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.

Value

L: The semi-projector $L$.
R: The semi-projector $R$.

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.

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

Author

David Morton de Lachapelle, http://people.epfl.ch/david.morton.

Examples


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$R)
#> 4 x 4 sparse Matrix of class "dsCMatrix"
#>             
#> [1,] 1 . . .
#> [2,] . 1 . .
#> [3,] . . 1 .
#> [4,] . . . 1
P <- crossprod(lr$R) # same as t(lr$R) %*% lr$R
# P is an orthogonal projector
isSymmetric(P)
#> [1] TRUE
sum((P %*% P - P)^2)
#> [1] 3.5206e-31

## 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)
#> [1] 1.110223e-16 1.110223e-16 1.110223e-16 1.110223e-16