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In this thesis, I study the spectral characteristics of large dynamic networks and formulate the spectral evolution model. The spectralevolution model applies to networks that evolve over time, and describes their spectral decompositions such as the eigenvalue and singular valuedecomposition. The spectral evolution model states that over time, the eigenvalues of anetwork change while its eigenvectors stay approximately constant.
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Web Science & Technologies
University of Koblenz ▪ Landau, Germany
On the Spectral Evolutionof Large Networks
Jérôme Kunegis
Jérôme [email protected] 2 / 29
Networks
…are everywhere
Communication
Authorship
Friendship
c
Interaction
Trust
Co-occurrence
Jérôme [email protected] 7 / 29
Recommender Systems
Predict who I will add as friend next
Facebook's algorithm: find friends-of-friends → Problem: Rest of the network is ignored!
:-(
me
Jérôme [email protected] 8 / 29
Outline
1. Algebraic Link Prediction2. Spectral Transformations3. Learning Link Prediction
Take into account the whole network
Jérôme [email protected] 9 / 29
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
Aij = 1 when i and j are connected
Aij = 0 when i and j are not connected
A is square and symmetric
1 2 4 5 6
3
A =
1
2
3
4
5
6
1 2 3 4 5 6
Jérôme [email protected] 10 / 29
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
1 2 4
30 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
1 0 1 1 0 00 3 1 1 1 01 1 2 1 1 01 1 1 3 0 10 1 1 0 2 00 0 0 1 0 1
=
2
A2 =
Jérôme [email protected] 11 / 29
Friend of a Friend of a Friend
Compute the number of friends-of-friends-of-friends:
Problem: A3 is not sparse!
1 2 4 5 6
3
1 2 3 4 5 61
23456
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
0 3 1 1 1 03 2 4 5 1 11 4 2 4 1 11 5 4 2 4 01 1 1 4 0 20 1 1 0 2 0
3
=A3 =
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Eigenvalue Decomposition
A = UΛUT
whereU are the eigenvectors UTU = IΛ are the eigenvalues Λ
ij = 0 when i ≠ j
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Computing A3
Use the eigenvalue decomposition A = UΛUT
A3 = UΛUT UΛUT UΛUT = UΛ3UT
Exploit U and Λ: UTU = I because U contains eigenvectors (Λk)
ii = Λ
ii
k because Λ contains eigenvalues
Result: Just cube all eigenvalues!
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Matrix Exponential
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
Jérôme [email protected] 15 / 29
Spectral Transformations
A2 = UΛ2UT
A3 = UΛ3UT
exp(A) = Uexp(Λ)UT
Friend of a friend
Friend of a friend of a friend
Matrix exponential
…are link prediction functions!
Jérôme [email protected] 16 / 29
Outline
1. Algebraic Link Prediction2. Spectral Transformations3. Learning Link Prediction
Why does it work?
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Looking at Real Facebook Data
Dataset: Facebook New Orleans (Viswanath et al. 2009)
63,731 persons1,545,686 friendship links with creation dates
Adjacency matrix At at time t (t = 1 . . . 75)
Compute all eigenvalue decompositions At = U
tΛtUtT
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Evolution of Eigenvalues
Growing eigenvalue
Constant eigenvalue(Λ
t)ii
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Eigenvector Evolution
Cosine similarity between (Ut)
•i and (U
t+x)
•i
Constant
Sudden change
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Eigenvector Permutation
Time split: old edges A = UΛUT
new edges B = VDVT
Eigenvectors permute
|U•i∙ V
•j|
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Outline
1. Algebraic Link Prediction2. Spectral Transformations3. Learning Link Prediction
a) Learning by Extrapolationb) Learning by Curve Fitting
What spectral transformation is best?
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a) Learning by Extrapolation CIKM 2010
Extrapolate the growth of the spectrum
Potential problem:overfitting
Good when growth is irregular
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Other Curves
Polynomiala + bx + cx2 + dx3 + ex4
Nonnegative polynomiala + . . . + hx7 a, . . ., h ≥ 0
Neumann kernelb / (1 − ax)
Matrix exponentialb exp(ax)
Rank reductionax if |x| ≥ x
0, 0 otherwise
ICML 2009
Friend of a Frienda + bx + cx2
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Evaluation Methodology
3-way split of edge set by edge creation time
Training set Ea ∪ Eb Test set Ec
Source set Ea Target set Eb
All edges E
Learn
Apply
Edge creation time
˙
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Experiments
Precision of link prediction (1 = perfect)
All datasets available at konect.uni-koblenz.de
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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
Thank You!
ACKNOWLEDGMENTS →
Jérôme [email protected] 30 / 29
Selected Publications
The Slashdot Zoo: Mining a social network with negative edgesJ. Kunegis, A. Lommatzsch and C. Bauckhage In 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 and visualizationJ. 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 [email protected] 31 / 29
References
B. Viswanath, A. Mislove, M. Cha, K. P. Gummadi, On the evolution of user interaction in Facebook. In Proc. Workshop on Online Social Networks, pp. 37–42, 2009.
