laplacian_matrix {igraph} | R Documentation |
Graph Laplacian
Description
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 |
sparse |
Logical scalar, whether to return the result as a sparse
matrix. The |
normalization |
The normalization method to use when calculating the Laplacian matrix. See the "Normalization methods" section on this page. |
normalized |
Deprecated, use |
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.
Value
A numeric matrix.
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}
.
Related documentation in the C library
igraph_get_laplacian_sparse()
, igraph_get_laplacian()
.
Author(s)
Gabor Csardi csardi.gabor@gmail.com
Examples
g <- make_ring(10)
laplacian_matrix(g)
laplacian_matrix(g, normalization = "unnormalized")
laplacian_matrix(g, normalization = "unnormalized", sparse = FALSE)