A convenience function to create NormalizedLaplacian S4 objects, which are useful for finding the normalized Laplacian of the adjacency matrix of a graph.
Usage
NormalizedLaplacian(A)
# S4 method for NormalizedLaplacian,sparseMatrix
transform(iform, A)
# S4 method for NormalizedLaplacian,sparseMatrix
inverse_transform(iform, A)Arguments
- A
A matrix to transform.
- iform
An Invertiform object describing the transformation.
Value
NormalizedLaplacian()creates a NormalizedLaplacian object.transform()returns the transformed matrix, typically as a Matrix.inverse_transform()returns the inverse transformed matrix, typically as a Matrix.
Details
We define the normalized Laplacian \(L(A)\) of an \(n \times n\) graph adjacency matrix \(A\) as
$$ L(A)_{ij} = \frac{A_{ij}}{\sqrt{d^{out}_i} \sqrt{d^{in}_j}} $$
where
$$ d^{out}_i = \sum_{j=1}^n \| A_{ij} \| $$
and
$$ d^{in}_j = \sum_{i=1}^n \| A_{ij} \|. $$
When \(A_{ij}\) denotes the present of an edge from node \(i\) to node \(j\), which is fairly standard notation, \(d^{out}_i\) denotes the (absolute) out-degree of node \(i\) and \(d^{in}_j\) denotes the (absolute) in-degree of node \(j\).
Note that this documentation renders most clearly at https://rohelab.github.io/invertiforms/.
Examples
library(igraph)
library(igraphdata)
data("karate", package = "igraphdata")
A <- get.adjacency(karate)
iform <- NormalizedLaplacian(A)
L <- transform(iform, A)
A_recovered <- inverse_transform(iform, L)
all.equal(A, A_recovered)
#> [1] TRUE