Supra-adjacency matrix — supra

Arranges the layers of a multiplex network, in which the same actors are connected by several relations, into a single matrix of actor-layer pairs (De Domenico et al., 2013; Kivela et al., 2014).

Usage

supra_adjacency(
  layers,
  coupling = c("categorical", "ordinal", "none"),
  weight = 1,
  sparse = FALSE
)

Arguments

layers: A list of square matrices of the same order, one for each layer, with the same actors in the same order
coupling: Whether the copies of an actor are joined in every pair of layers (categorical, default), only in consecutive layers (ordinal), or not at all (none)
weight: The value of the ties between the copies of the same actor
sparse: Whether to return a sparse matrix of the Matrix package

Value

This function returns the supra-adjacency matrix of the layers.

Details

The supra-adjacency matrix has one row and one column for each actor in each layer. The blocks of the diagonal are the layers, and the blocks outside it are the coupling between the layers, which joins the copies of the same actor.

coupling = "categorical" (default) joins the copies of an actor in every pair of layers, which is the usual choice when the layers are different relations. "ordinal" joins them only in consecutive layers, for layers that follow an order, such as time. "none" leaves the layers apart, and the result is block diagonal.

The weight of the coupling is the value given to those ties, and it sets how much the layers are held together in measures computed on the whole structure.

The rows and the columns are named after the actor and the layer, as actor_layer, taking the names of the matrices and of the list of layers when they have them.

References

De Domenico, M., Sole-Ribalta, A., Cozzo, E., Kivela, M., Moreno, Y., Porter, M. A., Gomez, S. and Arenas, A. (2013). Mathematical formulation of multilayer networks. Physical Review X, 3(4), 041022. doi:10.1103/PhysRevX.3.041022

Kivela, M., Arenas, A., Barthelemy, M., Gleeson, J. P., Moreno, Y. and Porter, M. A. (2014). Multilayer networks. Journal of Complex Networks, 2(3), 203-271. doi:10.1093/comnet/cnu016

Author

Alejandro Espinosa-Rada

Examples


A1 <- matrix(c(
  0, 1, 0,
  1, 0, 1,
  0, 1, 0
), byrow = TRUE, ncol = 3, dimnames = list(letters[1:3], letters[1:3]))

A2 <- matrix(c(
  0, 0, 1,
  0, 0, 0,
  1, 0, 0
), byrow = TRUE, ncol = 3, dimnames = list(letters[1:3], letters[1:3]))

supra_adjacency(list(advice = A1, friendship = A2))
#>              a_advice b_advice c_advice a_friendship b_friendship c_friendship
#> a_advice            0        1        0            1            0            0
#> b_advice            1        0        1            0            1            0
#> c_advice            0        1        0            0            0            1
#> a_friendship        1        0        0            0            0            1
#> b_friendship        0        1        0            0            0            0
#> c_friendship        0        0        1            1            0            0
#> attr(,"actors")
#> [1] "a" "b" "c"
#> attr(,"layers")
#> [1] "advice"     "friendship"

# Layers that follow an order are coupled only with the next one
supra_adjacency(list(A1, A2), coupling = "ordinal")
#>      a_L1 b_L1 c_L1 a_L2 b_L2 c_L2
#> a_L1    0    1    0    1    0    0
#> b_L1    1    0    1    0    1    0
#> c_L1    0    1    0    0    0    1
#> a_L2    1    0    0    0    0    1
#> b_L2    0    1    0    0    0    0
#> c_L2    0    0    1    1    0    0
#> attr(,"actors")
#> [1] "a" "b" "c"
#> attr(,"layers")
#> [1] "L1" "L2"