scg_eps: Error of the spectral coarse graining (SCG) approximation

Description

scg_eps computes \(\Vert v_i-Pv_i\Vert\), where
\(v_i\) is the \(i\)th eigenvector in V and \(P\) is the
projector corresponding to the mtype argument.

A numeric matrix of (eigen)vectors assumed normalized. The vectors
are to be stored column-wise in V).

groups

A vector of nrow(V) integers labeling each group vertex
in the partition.

mtype

The type of semi-projector used for the SCG. For now
“symmetric”, “laplacian” and “stochastic” are
available.

p

A probability vector of length nrow(V). p is the
stationary probability distribution of a Markov chain when mtype =
“stochastic”. This parameter is ignored otherwise.

norm

Either “row” or “col”. If set to “row” the
rows of the Laplacian matrix sum to zero and the rows of the stochastic
matrix sum to one; otherwise it is the columns.

Value

scg_eps returns with a numeric vector whose \(i\)th
component is \(\Vert v_i-Pv_i\Vert\) (see Details).

References

D. Morton de Lachapelle, D. Gfeller, and P. De Los Rios,
Shrinking Matrices while Preserving their Eigenpairs with Application to the
Spectral Coarse Graining of Graphs. Submitted to SIAM Journal on
Matrix Analysis and Applications, 2008.
http://people.epfl.ch/david.morton