# embed_laplacian_matrix

From igraph v1.0.0
by Gabor Csardi

##### Spectral Embedding of the Laplacian of a Graph

Spectral decomposition of Laplacian matrices of graphs.

- Keywords
- graphs

##### Usage

```
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)
```

##### 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 weighted
closeness. If the graph has a
`weight`

edge attribute, then this is used by default. - 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, andsa is the (algebraic) smallest eigenvalues. The de - 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; - 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`

.

##### 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.

##### 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.

##### 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

##### See Also

##### 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)
```

*Documentation reproduced from package igraph, version 1.0.0, License: GPL (>= 2)*

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