Incidence matrix of the nodes of a network and the simplices of its clique complex or of its neighbourhood complex.
Usage
simplicial_complexes(
A,
zero_simplex = TRUE,
projection = FALSE,
complex = c("clique", "neighbourhood"),
closed = FALSE,
valued = FALSE
)Arguments
- A
A square matrix of a network, with names.
- zero_simplex
Whether to include the isolated nodes as simplices of dimension 0, for
complex = "clique".- projection
Whether to return the links between the simplices through their shared nodes, and between the nodes through their shared simplices.
- complex
The complex: the maximal cliques (
clique, default) or the neighbourhoods (neighbourhood).- closed
Whether the neighbourhoods include the node itself, for
complex = "neighbourhood".- valued
Whether the projections count the shared nodes or simplices instead of indicating whether there are any.
Value
This function returns the incidence matrix of the nodes (rows) and the simplices (columns). With
projection = TRUE, a list with the incidence matrix (simplex), the projection of the simplices
(projection1) and the projection of the nodes (projection2).
Details
A simplex is a set of nodes, and a simplicial complex a collection of simplices that contains every face of its simplices (Atkin, 1974). The complex is represented by its maximal simplices, as the faces are implied by the simplices that contain them.
With complex = "clique" (default), the simplices are the maximal cliques of the underlying
undirected network, so that a clique of four nodes is a single simplex of dimension 3. The isolated
nodes are cliques of a single node, and they are included as simplices of dimension 0 when
zero_simplex = TRUE.
With complex = "neighbourhood", each node is the simplex of its neighbours, the rows of
A (the out-neighbours of a directed network), and with closed = TRUE the node is also a
vertex of its own simplex (Raj et al., 2024). The nodes without neighbours do not form a simplex
unless the neighbourhoods are closed.
The rows of the result are the nodes and the columns the simplices, named after their nodes
(a-b-c) for the clique complex, and after the node whose neighbourhood they are (N(a)
or N[a]) for the neighbourhood complex. q_analysis(t(S), simplicial_complex = TRUE) is
the Q-analysis of the simplices, and q_analysis(S, simplicial_complex = TRUE) that of the
conjugate complex, in which the nodes are connected through the simplices they share.
References
Atkin, R. H. (1974). Mathematical structure in human affairs. New York: Crane, Rusak.
Freeman, L. C. (1980). Q-analysis and the structure of friendship networks. International Journal of Man-Machine Studies, 12(4), 367–378. doi:10.1016/S0020-7373(80)80021-6
Raj, U., Banerjee, A., Ray, S. and Bhattacharya, S. (2024). Structure of higher-order interactions in social-ecological networks through Q-analysis of their neighbourhood and clique complex. PLOS ONE, 19(8), e0306409. doi:10.1371/journal.pone.0306409
Wasserman, S. and Faust, K. (1994). Social network analysis: Methods and applications. Cambridge University Press.
Examples
A <- matrix(c(
0, 1, 1, 0, 0, 0, 0, 1, 0,
1, 0, 1, 0, 0, 0, 0, 0, 0,
1, 1, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 1, 1, 0, 0, 0,
0, 0, 0, 1, 0, 0, 0, 0, 0,
0, 0, 0, 1, 0, 0, 1, 1, 0,
0, 0, 0, 0, 0, 1, 0, 1, 0,
1, 0, 0, 0, 0, 1, 1, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0
), byrow = TRUE, ncol = 9)
rownames(A) <- letters[1:nrow(A)]
colnames(A) <- rownames(A)
simplicial_complexes(A)
#> a-b-c f-g-h a-h d-e d-f i
#> a 1 0 1 0 0 0
#> b 1 0 0 0 0 0
#> c 1 0 0 0 0 0
#> d 0 0 0 1 1 0
#> e 0 0 0 1 0 0
#> f 0 1 0 0 1 0
#> g 0 1 0 0 0 0
#> h 0 1 1 0 0 0
#> i 0 0 0 0 0 1
simplicial_complexes(A, zero_simplex = FALSE)
#> a-b-c f-g-h a-h d-e d-f
#> a 1 0 1 0 0
#> b 1 0 0 0 0
#> c 1 0 0 0 0
#> d 0 0 0 1 1
#> e 0 0 0 1 0
#> f 0 1 0 0 1
#> g 0 1 0 0 0
#> h 0 1 1 0 0
#> i 0 0 0 0 0
simplicial_complexes(A, complex = "neighbourhood", closed = TRUE)
#> N[a] N[b] N[c] N[d] N[e] N[f] N[g] N[h] N[i]
#> a 1 1 1 0 0 0 0 1 0
#> b 1 1 1 0 0 0 0 0 0
#> c 1 1 1 0 0 0 0 0 0
#> d 0 0 0 1 1 1 0 0 0
#> e 0 0 0 1 1 0 0 0 0
#> f 0 0 0 1 0 1 1 1 0
#> g 0 0 0 0 0 1 1 1 0
#> h 1 0 0 0 0 1 1 1 0
#> i 0 0 0 0 0 0 0 0 1
simplicial_complexes(A, projection = TRUE, valued = TRUE)$projection2
#> a b c d e f g h i
#> a 0 1 1 0 0 0 0 1 0
#> b 1 0 1 0 0 0 0 0 0
#> c 1 1 0 0 0 0 0 0 0
#> d 0 0 0 0 1 1 0 0 0
#> e 0 0 0 1 0 0 0 0 0
#> f 0 0 0 1 0 0 1 1 0
#> g 0 0 0 0 0 1 0 1 0
#> h 1 0 0 0 0 1 1 0 0
#> i 0 0 0 0 0 0 0 0 0