Spectra of Adjacency and Laplacian Matrices

Transcription

1 Spectra of Adjacency and Laplacian Matrices Definition: University of Alicante (Spain) Matrix Computing (subject 3168 Degree in Maths) 30 hours (theory)) + 15 hours (practical assignment)

2 Contents 1. Spectra of the Adjacency Matrices 1. Ordinary Spectrum 2. Spectral decomposition 2. Spectral Connected Components 1. Analysis of the Perron-Frobenius eigenvector 2. Finding connected components 3. Courant-Fischer Theorem 1. Rayleing quotient and Rayleing-Ritz theorem as a particular case. 2. Courant-Fischer theorem for finding eigenvalues 4. Spectra of Laplacian Matrices 1. Laplacian matrices and their spectra 2. The Friedler number and vector 3. Properties and bounds 5. Laplacians and the Dirichlet Sum 1. Connections with Laplacian matrices and Courant-Fischer 2. Interpretation of eigenvectors 3. Interpretation of smoothers.

3 Spectra of Adjacency Matrices Graphs and Adjacency Matrices:

4 Spectra of Adjacency Matrices Ordinary spectrum: The spectrum of a nxn symmetric real-valued matrix is a set of n real eigenvalues. Each eigenvalue is associated to an eigenvector. The set of eigenvectors define an orthornormal basis. The largest eigenvalueλ n is associated to the principal eigenvector. In this case, the Generalized Perron-Frobenius theorem (nonnegative) ensures that A has a unique largest real eigenvalue and that the corresponding eigenvector has strictlly non-negative components. Spectral decomposition: The spectral decomposition of matrix A is given by:

5 Spectra of Adjacency Matrices Example #1: Perron-Frobenius eigenvalue Principal eigenvector Regular graphs yield eigenvalues and properties closely related. Non-regularity: eigenvalues dominated by degree

6 Spectra of Adjacency Matrices Example #2:

7 Spectra of Adjacency Matrices Example #2: Perron-Frobenius eigenvalue Principal eigenvector A small non 1-regularity yields eigenvalues between 1 and degree Analyzing the spectrum is impossible to know whether the graph is disconnected or how to find the connected components. Can this be done fromφ?

8 Spectral Connected Components Example #3:

9 Spectral Connected Components Example #3: Possitive eigenvalues CC#2 Possitive eigenvector (PE): all non-zero components have same sign, and the corresponding eigenvalue must be possitive. Connected components (CC): the non-zero components of the PE mean the degree of membership of a node to a CC. And the eigenvalue is the cohesiveness of the cluster (the greater the eigenvalue the greater the cohesiveness). Why? Courant-Fischer Theorem! CC#1

10 Courant-Fischer Theorem Rayleigh Quotient: If x is an eigenvector of A we have: If A is real and symmetric x can be posed as a linear combination of the n orthonormal eigenvectors of A:

11 Courant-Fischer Theorem Courant-Fischer Theorem: A particular case (Rayleigh-Ritz), derived from above yields: In general, if A is real and symmetric, the rest of eigenvalues come from the analysis of vectors orthogonal to subspaces X k of R n

12 Courant-Fischer Theorem Exercise #1: Show that if in a connected component, all the nodes in it form a complete subgraph, then the corresponding eigenvalue of the CC is the number of nodes in the CC minus one (see, for instance, example#3). A simpler example:

13 Spectra of Laplacian Matrices Laplacian of a Graph:

14 Spectra of Laplacian Matrices Properties: Possitive definiteness (all eigenvalues >= 0). The smallest eigenvalueλ 1 is 0 and its multiplicity is the number of connected components in the graph. The second eigenvalueλ 2 is the algebraic connectivity and it is only non-zero for connected graphs. Its associated eigenvector is called the Friedler vector. Close-to-zero values in the Friedler vector can be reduced to zero in practice inducing a new partition. The sign of the non-zero elements of the Friedler vector are useful for partitioning the graph (go back here when talking about graph cuts). The first non-zero eigenvalue The quantityλ k -λ 1 = λ k is called the spectral gap.

15 Spectra of Laplacian Matrices Example #4: Friedler vector Spectral Gap

16 Spectra of Laplacian Matrices Normalized Laplacian: Spectrum

17 Spectra of Laplacian Matrices Bounds (Normalized Laplacian Spectra): [Chung,97] 1. The sum of eigenvalues is bounded by the number of vertices n, with equality holding iff there are no isolated vertices. 2. For n>=2 we haveλ 2 <= n/(n-1) with equality holding iff is the complete graph of n vertices K n. 3. For n>=2 and no isolated vertices we have: λ n >= n/(n-1) 4. A non complete graph satisfiesλ 2 <= A connected graph has λ 2 > 1. Otherwise, the number of connected components is given by the multiplicity of the zero eigenvalue. 6. For all i <= n we have thanλ i <= 2, beingλ n = 2 iff a connected component of the graph is bipartite and non-trivial. 7. The spectrum of a graph is the union of the spectrum of its connected components. 8. A bipartite graph has i+1 connected components beingλ n-j+1 =2 for 1<=j<=i. 9. Being the diameter D of a graph the maximum distance between two nodes and the volume V the sum of all degrees we have: λ 1 >=1/(D*V)

18 Spectra of Laplacian Matrices Example #5: λ 2 <=[n/(n-1) = 1.143]<= λ n Sum=8 non-isolated vertices 3 CCs Friedler

19 Laplacians and the Dirichlet Sum Dirichlet Sum: Unnormalized Laplacian: Exercise#2 (proof) Normalized Laplacian:

20 Laplacians and the Dirichlet Sum Connections with Courant-Fischer: Connection with the Friedler value:

21 Laplacians and the Dirichlet Sum Exercise#3: Using the connections between the Dirichlet sum and eigenvalues, proof that λ n >= d, being d the degree of a given vertex in the graph. Finding eigenvectors (and connection with degree): Find the maximizing or minimizing f and then get as eigenvector: D 1/2 f The smoothing interpretation : The Dirichlet sum is interpreted in terms of the amount of variability of its function over the structure. Computing eigenvalues of the Laplacian is linked to maximize or minimize such amount of variability. The Dirichlet sum will be useful when we study the random walker and spectral learning.