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# Network Growth and the Spectral Evolution Model

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We introduce and study the spectral evolution model, which characterizes
the growth of large networks in terms of the eigenvalue decomposition of
their adjacency matrices: In large networks, changes over time result in
a change of a graph's spectrum, leaving the eigenvectors unchanged. We
validate this hypothesis for several large social, collaboration,
authorship, rating, citation, communication and tagging networks,
covering unipartite, bipartite, signed and unsigned graphs. Following
these observations, we introduce a link prediction algorithm based on
the extrapolation of a network's spectral evolution. This new link
prediction method generalizes several common graph kernels that can be
expressed as spectral transformations. In contrast to these graph
kernels, the spectral extrapolation algorithm does not make assumptions
about specific growth patterns beyond the spectral evolution model. We
thus show that it performs particularly well for networks with
irregular, but spectral, growth patterns.

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### Network Growth and the Spectral Evolution Model

1. 1. Web Science & Technologies University of Koblenz ▪ Landau, Germany Network Growth and the Spectral Evolution Model Jérôme Kunegis¹, Damien Fay², Christian Bauckhage³ ¹ University of Koblenz-Landau ² University of Cambridge ³ Fraunhofer IAIS CIKM 2010, Toronto, Canada
2. 2. Jérôme Kunegis kunegis@uni-koblenz.de CIKM 2010 2 / 24 Recommender Systems Example: Find friends of Facebook
3. 3. Jérôme Kunegis kunegis@uni-koblenz.de CIKM 2010 3 / 24 Graph Theory Known links (A) Unknown links (B) The users of Facebook are connected by friendship links, forming a graph. This graph is undirected. Let A be the set of links in the network. Let B be the set of links that will appear in the future. Task: Find a suitable function f(A) = B.
4. 4. Jérôme Kunegis kunegis@uni-koblenz.de CIKM 2010 4 / 24 Algebraic Graph Theory Known links (A) Unknown links (B) Use adjacency matrices A, B ∈ {0, 1}n×n : A = B = [ 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 ] [ 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 ]
5. 5. Jérôme Kunegis kunegis@uni-koblenz.de CIKM 2010 5 / 24 Spectral Graph Theory Use th eigenvalue decomposition : A = UΛUT B = VΣVT  U and V are orthogonal and contain eigenvectors  Λ and Σ are diagonal and contain eigenvalues Task : find an f of the following form : f(UΛUT ) = VΣVT In this talk :  The observation that U ≈ V  Extrapolation of Λ to Σ
6. 6. Jérôme Kunegis kunegis@uni-koblenz.de CIKM 2010 6 / 24 Outline  Eigenvalue evolution  Eigenvector evolution  Diagonality test  The spectral evolution model  Explanations  Control tests  Spectral extrapolation
7. 7. Jérôme Kunegis kunegis@uni-koblenz.de CIKM 2010 7 / 24 Eigenvalue Evolution Wikipedia Facebook  Eigenvectors grow  Not all eigenvectors grow at the same speed, even in a single network
8. 8. Jérôme Kunegis kunegis@uni-koblenz.de CIKM 2010 8 / 24 Eigenvector Evolution Wikipedia Facebook Compute the cosine over time between eigenvectors and their initial value.  Some eigenvectors stay constant  Some eigenvectors change suddenly
9. 9. Jérôme Kunegis kunegis@uni-koblenz.de CIKM 2010 9 / 24 Eigenvector Permutation Wikipedia Facebook Compute the cosine between all eigenvector pairs at two times. Eigenvalues get permuted : A = UΛUT B = VΣVT Ui = Vj
