Spectral decomposition of the adjacency matrices of graphs.

```
embed_adjacency_matrix(
graph,
no,
weights = NULL,
which = c("lm", "la", "sa"),
scaled = TRUE,
cvec = graph.strength(graph, weights = weights)/(vcount(graph) - 1),
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. In a weighted embedding, the edge weights are used instead
of the binary adjacencny matrix.

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.

scaled

Logical scalar, if `FALSE`

, then \(U\) and \(V\) are
returned instead of \(X\) and \(Y\).

cvec

A numeric vector, its length is the number vertices in the graph. This vector is added to the diagonal of the adjacency matrix.

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 adjacency matrix, \(A\). This representation is
computed via the singular value decomposition of the adjacency matrix,
\(A=UDV^T\).In the case, where the graph is a random dot product graph
generated using latent position vectors in \(R^{no}\) for each vertex, the
embedding will provide an estimate of these latent vectors.

For undirected graphs the latent positions are calculated as
\(X=U^{no}D^{1/2}\), where \(U^{no}\) equals
to the first `no`

columns of \(U\), and \(D^{1/2}\) is
a diagonal matrix containing the top `no`

singular values on the
diagonal.

For directed graphs the embedding is defined as the pair \(X=U^{no}D^{1/2}\) and \(Y=V^{no}D^{1/2}\). (For undirected graphs \(U=V\), so it is enough to keep one of them.)

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_adjacency_matrix(RDP, 5)
# }
```

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