nd.extremal {NetworkDistance}

Extremal distance with top-k eigenvalues

Description

Extremal distance (nd.extremal) is a type of spectral distance measures on two graphs' graph Laplacian,

L := D-A

where A is an adjacency matrix and D_{ii}=\sum_j A_{ij}. It takes top-k eigenvalues from graph Laplacian matrices and take normalized sum of squared differences as metric. Note that it is 1. non-negative, 2. separated, 3. symmetric, and satisfies 4. triangle inequality in that it is indeed a metric.

Usage

nd.extremal(A, out.dist = TRUE, k = ceiling(nrow(A)/5))

Arguments

A

a list of length N containing adjacency matrices.

out.dist

a logical; TRUE for computed distance matrix as a dist object.

k

the number of largest eigenvalues to be used.

Value

a named list containing

D

an (N\times N) matrix or dist object containing pairwise distance measures.

spectra

an (N\times k) matrix where each row is top-k Laplacian eigenvalues.

References

Jakobson D, Rivin I (2002). “Extremal metrics on graphs I.” Forum Mathematicum, 14(1). ISSN 0933-7741, 1435-5337.

Examples

## load data
data(graph20)

## compute distance matrix
output = nd.extremal(graph20, out.dist=FALSE, k=2)

## visualize
opar = par(no.readonly=TRUE)
par(pty="s")
image(output$D[,20:1], main="two group case", col=gray(0:32/32), axes=FALSE)
par(opar)