10. 10. Jérôme Kunegis kunegis@uni-koblenz.de CIKM 2010 10 / 24 Diagonality Test A = UΛUT B = VΣVT Diagonalize B using U. B = UDUT D = U−1 B(UT )−1 D = UT BU UT BU should be diagonal!
11. 11. Jérôme Kunegis kunegis@uni-koblenz.de CIKM 2010 11 / 24 Diagonality Test Wikipedia Facebook  D is nearly diagonal  The diagonal of D is irregular
12. 12. Jérôme Kunegis kunegis@uni-koblenz.de CIKM 2010 12 / 24 The Spectral Evolution Model Networks grow spectrally  Eigenvectors stay constant  Eigenvalues change Why?
13. 13. Jérôme Kunegis kunegis@uni-koblenz.de CIKM 2010 13 / 24 Explanation : Matrix Powers Known links (A) Unknown links (B) The square of A constains the number of paths of length two between any node pair: A² = Generally, Ak contains the number of k-paths between any node pair. [ 1 0 1 0 0 0 2 0 1 0 1 0 2 0 1 0 1 0 2 0 0 0 1 0 1 ]
14. 14. Jérôme Kunegis kunegis@uni-koblenz.de CIKM 2010 14 / 24 Explanation: Polynomials p(A) = αA² + βA³ + γA + …⁴ Polynomials are good link prediction functions :  Count parallel paths  Weight paths by length (α > β > γ > …) The matrix power is a spectral transformation, e.g.: A² = (UΛUT )(UΛUT ) = UΛ²UT Polynomials are spectral transformations: p(A) = p(UΛUT ) = Up(Λ)UT
15. 15. Jérôme Kunegis kunegis@uni-koblenz.de CIKM 2010 15 / 24 Explanation: Graph Kernels Matrix exponential exp(A) = I + A + ½ A² + … = Uexp(Λ)UT Von Neumann kernel (I − αA)−1 = I + αA + α²A² + … = U(I − αΛ)−1 UT
16. 16. Jérôme Kunegis kunegis@uni-koblenz.de CIKM 2010 16 / 24 Explanation: Preferential Attachment Write A as a sum of rank-1 matrices: A = A1 + A2 + … Ai = λi ui ui T  Interpret each Ai as the adjacency matrix of one weighted graph  In Ai , vertex j has degree Σk λi uij uik ~ uij Consider the process of preferential attachment in each latent dimension separately: Σi ui ui T = λi ui ui T + εi ui ui T pa(A) = U(Λ + Ε)UT
17. 17. Jérôme Kunegis kunegis@uni-koblenz.de CIKM 2010 17 / 24 Control: Matrix Perturbation Add edges at random to A = UΛUT . The evolution of A should then be : A + E = Ũ Λ ŨT || Λ − Λ ||F = O(ε²) | U.k T Ũ.k | = O(ε) Using ||E||2 = ε.  Random growth is not spectral. ~ ~
18. 18. Jérôme Kunegis kunegis@uni-koblenz.de CIKM 2010 18 / 24 Control: Random Sampling Split an Erdős–Rényi random graph into A + B. Apply the diagonality test for transforming A into B.  Only one latent dimension is preserved.
19. 19. Jérôme Kunegis kunegis@uni-koblenz.de CIKM 2010 19 / 24 Spectral Extrapolation To predict links, extrapolate the evolution of eigenvalues.
20. 20. Jérôme Kunegis kunegis@uni-koblenz.de CIKM 2010 20 / 24 Experiments Methodology :  Retain newest edges as training set  Compute link prediction scores  Evaluate using the mean average precision (MAP)  User over a hundred datasets
21. 21. Jérôme Kunegis kunegis@uni-koblenz.de CIKM 2010 21 / 24 Experiments: Symmetric Networks
22. 22. Jérôme Kunegis kunegis@uni-koblenz.de CIKM 2010 22 / 24 Experiments: Weighted Networks Use the weighted adjacency matrix A.
23. 23. Jérôme Kunegis kunegis@uni-koblenz.de CIKM 2010 23 / 24 Experiments: Bipartite and Directed Networks Use the singular value decomposition A = UΛVT .
24. 24. Jérôme Kunegis kunegis@uni-koblenz.de CIKM 2010 24 / 24 Summary & Discussion  Eigenvectors remain constant  Eigenvalues grow irregularly  Extrapolate the eigenvalues to predict links Experimental results:  Extrapolation works best for bipartite and directed networks