Spectra of Adjacency and Laplaciansco/Spectral/Lesson1Spectra.pdf · Spectra of Laplacian Matrices...

Spectra of Adjacency and Laplacian Matrices [email protected]

Francisco Escolano, PhD http://www.rvg.ua.es/~sco

� Definition:

Spectra of Adjacency and Laplacian

Matrices

Francisco Escolano, PhDUniversity of Alicante (Spain)

http://www.rvg.ua.es/~sco

[email protected]

Matrix Matrix ComputingComputing ((subjectsubject 3168 3168 –– DegreeDegree in in MathsMaths) )

30 30 hourshours ((theorytheory) + 15 ) + 15 hourshours ((practicalpractical assignmentassignment))



Contents

1. Spectra of the Adjacency Matrices1. Ordinary Spectrum

2. Spectral decomposition

2. Spectral Connected Components1. Analysis of the Perron-Frobenius eigenvector

2. Finding connected components

3. Courant-Fischer Theorem1. 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 Sum1. Connections with Laplacian matrices and Courant-Fischer

2. Interpretation of eigenvectors

3. Interpretation of “smoothers”.



Spectra of Adjacency Matrices

� Graphs and 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 eigenvalue λn is associated to the principal eigenvector.

� In this case, the Generalized Perron-Frobenius theorem (non-

negative) 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:




� Example #1:

� Regular graphs yield eigenvalues and properties closely related.

� Non-regularity: eigenvalues dominated by degree

Perron-Frobenius eigenvalue

Principal eigenvector




� Example #2:

2 10

11

12




� Example #2:

� A small non 1-regularity yields eigenvalues between 1 and degree

� Analyzing the spectrum is impossible to know whether the graph isdisconnected or how to find the connected components.

Can this be done from Φ?

Perron-Frobenius eigenvalue

Principal eigenvector



Spectral Connected Components

� Example #3:

2 10

11

12



Spectral Connected Components

� Example #3:

� Possitive eigenvector (PE): all non-zero components have same sign, and thecorresponding eigenvalue must be possitive.

� Connected components (CC): the non-zero components of the PE mean the degreeof membership of a node to a CC. And the eigenvalue is the cohesiveness of thecluster (the greater the eigenvalue the greater the cohesiveness).

� Why? Courant-Fischer Theorem!

Possitive eigenvalues

CC#1

CC#2



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:




� 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 Xk of Rn




� 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:



Spectra of Laplacian Matrices

� Laplacian of a Graph:




� Properties:

� Possitive definiteness (all eigenvalues >= 0).

� The smallest eigenvalue λ1 is 0 and its multiplicity is the number of

connected components in the graph.

� The second eigenvalue λ2 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 quantity λk-λ1= λk is called the

spectral gap.




� Example #4:

Friedler vector

Spectral Gap




� Normalized Laplacian:

Spectrum




� 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 have λ2 <= n/(n-1) with equality holding iff is the complete

graph of n vertices Kn.

3. For n>=2 and no isolated vertices we have: λn >= n/(n-1)

4. A non complete graph satisfies λ2 <= 1.

5. A connected graph has λ2 > 1. Otherwise, the number of connected

components is given by the multiplicity of the zero eigenvalue.

6. For all i <= n we have than λi <= 2, being λn = 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 being λn-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: λ1>=1/(D*V)




� Example #5: 1 2 3 4

5 6 7 8Sum=8 non-isolated vertices

3 CCs

λ2 <=[n/(n-1) = 1.143]<= λn

Friedler



Laplacians and the Dirichlet Sum

� Dirichlet Sum:

� Unnormalized Laplacian: Exercise#2 (proof)

� Normalized Laplacian:




� Connections with Courant-Fischer:

� Connection with the Friedler value:




� Exercise#3:

� Using the connections between the Dirichlet sum and eigenvalues, proof thatλn >= 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: D1/2f

� The “smoothing interpretation”:

� The Dirichlet sum is interpreted in terms of the amount of variability of itsfunction over the structure.

� Computing eigenvalues of the Laplacian is linked to maximize or minimizesuch amount of variability.

� The Dirichlet sum will be useful when we study the random walker andspectral learning.

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