# scg_semi_proj

##### Semi-Projectors

A function to compute the \(L\) and \(R\) semi-projectors for a given partition of the vertices.

##### Usage

```
scg_semi_proj(groups, mtype = c("symmetric", "laplacian", "stochastic"),
p = NULL, norm = c("row", "col"), sparse = igraph_opt("sparsematrices"))
```

##### Arguments

- groups
A vector of

`nrow(X)`

or`vcount(X)`

integers giving the group label of every vertex in the partition.- mtype
The type of semi-projectors. For now “symmetric”, “laplacian” and “stochastic” are available.

- p
A probability vector of length

`length(gr)`

.`p`

is the stationary probability distribution of a Markov chain when`mtype`

= “stochastic”. This parameter is ignored in all other cases.- norm
Either “row” or “col”. If set to “row” the rows of the Laplacian matrix sum up to zero and the rows of the stochastic sum up to one; otherwise it is the columns.

- sparse
Logical scalar, whether to return sparse matrices.

##### Details

The three types of semi-projectors are defined as follows. Let \(\gamma(j)\) label the group of vertex \(j\) in a partition of all the vertices.

The symmetric semi-projectors are defined as $$L_{\alpha j}=R_{\alpha
j}= $$$$
\frac{1}{\sqrt{|\alpha|}}\delta_{\alpha\gamma(j)},$$ the (row) Laplacian
semi-projectors as $$L_{\alpha
j}=\frac{1}{|\alpha|}\delta_{\alpha\gamma(j)}\,\,\,\, $$$$ \textrm{and}\,\,\,\, R_{\alpha
j}=\delta_{\alpha\gamma(j)},$$ and the (row) stochastic
semi-projectors as $$L_{\alpha
j}=\frac{p_{1}(j)}{\sum_{k\in\gamma(j)}p_{1}(k)}\,\,\,\, $$$$ \textrm{and}\,\,\,\, R_{\alpha
j}=\delta_{\alpha\gamma(j)\delta_{\alpha\gamma(j)}},$$ where \(p_1\) is the (left) eigenvector
associated with the one-eigenvalue of the stochastic matrix. \(L\) and
\(R\) are defined in a symmetric way when `norm = col`

. All these
semi-projectors verify various properties described in the reference.

##### Value

The semi-projector \(L\).

The semi-projector \(R\).

##### References

D. Morton de Lachapelle, D. Gfeller, and P. De Los Rios,
Shrinking Matrices while Preserving their Eigenpairs with Application to the
Spectral Coarse Graining of Graphs. Submitted to *SIAM Journal on
Matrix Analysis and Applications*, 2008.
http://people.epfl.ch/david.morton

##### See Also

scg-method for a detailed introduction. `scg`

,
`scg_eps`

, `scg_group`

##### Examples

```
# NOT RUN {
library(Matrix)
# compute the semi-projectors and projector for the partition
# provided by a community detection method
g <- sample_pa(20, m = 1.5, directed = FALSE)
eb <- cluster_edge_betweenness(g)
memb <- membership(eb)
lr <- scg_semi_proj(memb)
#In the symmetric case L = R
tcrossprod(lr$R) # same as lr$R %*% t(lr$R)
P <- crossprod(lr$R) # same as t(lr$R) %*% lr$R
#P is an orthogonal projector
isSymmetric(P)
sum( (P %*% P-P)^2 )
## use L and R to coarse-grain the graph Laplacian
lr <- scg_semi_proj(memb, mtype="laplacian")
L <- laplacian_matrix(g)
Lt <- lr$L %*% L %*% t(lr$R)
## or better lr$L %*% tcrossprod(L,lr$R)
rowSums(Lt)
# }
```

*Documentation reproduced from package igraph, version 1.2.2, License: GPL (>= 2)*