HLIBpro  2.9
Spectrum of Graph Laplacian
Note
The example is based on the Diploma thesis "Effiziente, approximative Berechnung des Spektrums von Graphen mittels hierarchischer Matrizen" by Jan Nitzschmann.

# Problem Description

Given is a Graph $$G = (V,E)$$ with nodes $$V = \left\{v_1,\ldots,v_n\right\}$$ and edges $$E \subseteq V \times V$$. Let $$\mathcal{L} := I - D^{-1/2} A D^{-1/2}$$ be the normalised Laplacian of $$G$$ with diagonal $$D, d_{ii} := \textrm{degree}(v_i)$$, and adjacency matrix $$A$$,

$a_{ij} := \left\{ \begin{array}{ll} 1 & (i,j) \in E \\ 0 & \textrm{otherwise} \end{array} . \right.$

We are looking for the spectrum $$\Lambda = \left\{ \lambda_1, \ldots, \lambda_n \right\}$$ of $$\mathcal{L}$$. Please note, that $$\Lambda \subset [0,2]$$ with $$\lambda_1 = 0$$.

Computing all eigenvalues $$\lambda_i$$ is usually extremly costly. However, sometimes not the exact spectrum is of interest but the general distribution of eigenvalues, which permits to characterise certain properties of matrices and especially graphs.

Applying the convolution

$f_{\mu}(\lambda_0) := \sum_{\lambda_i} \int k_{\mu}(\lambda_0,\lambda') \delta(\lambda' - \lambda_i) d\lambda'$

with

$k_{\mu}(\lambda_0,\lambda') := \frac{\mu}{(\lambda' - \lambda_0)^2 + \mu^2}$

and $$\delta$$ being the usual delta distribution, one obtains a graphical representation of the spectrum. An example for this is shown in the following picture for a small graph (left) and the convoluted spectrum of its Laplacian together with the eigenvalues.

 Graph Spectrum

Note, that the above $$f_{\mu}(\lambda_0)$$ only represents a single sample for the whole spectrum. Depending on the sample frequency and the choice of $$\mu$$, the final image shows more or less details of the spectrum. The following picture show two other versions of the above spectrum with a coarse (left) and fine (right) sampling.

# Computation of the Spectrum

The above definition of $$f_{\mu}(\lambda_0)$$ contains the eigenvalues, which are not available. Hence, another way to compute $$f_{\mu}(\lambda_0)$$ is needed.

We start with the computation of the determinant of $$\mathcal{L}-\lambda I$$ using LU factorisation. Since the determinant of triangular matrices is the product of their diagonal entries and since the diagonal entries of $$L$$ are one, this can be restricted to the diagonal entries of $$U$$:

$|\textrm{det}(\mathcal{L}-\lambda I)| = |\textrm{det}(LU)| = |\textrm{det}(U)| = |\prod_i u_{ii}|.$

Applying the logarithm to both sides of the equation, we can change the product into a sum:

$\log |\textrm{det}(\mathcal{L}-\lambda I)| = \log |\prod_i u_{ii}| = \sum_i \log |u_{ii}|.$

Replacing $$\lambda$$ by $$\lambda_0 + i\mu$$, one obtains

\begin{eqnarray*} \sum_i \log |u_{ii}| & = & \log |\textrm{det}(\mathcal{L}-(\lambda_0 + i\mu) I)| \\ & = & \log \prod_{\lambda_i} |\lambda_i - \lambda_0 - i\mu| \\ & = & \sum_{\lambda_i} \log |\lambda_i - \lambda_0 - i\mu| \end{eqnarray*}

Here we have used the identity of the determinant with the corresponding characteristic polynomial $$\prod_{\lambda_i} (\lambda_i - *)$$.

Finally, differentiation with respect to $$\mu$$ yields

\begin{eqnarray*} \frac{d}{d\mu} \sum_i \log |u_{ii}| & = & \sum_{\lambda_i} \frac{d}{d\mu} \log |\lambda_i - \lambda_0 - i\mu| \\ & = & \sum_{\lambda_i} \frac{d}{d\mu} \frac{1}{2} \log | (\lambda_i - \lambda_0)^2 + \mu^2| \\ & = & \sum_{\lambda_i} \frac{\mu}{(\lambda_i - \lambda_0)^2 + \mu^2} \\ & = & f_{\mu}(\lambda_0). \end{eqnarray*}

Replacing the differentation by a (central) difference quotient makes the problem computable with a series of LU factorisations for $$\mathcal{L}-(\lambda_0+i\mu)I$$ for the different values of $$\lambda_0$$. Furthermore, instead of an exact LU decomposition, the H-LU factorisation is used to obtain log-linear costs.

