| NormalizedLaplacian {invertiforms} | R Documentation |
Construct and use the Normalized Laplacian
Description
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 signature 'NormalizedLaplacian,sparseMatrix'
transform(iform, A)
## S4 method for signature 'NormalizedLaplacian,sparseMatrix'
inverse_transform(iform, A)
Arguments
A |
A matrix to transform. |
iform |
An Invertiform object describing the transformation. |
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/.
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.
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)