Block densities and image matrix

Densities of the blocks given by a partition of the nodes, and the image matrix that summarises them (Lorrain and White, 1971; Wasserman and Faust, 1994).

Usage

block_density(A, partition, cutoff = "density", loops = FALSE)

Arguments

A

A square matrix

partition

A vector with the position of each node

cutoff

Density above which a block is a one in the image matrix: a number, or density (default) for the density of the whole network

loops

Whether the loops are counted as possible ties in the diagonal blocks

Value

This function returns the density of each block, the image matrix and the density of the network.

Details

A partition of the nodes divides the matrix into blocks. The density of a block is the proportion of the possible ties that are present, and the image matrix assigns a one to the blocks whose density is at least the cutoff. The usual criterion is the density of the whole network (cutoff = "density"), so a block is one when it is denser than the network as a whole.

The diagonal blocks contain the ties within a position, where the possible ties exclude the loops unless loops = TRUE.

References

Lorrain, F. and White, H. C. (1971). Structural equivalence of individuals in social networks. Journal of Mathematical Sociology, 1(1), 49–80. doi:10.1080/0022250X.1971.9989788

Wasserman, S. and Faust, K. (1994). Social network analysis: Methods and applications. Cambridge University Press.

Author

Alejandro Espinosa-Rada

Examples

A <- matrix(c(
  0, 1, 1, 0, 0, 0,
  1, 0, 1, 0, 0, 0,
  1, 1, 0, 1, 1, 1,
  0, 0, 1, 0, 0, 0,
  0, 0, 1, 0, 0, 0,
  0, 0, 1, 0, 0, 0
), byrow = TRUE, ncol = 6)
rownames(A) <- letters[1:nrow(A)]
colnames(A) <- rownames(A)

block_density(A, partition = c(1, 1, 1, 2, 2, 2))
#> $densities
#>           1         2
#> 1 1.0000000 0.3333333
#> 2 0.3333333 0.0000000
#> 
#> $image
#>   1 2
#> 1 1 0
#> 2 0 0
#> 
#> $density
#> [1] 0.4
#>