Spectral decomposition of Laplacian matrices of graphs.

```
embed_laplacian_matrix(
graph,
no,
weights = NULL,
which = c("lm", "la", "sa"),
degmode = c("out", "in", "all", "total"),
type = c("default", "D-A", "DAD", "I-DAD", "OAP"),
scaled = TRUE,
options = igraph.arpack.default
)
```

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 stregth (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.

degmode

TODO

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 undireted 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 `igraph.arpack.default`

.

A list containing with entries:

Estimated latent positions,
an `n`

times `no`

matrix, `n`

is the number of vertices.

`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.

The eigenvalues (for undirected graphs) or the singular values (for directed graphs) calculated by the algorithm.

A named list, information about the underlying ARPACK
computation. See `arpack`

for the 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.

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

```
# NOT RUN {
## 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)
# }
```

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