Package 'invertiforms'

Construct and use DoubleCenter transformations

Description

A convenience function to create DoubleCenter S4 objects, which are useful for simultaneously row and column centering a matrix.

Usage

DoubleCenter(A)

## S4 method for signature 'DoubleCenter,sparseMatrix'
transform(iform, A)

## S4 method for signature 'DoubleCenter,sparseLRMatrix'
inverse_transform(iform, A)

## S4 method for signature 'DoubleCenter,vsp_fa'
inverse_transform(iform, A)

Arguments

A	A matrix to transform.
iform	An Invertiform object describing the transformation.

Value

• DoubleCenter() creates a DoubleCenter object.

• transform() returns the transformed matrix, typically as a sparseLRMatrix::sparseLRMatrix.

• inverse_transform() returns the inverse transformed matrix, typically as a sparseLRMatrix::sparseLRMatrix in most cases. When possible reduces the sparseLRMatrix::sparseLRMatrix to a Matrix::sparseMatrix().

Examples

library(igraph)
library(igraphdata)

data("karate", package = "igraphdata")

A <- get.adjacency(karate)

iform <- DoubleCenter(A)

A_tilde <- transform(iform, A)
A_recovered <- inverse_transform(iform, A_tilde)

all.equal(A, A_recovered)

---

Row and column centering transformation

Description

Row and column centering transformation

Slots

row_means

numeric.

col_means

numeric.

overall_mean

numeric.

---

Apply the inverse of an invertible transformation

Description

Apply the inverse of an invertible transformation

Usage

inverse_transform(iform, A)

Arguments

iform	An Invertiform object describing the transformation.
A	A matrix to inverse transform.

Value

The inverse transformed matrix.

---

An abstract S4 class representing an invertible transformation

Description

An abstract S4 class representing an invertible transformation

---

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::Matrix.

• inverse_transform() returns the inverse transformed matrix, typically as a Matrix::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)

---

Normalized graph Laplacian transformation

Description

Normalized graph Laplacian transformation

Slots

rsA

numeric.

csA

numeric.

---

Construct and use the Perturbed Laplacian

Description

Construct and use the Perturbed Laplacian

Usage

PerturbedLaplacian(A, tau = NULL)

## S4 method for signature 'PerturbedLaplacian,sparseMatrix'
transform(iform, A)

## S4 method for signature 'PerturbedLaplacian,sparseLRMatrix'
inverse_transform(iform, A)

Arguments

A	A matrix to transform.
tau	Additive regularizer for row and column sums of abs(A). Typically this corresponds to inflating the (absolute) out-degree and the (absolute) in-degree of each node by tau. Defaults to NULL, in which case we set tau to the mean value of abs(A).
iform	An Invertiform object describing the transformation.

Details

We define the perturbed Laplacian $L^\tau(A)$ of an $n \times n$ graph adjacency matrix $A$ as

$L^\tau(A)_{ij} = \frac{A_{ij} + \frac{\tau}{n}}{\sqrt{d^{out}_i + \tau} \sqrt{d^{in}_j + \tau}}$

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 more clearly at https://rohelab.github.io/invertiforms/.

Value

• PerturbedLaplacian() creates a PerturbedLaplacian object.

• transform() returns the transformed matrix, typically as a Matrix::Matrix.

• inverse_transform() returns the inverse transformed matrix, typically as a Matrix::Matrix.

Examples

library(igraph)
library(igraphdata)

data("karate", package = "igraphdata")

A <- get.adjacency(karate)

iform <- PerturbedLaplacian(A)

L <- transform(iform, A)
L

## Not run: 
A_recovered <- inverse_transform(iform, L)
all.equal(A, A_recovered)

## End(Not run)

---

Perturbed graph Laplacian transformation

Description

Perturbed graph Laplacian transformation

Slots

tau

numeric.

rsA

numeric.

csA

numeric.

tau_choice

character.

---

Construct and use the Regularized Laplacian

Description

Construct and use the Regularized Laplacian

Usage

RegularizedLaplacian(A, tau_row = NULL, tau_col = NULL)

## S4 method for signature 'RegularizedLaplacian,Matrix'
transform(iform, A)

## S4 method for signature 'RegularizedLaplacian,matrix'
transform(iform, A)

## S4 method for signature 'RegularizedLaplacian,sparseLRMatrix'
transform(iform, A)

## S4 method for signature 'RegularizedLaplacian,Matrix'
inverse_transform(iform, A)

## S4 method for signature 'RegularizedLaplacian,matrix'
inverse_transform(iform, A)

## S4 method for signature 'RegularizedLaplacian,vsp_fa'
inverse_transform(iform, A)

Arguments

A	A matrix to transform.
tau_row	Additive regularizer for row sums of abs(A). Typically this corresponds to inflating the (absolute) out-degree of each node by tau_row. Defaults to NULL, in which case we set tau_row to the mean (absolute) row sum of A.
tau_col	Additive regularizer for column sums of abs(A). Typically this corresponds to inflating the (absolute) in-degree of each node by tau_col. Defaults to NULL, in which case we set tau_col to the mean (absolute) column sum of A.
iform	An Invertiform object describing the transformation.

Details

We define the regularized Laplacian $L^\tau(A)$ of an $n \times n$ graph adjacency matrix $A$ as

$L^\tau(A)_{ij} = \frac{A_{ij}}{\sqrt{d^{out}_i + \tau_{row}} \sqrt{d^{in}_j + \tau_{col}}}$

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 $j$. Then $\tau_{row}$ is an additive out-degree regularizer and $\tau_{col}$ is an additive in-degree regularizer.

Note that this documentation renders more clearly at https://rohelab.github.io/invertiforms/.

Value

• RegularizedLaplacian() creates a RegularizedLaplacian object.

• transform() returns the transformed matrix, typically as a Matrix::Matrix.

• inverse_transform() returns the inverse transformed matrix, typically as a Matrix::Matrix.

Examples

library(igraph)
library(igraphdata)

data("karate", package = "igraphdata")

A <- get.adjacency(karate)

iform <- RegularizedLaplacian(A)

L <- transform(iform, A)
L

A_recovered <- inverse_transform(iform, L)

all.equal(A, A_recovered)

---

Regularized graph Laplacian transformation

Description

Regularized graph Laplacian transformation

Slots

tau_row

numeric.

tau_col

numeric.

rsA

numeric.

csA

numeric.

tau_choice_row

character.

tau_choice_col

character.

---

Apply an invertible transformation

Description

Apply an invertible transformation

Usage

transform(iform, A)

Arguments

iform	An Invertiform object describing the transformation.
A	A matrix to transform.

Value

The transformed matrix.

---