MMOC (version 0.1.1.0)

Laplacian: Calculate the graph Laplacian from a given adjacency matrix

Description

Calculate the graph laplacian from a given kernel matrix that represents the full graph weighted adjacency matrix

Usage

Laplacian(
  A,
  laplacian = c("shift", "Ng", "sym", "rw"),
  grf.type = c("full", "knn", "e-graph"),
  k = 5,
  rho = NULL,
  epsilon = 0,
  mutual = FALSE,
  binary.grf = FALSE,
  plots = TRUE
)

Value

An n\(\times\)n matrix where n is the number of rows in dat.

Arguments

A

An n by n kernel matrix, where n is the sample size, that represents your initial adjacency matrix. Kernel matrices are symmetric, positive semi-definite distance matrices

laplacian

One of "shift", "Ng", "rw" or "sym". See details for description

grf.type

Type of graph to calculate: "full" for adjacency matrix equal to the kernel, "knn" for a k-nearest neighbors graph, "e-graph" for an "epsilon graph"

k

An integer value for k in the k-nearest neighbors graph. Only the k largest edges (most similar neighbors) will be kept

rho

A value for the dispersion parameter in the Gaussian kernel. It is in the denominator of the exponent, so higher values correspond to lower similarity. By default it is the median pairwise Gaussian distance

epsilon

The cutoff value for the e-graph. Edges lower than this value will be removed

mutual

Make a "mutual" knn graph. Only keeps edges when two nodes are both in each others k-nearest set

binary.grf

Set all edges >0 to 1

plots

Whether or not to plot the final graph, a heatmap of calculated kernel, and the eigen values of the Laplacian

Details

The four Lapalacians are defined as \(L_{shift}=I+D^{-1/2}AD^{-1/2}\), \(L_{Ng}=D^{-1/2}AD^{-1/2}\), \(L_{sym}=I-D^{-1/2}AD^{-1/2}\), and \(L_{rw}=I-D^{-1}A\). The shifted Laplacian, \(L_{shift}=I+D^{-1/2}AD^{-1/2}\), is recommended for multi-view spectral clustering.

References

https://academic.oup.com/bioinformatics/article/36/4/1159/5566508#199177546

Examples

## Generating data with 3 distinct clusters
## Note that 'clustStruct' returns a list
dd <- clustStruct(n=120, p=30, k=3, noiseDat='random')[[1]]

## Gaussian kernel
rho <- median(dist(dd))
A <- exp(-(1/rho)*as.matrix(dist(dd, method = "euclidean", upper = TRUE)^2))

Laplacian(A, laplacian='shift', grf.type = 'knn')