# Sparse to H-Matrix Conversion

The program starts by converting a sparse matrix, which is assumed to represent a graph laplacian, into an H-matrix:

#include <iostream>
using namespace std;
using namespace HLIB;
int
main ( int argc,
char ** argv )
{
std::string matrixfile = "graph.mat";
double mu = 1e-2;
double d = 1e-3;
uint nsteps = 100;
double eps = 1e-4;
try
{
HLIB::INIT();
auto M = read_matrix( matrixfile.c_str() );
if ( ! IS_TYPE( M, TSparseMatrix ) )
{
std::cout << "input matrix is not sparse" << std::endl;
return 1;
}
auto S = ptrcast( M.get(), TSparseMatrix );
S->set_unsymmetric();
TMETISAlgPartStrat part_strat;
TAlgCTBuilder base_ct_builder( & part_strat, 100 );
TAlgNDCTBuilder nd_ct_builder( & base_ct_builder );
auto ct = nd_ct_builder.build( S );
TBCBuilder bct_builder;
auto bct = bct_builder.build( ct.get(), ct.get(), & adm_cond );
TSparseMBuilder h_builder( S, ct->perm_i2e(), ct->perm_e2i() );
auto L = h_builder.build( bct->root(), acc_exact );

In addition to $$L$$ also the identity matrix is needed to compute $$\mathcal{L}-(\lambda_0+i\mu)I$$:

auto Id = identity( bct.get() );
L->set_complex( true );
Id->set_complex( true );

Both matrices need to be complex valued since $$\lambda_0+i\mu$$ is, except for $$\mu = 0$$, which represents the limit case with exact representation of the spectrum.

# Computing the Difference Quotient

For the computation of the spectrum in the sample space $$[0,2]$$ equally distant sample points are chosen as defined by the number of steps nsteps, i.e., $$\lambda_0 = 0, h_s, 2 \cdot h_s, ..., 2$$ with $$h_s = 2/\textrm{nsteps}$$.

Afterwards, for each value of $$\lambda_0$$ the difference quotient is computed and $$\lambda_0$$ as well as the result of the difference quotient are stored:

const auto lambda_0 = i * stepwidth;
const auto f_mu_plus_h = calc_abs_det( L, Id, lambda_0, mu+h, eps );
const auto f_mu_minus_h = calc_abs_det( L, Id, lambda_0, mu-h, eps );
x(i) = lambda_0;
y(i) = (f_mu_plus_h - f_mu_minus_h) / (2 * h);

Since computation for different values of $$\lambda_0$$ are independent, they may be computed in parallel. This is done by using the parallel_for algorithm from TBB:

tbb::parallel_for( uint(0), nsteps+1,
[=,&x,&y] ( const uint i )
{
...
} );
Remarks
Although H-LU scales very good with the number of cores, it is even more efficient to perform factorisations is parallel since there are still some portions of the algorithm implemented sequentially.

The result of the computation is stored in two objects of type BLAS::Vector and returned as a std::pair, which completes the function calc_y:

std::pair< BLAS::Vector< double >,
BLAS::Vector< double > >
calc_y ( const TMatrix * L,
const TMatrix * Id,
const double mu,
const double h,
const uint nsteps,
const double eps )
{
BLAS::Vector< double > x( nsteps+1 );
BLAS::Vector< double > y( nsteps+1 );
const double stepwidth = 2.0 / double(nsteps);
tbb::parallel_for( uint(0), nsteps+1,
[=,&x,&y] ( const uint i )
{
const auto lambda_0 = i * stepwidth;
const auto f_mu_plus_h = calc_abs_det( L, Id, lambda_0, mu+h, eps );
const auto f_mu_minus_h = calc_abs_det( L, Id, lambda_0, mu-h, eps );
x(i) = lambda_0;
y(i) = (f_mu_plus_h - f_mu_minus_h) / (2 * h);
} );
return std::pair< BLAS::Vector< double >,
BLAS::Vector< double > >( std::move( x ),
std::move( y ) );
}

This is then called from the main function via

auto spectrum = calc_y( L.get(), Id.get(), mu, d, nsteps, eps );

For visualization, both arrays are written to files in Matlab format, which finishes the program:

TMatlabVectorIO vio;
vio.write( spectrum.first, "x.mat", "x" );
vio.write( spectrum.second, "y.mat", "y" );
HLIB::DONE();
}
catch ( std::exception & e )
{
std::cout << e.what() << std::endl;
}
}

# Computing the Determinant

Left open is the computation of the determinant of $$\mathcal{L}-(\lambda_0+i\mu)I$$ implemented in calc_abs_det via H-LU factorisation.

