Cohesion Matrix — cohesion

Creates a matrix of (pairwise) cohesion values from a matrix of pairwise distances or a dist object.

Usage

cohesion_matrix(d)

Arguments

d: A matrix of pairwise distances or a dist object.

Value

The matrix of cohesion values. An object of class cohesion_matrix.

Details

Computes the matrix of (pairwise) cohesion values, C_xw, from a matrix of pairwise distances or a dist object. Cohesion is an interpretable probability that reflects the strength of alignment of a point, w, to another point, x. The rows of the cohesion matrix can be seen as providing neighborhood weights. These values may be used for defining associated weighted graphs (for the purpose of community analysis) as in Berenhaut, Moore, and Melvin (2022).

Given an n x n distance matrix, the sum of the entries in the resulting cohesion matrix is always equal to n/2. Cohesion is partitioned local depth (see local_depths) and thus the row sums of the cohesion matrix provide a measure of local depth centrality.

If you have a matrix that is already a cohesion matrix and you would like to add the class, see as_cohesion_matrix().

References

K. S. Berenhaut, K. E. Moore, R. L. Melvin, A social perspective on perceived distances reveals deep community structure. Proc. Natl. Acad. Sci., 119(4), 2022.

Examples


plot(exdata1)
text(exdata1 + .08, lab = 1:8)


D <- dist(exdata1)
C <- cohesion_matrix(D)
C
#>            1          2          3          4          5          6          7
#> 1 0.16581633 0.11989796 0.11989796 0.01785714 0.01785714 0.00000000 0.00000000
#> 2 0.13061224 0.17653061 0.01785714 0.11989796 0.00000000 0.00000000 0.02040816
#> 3 0.11870748 0.02040816 0.16462585 0.10544218 0.03826531 0.01785714 0.00000000
#> 4 0.01785714 0.10425170 0.10425170 0.17874150 0.06207483 0.09540816 0.06207483
#> 5 0.01785714 0.00000000 0.03571429 0.05612245 0.18707483 0.12159864 0.07993197
#> 6 0.00000000 0.00000000 0.02040816 0.07653061 0.15986395 0.20748299 0.15986395
#> 7 0.00000000 0.01785714 0.00000000 0.05952381 0.07993197 0.12159864 0.18707483
#> 8 0.00000000 0.07993197 0.00000000 0.00000000 0.00000000 0.01785714 0.03571429
#>            8
#> 1 0.00000000
#> 2 0.07653061
#> 3 0.00000000
#> 4 0.00000000
#> 5 0.00000000
#> 6 0.01785714
#> 7 0.03571429
#> 8 0.15136054
#> attr(,"class")
#> [1] "cohesion_matrix" "matrix"          "array"          

## neighbor weights (provided by cohesion) for the 8th point in exdata1
C[8, ]
#>          1          2          3          4          5          6          7 
#> 0.00000000 0.07993197 0.00000000 0.00000000 0.00000000 0.01785714 0.03571429 
#>          8 
#> 0.15136054 
localdepths <- rowSums(C)