Spectral Embedding of the Laplacian of a Graph

Spectral decomposition of Laplacian matrices of graphs.

Usage

embed_laplacian_matrix(
  graph,
  no,
  weights = NULL,
  which = c("lm", "la", "sa"),
  type = c("default", "D-A", "DAD", "I-DAD", "OAP"),
  scaled = TRUE,
  options = arpack_defaults()
)

Arguments

graph

The input graph, directed or undirected.

no

An integer scalar. This value is the embedding dimension of the spectral embedding. Should be smaller than the number of vertices. The largest no-dimensional non-zero singular values are used for the spectral embedding.

weights

Optional positive weight vector for calculating a weighted embedding. If the graph has a weight edge attribute, then this is used by default. For weighted embedding, edge weights are used instead of the binary adjacency matrix, and vertex strength (see strength()) is used instead of the degrees.

which

Which eigenvalues (or singular values, for directed graphs) to use. ‘lm’ means the ones with the largest magnitude, ‘la’ is the ones (algebraic) largest, and ‘sa’ is the (algebraic) smallest eigenvalues. The default is ‘lm’. Note that for directed graphs ‘la’ and ‘lm’ are the equivalent, because the singular values are used for the ordering.

type

The type of the Laplacian to use. Various definitions exist for the Laplacian of a graph, and one can choose between them with this argument.

Possible values: D-A means \(D-A\) where \(D\) is the degree matrix and \(A\) is the adjacency matrix; DAD means \(D^{1/2}\) times \(A\) times \(D^{1/2}{D^1/2}\), \(D^{1/2}\) is the inverse of the square root of the degree matrix; I-DAD means \(I-D^{1/2}\), where \(I\) is the identity matrix. OAP is \(O^{1/2}AP^{1/2}\), where \(O^{1/2}\) is the inverse of the square root of the out-degree matrix and \(P^{1/2}\) is the same for the in-degree matrix.

OAP is not defined for undirected graphs, and is the only defined type for directed graphs.

The default (i.e. type default) is to use D-A for undirected graphs and OAP for directed graphs.

scaled

Logical scalar, if FALSE, then \(U\) and \(V\) are returned instead of \(X\) and \(Y\).

options

A named list containing the parameters for the SVD computation algorithm in ARPACK. By default, the list of values is assigned the values given by arpack_defaults().

Value

A list containing with entries:

X: Estimated latent positions, an n times no matrix, n is the number of vertices.
Y: NULL for undirected graphs, the second half of the latent positions for directed graphs, an n times no matrix, n is the number of vertices.
D: The eigenvalues (for undirected graphs) or the singular values (for directed graphs) calculated by the algorithm.
options: A named list, information about the underlying ARPACK computation. See arpack() for the details.

Details

This function computes a no-dimensional Euclidean representation of the graph based on its Laplacian matrix, \(L\). This representation is computed via the singular value decomposition of the Laplacian matrix.

They are essentially doing the same as embed_adjacency_matrix(), but work on the Laplacian matrix, instead of the adjacency matrix.

References

Sussman, D.L., Tang, M., Fishkind, D.E., Priebe, C.E. A Consistent Adjacency Spectral Embedding for Stochastic Blockmodel Graphs, Journal of the American Statistical Association, Vol. 107(499), 2012

Author

Gabor Csardi csardi.gabor@gmail.com

igraph_laplacian_spectral_embedding().

Examples


## A small graph
lpvs <- matrix(rnorm(200), 20, 10)
lpvs <- apply(lpvs, 2, function(x) {
  return(abs(x) / sqrt(sum(x^2)))
})
RDP <- sample_dot_product(lpvs)
embed <- embed_laplacian_matrix(RDP, 5)