérôme Kunegis kunegis@uni-koblenz.de 10 / 29
  11. 11. Friend of a Friend of a Friend 3 1 2 4 5 6Compute the number of friends-of-friends-of-friends: 1 2 3 4 5 6 3 0 1 0 0 0 0 0 3 1 1 1 0 1 1 0 1 1 0 0 3 2 4 5 1 1 2 A3 = 0 0 1 1 0 1 1 0 0 1 0 0 = 1 1 4 5 2 4 4 2 1 4 1 0 3 4 0 0 0 1 0 1 1 1 1 4 0 2 5 0 0 0 0 1 0 0 1 1 0 2 0 6 Problem: A3 is not sparse! Jérôme Kunegis kunegis@uni-koblenz.de 11 / 29
  12. 12. Eigenvalue Decomposition A = UΛUTwhere U are the eigenvectors U TU = I Λ are the eigenvalues Λij = 0 when i ≠ j Jérôme Kunegis kunegis@uni-koblenz.de 12 / 29
  13. 13. Computing A3Use the eigenvalue decomposition A = UΛUT A3 = UΛUT UΛUT UΛUT = UΛ3UTExploit U and Λ: U TU = I because U contains eigenvectors (Λ ) = Λiik because Λ contains eigenvalues k iiResult: Just cube all eigenvalues! Jérôme Kunegis kunegis@uni-koblenz.de 13 / 29
  14. 14. Matrix Exponential 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 kunegis@uni-koblenz.de 14 / 29
  15. 15. Spectral Transformations A2 = UΛ2UT Friend of a friend A = UΛ UT 3 3 Friend of a friend of a friendexp(A) = Uexp(Λ)UT Matrix exponential …are link prediction functions! Jérôme Kunegis kunegis@uni-koblenz.de 15 / 29
  16. 16. Outline 1. Algebraic Link Prediction 2. Spectral Transformations 3. Learning Link Prediction Why does it work? Jérôme Kunegis kunegis@uni-koblenz.de 16 / 29
  17. 17. Looking at Real Facebook DataDataset: Facebook New Orleans (Viswanath et al. 2009)63,731 persons1,545,686 friendship links with creation datesAdjacency matrix At at time t (t = 1 . . . 75)Compute all eigenvalue decompositions At = UtΛtUtT Jérôme Kunegis kunegis@uni-koblenz.de 17 / 29
  18. 18. Evolution of Eigenvalues Constant eigenvalue (Λt)ii Growing eigenvalue Jérôme Kunegis kunegis@uni-koblenz.de 18 / 29
  19. 19. Eigenvector EvolutionCosine similarity between (Ut)•i and (Ut+x)•i Constant Sudden change Jérôme Kunegis kunegis@uni-koblenz.de 19 / 29
  20. 20. Eigenvector PermutationTime split: old edges A = UΛUT new edges B = VDVTEigenvectors permute |U•i∙ V•j| Jérôme Kunegis kunegis@uni-koblenz.de 20 / 29
  21. 21. Outline1. Algebraic Link Prediction2. Spectral Transformations3. Learning Link Prediction a) Learning by Extrapolation b) Learning by Curve FittingWhat spectral transformation is best? Jérôme Kunegis kunegis@uni-koblenz.de 21 / 29
  22. 22. a) Learning by Extrapolation CIKM 2010 Extrapolate the growth of the spectrum Good when growth is irregular Potential problem: overfitting Jérôme Kunegis kunegis@uni-koblenz.de 22 / 29
  23. 23. b) Learning by Curve Fitting f A B UΛUT B Λ UTBU f Diagonal Jérôme Kunegis kunegis@uni-koblenz.de 23 / 29
  24. 24. Curve Fitting (UTBU)ii Λii Jérôme Kunegis kunegis@uni-koblenz.de 24 / 29
  25. 25. Polynomial Curve FittingFit a polynomial a + bx + cx2 + dx3 + ex4 Jérôme Kunegis kunegis@uni-koblenz.de 25 / 29
  26. 26. Other Curves ICML 2009 Friend of a Friend a + bx + cx2 Polynomial a + bx + cx2 + dx3 + ex4 Nonnegative polynomial a + . . . + hx7 a, . . ., h ≥ 0 Matrix exponential b exp(ax) Neumann kernel b / (1 − ax) Rank reduction ax if |x| ≥ x0, 0 otherwise Jérôme Kunegis kunegis@uni-koblenz.de 26 / 29
  27. 27. Evaluation Methodology All edges E Training set Ea ∪ Eb ˙ Apply Test set Ec Source set Ea Learn Target set Eb Edge creation time 3-way split of edge set by edge creation time Jérôme Kunegis kunegis@uni-koblenz.de 27 / 29
  28. 28. Experiments Precision of link prediction (1 = perfect) All datasets available at konect.uni-koblenz.de Jérôme Kunegis kunegis@uni-koblenz.de 28 / 29
  29. 29. Conclusion● Observation ● Eigenvalue change, eigenvectors are constant● Why? ● Graph kernels, triangle closing, the sum-over-paths model, rank reduction, etc.● Application to recommender systems ● By learning the spectral transformation for a given dataset ACKNOWLEDGMENTS → Thank You! Jérôme Kunegis kunegis@uni-koblenz.de 29 / 29
  30. 30. Selected PublicationsThe Slashdot Zoo: Mining a social network with negative edgesJ. Kunegis, A. Lommatzsch and C. BauckhageIn Proc. World Wide Web Conf., pp. 741–750, 2009.Learning spectral graph transformations for link predictionJ. Kunegis and A. LommatzschIn Proc. Int. Conf. on Machine Learning, pp. 561–568, 2009.Spectral analysis of signed graphs for clustering, prediction andvisualizationJ. Kunegis, S. Schmidt, A. Lommatzsch and J. LernerIn Proc. SIAM Int. Conf. on Data Mining, pp. 559–570, 2010.Network growth and the spectral evolution modelJ. Kunegis, D. Fay and C. BauckhageIn Proc. Conf. on Information and Knowledge Management,pp. 739–748, 2010. Jérôme Kunegis kunegis@uni-koblenz.de 30 / 29
  31. 31. ReferencesB. Viswanath, A. Mislove, M. Cha, K. P. Gummadi, Onthe evolution of user interaction in Facebook. In Proc.Workshop on Online Social Networks, pp. 37–42, 2009. Jérôme Kunegis kunegis@uni-koblenz.de 31 / 29

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