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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.

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. “unnormalized” Unnormalized Laplacian. “symmetric” Symmetric normalized Laplacian. “left” Left-stochastic normalized Laplacian. “right” Right-stochastic normalized Laplacian.

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.

Author

Gabor Csardi csardi.gabor@gmail.com

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