# distances: Distances Between Networks In ilovato/nevada: NEtwork-VAlued Data Analysis

## Description

This is a collection of functions computing the distance between two networks.

## Usage

 1 2 3 4 5 6 7 dist_hamming(x, y, representation = "laplacian") dist_frobenius(x, y, representation = "laplacian") dist_spectral(x, y, representation = "laplacian") dist_root_euclidean(x, y, representation = "laplacian") 

## Arguments

 x An igraph object or a matrix representing an underlying network. y An igraph object or a matrix representing an underlying network. Should have the same number of vertices as x. representation A string specifying the desired type of representation, among: "adjacency", "laplacian" [default], "modularity" or "graphon".

## Details

Let X be the matrix representation of network x and Y be the matrix representation of network y. The Hamming distance between x and y is given by

\frac{1}{N(N-1)} ∑_{i,j} |X_{ij} - Y_{ij}|,

where N is the number of vertices in networks x and y. The Frobenius distance between x and y is given by

√{∑_{i,j} (X_{ij} - Y_{ij})^2}.

The spectral distance between x and y is given by

√{∑_i (a_i - b_i)^2},

where a and b of the eigenvalues of X and Y, respectively. This distance gives rise to classes of equivalence. Consider the spectral decomposition of X and Y:

X=VAV^{-1}

and

Y = UBU^{-1},

where V and U are the matrices whose colums are the eigenvectors of X and Y, respectively and A and B are the diagonal matrices with elements the eigenvalues of X and Y, respectively. The root-Euclidean distance between x and y is given by

√{∑_i (V √{A} V^{-1} - U √{B} U^{-1})^2}.

Root-Euclidean distance can used only with the laplacian matrix representation.

## Value

A scalar measuring the distance between the two input networks.

## Examples

 1 2 3 4 5 6 g1 <- igraph::sample_gnp(20, 0.1) g2 <- igraph::sample_gnp(20, 0.2) dist_hamming(g1, g2, "adjacency") dist_frobenius(g1, g2, "adjacency") dist_spectral(g1, g2, "laplacian") dist_root_euclidean(g1, g2, "laplacian") 

ilovato/nevada documentation built on May 30, 2019, 9:47 p.m.