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

## Usage

```
dist_hamming(x, y, representation = "laplacian")
dist_frobenius(
x,
y,
representation = "laplacian",
matching_iterations = 0,
target_matrix = NULL
)
dist_spectral(x, y, representation = "laplacian")
dist_root_euclidean(x, y, representation = "laplacian")
```

## Arguments

- x
An

`igraph::igraph`

object or a matrix representing an underlying network.- y
An

`igraph::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"`

,`"modularity"`

or`"graphon"`

. Default is`"laplacian"`

.- matching_iterations
An integer value specifying the maximum number of runs when looking for the optimal permutation for graph matching. Defaults to

`0L`

in which case no matching is done.- target_matrix
A square numeric matrix of size

`n`

equal to the order of the graphs specifying a target matrix towards which the initial doubly stochastic matrix is shrunk each time the graph matching algorithm fails to provide a good minimum. Defaults to`NULL`

in which case the target matrix is automatically chosen between the identity matrix or the uniform matrix on the n-simplex.

## 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 $$\frac{1}{N(N-1)} \sum_{i,j} |X_{ij} - Y_{ij}|,$$ where \(N\) is the number of vertices in networks \(x\) and \(y\). The Frobenius distance between \(x\) and \(y\) is given by $$\sqrt{\sum_{i,j} (X_{ij} - Y_{ij})^2}.$$ The spectral distance between \(x\) and \(y\) is given by $$\sqrt{\sum_i (a_i - b_i)^2},$$ 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 columns 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 $$\sqrt{\sum_i (V \sqrt{A} V^{-1} - U \sqrt{B} U^{-1})^2}.$$ Root-Euclidean distance can used only with the laplacian matrix representation.

## Examples

```
g1 <- igraph::sample_gnp(20, 0.1)
g2 <- igraph::sample_gnp(20, 0.2)
dist_hamming(g1, g2, "adjacency")
#> [1] 0.3105263
dist_frobenius(g1, g2, "adjacency")
#> [1] 10.86278
dist_spectral(g1, g2, "laplacian")
#> [1] 13.81045
dist_root_euclidean(g1, g2, "laplacian")
#> [1] 11.22648
```