Graph Laplacian

The Laplacian of a graph.

Usage

laplacian_matrix(
  graph,
  weights = NULL,
  sparse = igraph_opt("sparsematrices"),
  normalization = c("unnormalized", "symmetric", "left", "right"),
  normalized
)

Arguments

graph: The input graph.
weights: An optional vector giving edge weights for weighted Laplacian matrix. If this is NULL and the graph has an edge attribute called weight, then it will be used automatically. Set this to NA if you want the unweighted Laplacian on a graph that has a weight edge attribute.
sparse: Logical scalar, whether to return the result as a sparse matrix. The Matrix package is required for sparse matrices.
normalization: The normalization method to use when calculating the Laplacian matrix. See the "Normalization methods" section on this page.
normalized: Deprecated, use normalization instead.

Value

A numeric matrix.

Details

The Laplacian Matrix of a graph is a symmetric matrix having the same number of rows and columns as the number of vertices in the graph and element (i,j) is d[i], the degree of vertex i if if i==j, -1 if i!=j and there is an edge between vertices i and j and 0 otherwise.

The Laplacian matrix can also be normalized, with several conventional normalization methods. See the "Normalization methods" section on this page.

The weighted version of the Laplacian simply works with the weighted degree instead of the plain degree. I.e. (i,j) is d[i], the weighted degree of vertex i if if i==j, -w if i!=j and there is an edge between vertices i and j with weight w, and 0 otherwise. The weighted degree of a vertex is the sum of the weights of its adjacent edges.

Normalization methods

The Laplacian matrix \(L\) is defined in terms of the adjacency matrix \(A\) and a diagonal matrix \(D\) containing the degrees as follows:

"unnormalized": Unnormalized Laplacian, \(L = D - A\).
"symmetric": Symmetrically normalized Laplacian, \(L = I - D^{-\frac{1}{2}} A D^{-\frac{1}{2}}\).
"left": Left-stochastic normalized Laplacian, \({L = I - D^{-1} A}\).
"rigth": Right-stochastic normalized Laplacian, \(L = I - A D^{-1}\).

Author

Gabor Csardi csardi.gabor@gmail.com

igraph_get_laplacian_sparse(), igraph_get_laplacian().

Examples


g <- make_ring(10)
laplacian_matrix(g)
#> 10 x 10 sparse Matrix of class "dgCMatrix"
#>                                    
#>  [1,]  2 -1  .  .  .  .  .  .  . -1
#>  [2,] -1  2 -1  .  .  .  .  .  .  .
#>  [3,]  . -1  2 -1  .  .  .  .  .  .
#>  [4,]  .  . -1  2 -1  .  .  .  .  .
#>  [5,]  .  .  . -1  2 -1  .  .  .  .
#>  [6,]  .  .  .  . -1  2 -1  .  .  .
#>  [7,]  .  .  .  .  . -1  2 -1  .  .
#>  [8,]  .  .  .  .  .  . -1  2 -1  .
#>  [9,]  .  .  .  .  .  .  . -1  2 -1
#> [10,] -1  .  .  .  .  .  .  . -1  2
laplacian_matrix(g, normalization = "unnormalized")
#> 10 x 10 sparse Matrix of class "dgCMatrix"
#>                                    
#>  [1,]  2 -1  .  .  .  .  .  .  . -1
#>  [2,] -1  2 -1  .  .  .  .  .  .  .
#>  [3,]  . -1  2 -1  .  .  .  .  .  .
#>  [4,]  .  . -1  2 -1  .  .  .  .  .
#>  [5,]  .  .  . -1  2 -1  .  .  .  .
#>  [6,]  .  .  .  . -1  2 -1  .  .  .
#>  [7,]  .  .  .  .  . -1  2 -1  .  .
#>  [8,]  .  .  .  .  .  . -1  2 -1  .
#>  [9,]  .  .  .  .  .  .  . -1  2 -1
#> [10,] -1  .  .  .  .  .  .  . -1  2
laplacian_matrix(g, normalization = "unnormalized", sparse = FALSE)
#>       [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
#>  [1,]    2   -1    0    0    0    0    0    0    0    -1
#>  [2,]   -1    2   -1    0    0    0    0    0    0     0
#>  [3,]    0   -1    2   -1    0    0    0    0    0     0
#>  [4,]    0    0   -1    2   -1    0    0    0    0     0
#>  [5,]    0    0    0   -1    2   -1    0    0    0     0
#>  [6,]    0    0    0    0   -1    2   -1    0    0     0
#>  [7,]    0    0    0    0    0   -1    2   -1    0     0
#>  [8,]    0    0    0    0    0    0   -1    2   -1     0
#>  [9,]    0    0    0    0    0    0    0   -1    2    -1
#> [10,]   -1    0    0    0    0    0    0    0   -1     2