Before the factorisation, the matrix first has to be computed:

double
calc_abs_det ( const TMatrix * L,
const TMatrix * Id,
const double lambda_0,
const double mu,
const double eps )
{
auto L_lmu_Id = L->copy();
add( HLIB::complex( -lambda_0, -mu ), Id, HLIB::complex( 1, 0 ), L_lmu_Id.get(), acc_exact );

By default, ๐ง๐ซ๐จ๐ก๐๐๐ uses block-wise LU factorisation, i.e., computes the inverse of diagonal blocks. In this case however, the point-wise LU factorisation is needed since the diagonal entries are used.

fac_options_t opts;
opts.eval = point_wise;
factorise( L_lmu_Id.get(), fixed_prec( eps ), opts );

After the factorisation, these diagonal entries only need to be extracted from the matrix. Note, that L_lmu_id stores $$L$$ and $$U$$ but since the diagonal entries of $$L$$ are one and therefore not explicitly stored, the diagonal of L_lmu_id is equal to the diagonal of $$U$$.

To obtain the diagonal entries the function diagonal is available, which returns a vector containing the coefficients.

In order to prevent stability issues, the function tests for very small values and aborts further computation if such values are present:

auto diag = diagonal( L_lmu_Id.get() );
double sum = 0.0;
for ( auto i : diag->is() )
{
const auto u_ii = diag->centry( i );
const auto norm_uii = std::sqrt( norm( u_ii ) );
if ( norm_uii < 1e-12 )
{
std::cout << "detected near zero entry" << std::endl;
return 0;
}
else
sum += std::log( norm_uii );
}
return sum;
}

# The Plain Program

#include <iostream>
#include <tbb/parallel_for.h>
#include "hlib.hh"
using namespace std;
using namespace HLIB;
double
calc_abs_det ( const TMatrix * L,
const TMatrix * Id,
const double lambda_0,
const double mu,
const double eps )
{
auto L_lmu_Id = L->copy();
opts.eval = point_wise;
add( HLIB::complex( -lambda_0, -mu ), Id, HLIB::complex( 1, 0 ), L_lmu_Id.get(), acc_exact );
factorise( L_lmu_Id.get(), fixed_prec( eps ), opts );
auto diag = diagonal( L_lmu_Id.get() );
double sum = 0.0;
for ( auto i : diag->is() )
{
const auto u_ii = diag->centry( i );
const auto norm_uii = std::sqrt( norm( u_ii ) );
if ( norm_uii < 1e-12 )
{
std::cout << "detected near zero entry" << std::endl;
return 0;
}
else
sum += std::log( norm_uii );
}
return sum;
}
std::pair< BLAS::Vector< double >,
calc_y ( const TMatrix * L,
const TMatrix * Id,
const double mu,
const double d,
const uint nsteps,
const double eps )
{
BLAS::Vector< double > x( nsteps+1 );
BLAS::Vector< double > y( nsteps+1 );
const double stepwidth = 2.0 / double(nsteps);
tbb::parallel_for( uint(0), nsteps+1,
[=,&x,&y] ( const uint i )
{
const auto lambda_0 = i * stepwidth;
const auto f_mu_plus_d = calc_abs_det( L, Id, lambda_0, mu+d, eps );
const auto f_mu_minus_d = calc_abs_det( L, Id, lambda_0, mu-d, eps );
x(i) = lambda_0;
y(i) = (f_mu_plus_d - f_mu_minus_d) / (2 * d);
} );
return std::pair< BLAS::Vector< double >,
BLAS::Vector< double > >( std::move( x ),
std::move( y ) );
}
int
main ( int argc,
char ** argv )
{
std::string matrixfile = "graph.mat";
double mu = 1e-2;
double h = 1e-3;
uint nsteps = 100;
double eps = 1e-4;
try
{
HLIB::INIT();
auto M = read_matrix( matrixfile.c_str() );
if ( ! IS_TYPE( M, TSparseMatrix ) )
{
std::cout << "input matrix is not sparse" << std::endl;
return 1;
}
auto S = ptrcast( M.get(), TSparseMatrix );
S->set_unsymmetric();
TMETISAlgPartStrat part_strat;
TAlgCTBuilder base_ct_builder( & part_strat, 100 );
TAlgNDCTBuilder nd_ct_builder( & base_ct_builder );
auto ct = nd_ct_builder.build( S );
TBCBuilder bct_builder;
auto bct = bct_builder.build( ct.get(), ct.get(), & adm_cond );
TSparseMBuilder h_builder( S, ct->perm_i2e(), ct->perm_e2i() );
auto L = h_builder.build( bct->root(), acc_exact );
auto Id = identity( bct.get() );
L->set_complex( true );
Id->set_complex( true );
auto spectrum = calc_y( L.get(), Id.get(), mu, h, nsteps, eps );
vio.write( spectrum.first, "x.mat", "x" );
vio.write( spectrum.second, "y.mat", "y" );
HLIB::DONE();
}
catch ( std::exception & e )
{
std::cout << e.what() << std::endl;
}
}

