Q-structure of a simplicial complex.
Arguments
- A
An incidence matrix
- simplicial_complex
Whether the incidence matrix is a simplices or simplicial complexes representation
- dimensions
Return the successively chains from high to low dimensions ($q$) and the number of components ($Q_p$)
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. https://doi.org/10.1016/S0020-7373(80)80021-6
Examples
A <- matrix(c(
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0,
0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0,
0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0,
0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,
0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0,
0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0
), byrow = TRUE, ncol = 19)
colnames(A) <- letters[1:ncol(A)]
rownames(A) <- 1:nrow(A)
q_analysis(A, simplicial_complex = TRUE)
#> $`2`
#> component node
#> 1 1 3
#> 2 2 13
#>
#> $`6`
#> component node
#> 1 1 3
#> 2 2 13
#> 3 3 2
#> 4 4 19
#> 5 5 20
#> 6 6 21
#>
#> $`9`
#> component node
#> 1 1 3
#> 10 2 15
#> 11 3 25
#> 5 4 20
#> 7 5 4
#> 8 6 9
#> 9 7 11
#> 2 8 13
#> 4 8 19
#> 6 8 21
#> 3 9 2
#> 12 9 28
#>
#> $`3`
#> component node
#> 1 1 3
#> 2 1 13
#> 3 1 2
#> 4 1 19
#> 5 1 20
#> 6 1 21
#> 7 1 4
#> 8 1 9
#> 9 1 11
#> 10 1 15
#> 11 1 25
#> 12 1 28
#> 14 1 6
#> 15 1 8
#> 16 1 10
#> 18 1 14
#> 19 1 16
#> 20 1 17
#> 23 1 24
#> 24 1 26
#> 25 1 29
#> 13 2 5
#> 21 2 18
#> 17 3 12
#> 22 3 23
#>