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.

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. Jérôme Kunegis
kunegis@uni-koblenz.de
CIKM 2010
2 / 24
Recommender Systems
Example: Find friends of Facebook

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. 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. 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. 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. 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. 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. 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. 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. 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. Jérôme Kunegis
kunegis@uni-koblenz.de
CIKM 2010
12 / 24
The Spectral Evolution Model
Networks grow spectrally

Eigenvectors stay constant

Eigenvalues change
Why?

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. 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. 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. 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. 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. 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. Jérôme Kunegis
kunegis@uni-koblenz.de
CIKM 2010
19 / 24
Spectral Extrapolation
To predict links, extrapolate the evolution of eigenvalues.

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. Jérôme Kunegis
kunegis@uni-koblenz.de
CIKM 2010
21 / 24
Experiments: Symmetric Networks

22. Jérôme Kunegis
kunegis@uni-koblenz.de
CIKM 2010
22 / 24
Experiments: Weighted Networks
Use the weighted adjacency matrix A.

23. Jérôme Kunegis
kunegis@uni-koblenz.de
CIKM 2010
23 / 24
Experiments: Bipartite and Directed Networks
Use the singular value decomposition A = UΛVT
.

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