# Example Matrices

The Sparse Matrix Collection from the University of Florida provides a wide range of sparse matrices.

The following Matlab function will compute the normalised Laplacian from a given sparse matrix, assuming the matrix contains the adjacency matrix of the graph (loops are ignored!).

function [ Ln ] = laplacian ( A )
s = size(A);
n = s(1);
sums = sum( A );
D=sparse([1:n],[1:n],sums,n,n);
L=tril(A,-1)+D+triu(A,1);
invsums = 1./sqrt(sums);
D=sparse([1:n],[1:n],invsums,n,n);
Ln = D*L*D;
end
HLIB::TMatrix::build
virtual void build(TByteStream &s)
use data from stream s to build matrix
HLIB::factorise
std::unique_ptr< TFacMatrix > factorise(TMatrix *A, const TTruncAcc &acc, const fac_options_t &options=fac_options_t())
compute factorisation of A
HLIB::TBCBuilder
Recursively build block cluster tree with supplied admissibility condition.
Definition: TBCBuilder.hh:27
HLIB::TMETISAlgPartStrat
Graph partitioning using METIS.
Definition: TAlgPartStrat.hh:77
HLIB::TBCBuilder::build
virtual std::unique_ptr< TBlockClusterTree > build(const TClusterTree *rowct, const TClusterTree *colct, const TAdmCondition *ac) const
HLIB::TMatlabVectorIO::write
virtual void write(const TVector *v, const std::string &fname) const
write vector v with name "v" to file fname
HLIB::TMatlabVectorIO
Class for vector I/O in Matlab format.
Definition: TVectorIO.hh:199
HLIB::TClusterTree::perm_e2i
const TPermutation * perm_e2i() const
return external to internal permutation
Definition: TClusterTree.hh:73
HLIB::TClusterTree::perm_i2e
const TPermutation * perm_i2e() const
return internal to external permutation
Definition: TClusterTree.hh:76
HLIB::TMatrix::copy
virtual auto copy() const -> std::unique_ptr< TMatrix >
return copy of matrix
HLIB::BLAS::Vector
Standard vector in basic linear algebra, i.e. BLAS/LAPACK.
Definition: Vector.hh:38
void add(const T_value alpha, const TMatrix *A, const T_value beta, TMatrix *C, const TTruncAcc &acc)
C โ ฮฑยทA + ฮฒยทC.
HLIB::TMatrix::set_complex
void set_complex(const bool b, const bool force=false)
Definition: TMatrix.hh:292
HLIB::TSparseMatBuilder
Creates H-matrices out of sparse matrices.
Definition: TMatBuilder.hh:197
HLIB::TBlockClusterTree::root
const TBlockCluster * root() const
return root of block cluster tree
Definition: TBlockClusterTree.hh:67
HLIB::fac_options_t
options for matrix factorisations
Definition: mat_fac.hh:44
HLIB::TAlgCTBuilder
Base class for cluster tree construction algorithms based on graph partitioning with graph defined by...
Definition: TAlgCTBuilder.hh:30
HLIB::TMatrix
Base class for all matrices, defining basic properties, e.g. underlying block index and processor set...
Definition: TMatrix.hh:57
HLIB::fac_options_t::eval
eval_type_t eval
factorise block (default) or point wise
Definition: mat_fac.hh:50
HLIB::TSparseMatrix
Class for a sparse matrix stored in compressed row storage format.
Definition: TSparseMatrix.hh:29
HLIB::TAlgNDCTBuilder
Enhances algebraic clustering by nested dissection.
Definition: TAlgCTBuilder.hh:125