Alter homophily

E-I index and Yule's Q of each node, for categories that can overlap (Everett and Borgatti, 2026).

Usage

alter_homophily(
  A,
  B,
  method = c("ei", "yule"),
  similarity = c("product", "minimum", "cosine")
)

Arguments

A

A square matrix

B

A matrix of the membership of the nodes (rows) in the categories (columns), or a vector with the category of each node

method

The index: ei (default) or yule

similarity

The similarity of the memberships: product (default), minimum or cosine

Value

This function returns a vector with the index of each node.

Details

The internal ties of a node \(i\) are \(I = \sum_j A_{ij} S_{ij}\), where \(S_{ij}\) is the similarity of the memberships of \(i\) and \(j\), and the external ties are \(E = D - I\), where \(D\) is the degree. The E-I index is \((E - I) / (E + I)\) (Krackhardt and Stern, 1988): -1 when all the alters are in the categories of ego (homophily) and +1 when none is (heterophily).

Yule's Q also uses the nodes that are not alters, to take into account how many nodes of each category are available: \(a = I\) and \(b = E\) for the alters, and \(c\) and \(d\) are the same quantities for the other nodes. Then \(Q = (ad - bc) / (ad + bc)\), which is positive for homophily, and zero when the ties do not depend on the categories.

The similarity of two memberships can be defined in three ways:

similarity = "product" (default), \(S_{ij} = \sum_k B_{ik} B_{jk}\). It reduces to the usual indices when the categories are a partition. If the categories are, for instance, the proportion of time spent in each of several places, it is the probability that two nodes are in the same place.

similarity = "minimum", \(S_{ij} = \sum_k \min(B_{ik}, B_{jk})\), the trait version (\(E_s-I_s\) and \(Q_s\)). Two nodes with identical memberships are fully similar even when they split their time among categories, which fits categories that are traits such as skills or interests.

similarity = "cosine", the cosine of the memberships, which favours two nodes that concentrate in the same categories, such as two specialists with the same specialty.

The ties are binary and the loops are ignored. For a directed network the alters are the out-neighbours. Nodes without alters, and Yule's Q with \(ad + bc = 0\), are NA.

References

Everett, M. G. and Borgatti, S. P. (2026). Alter composition with overlapping group memberships. Social Networks, 85, 80–88. doi:10.1016/j.socnet.2025.12.001

Krackhardt, D. and Stern, R. N. (1988). Informal networks and organizational crises: An experimental simulation. Social Psychology Quarterly, 51(2), 123–140. doi:10.2307/2786835

Author

Alejandro Espinosa-Rada

Examples

A <- matrix(c(
  0, 1, 1, 0,
  1, 0, 1, 1,
  1, 1, 0, 0,
  0, 1, 0, 0
), byrow = TRUE, ncol = 4)
rownames(A) <- colnames(A) <- c("a", "b", "c", "d")
B <- matrix(c(
  10, 0,
  5, 5,
  0, 8,
  2, 6
), byrow = TRUE, ncol = 2)

alter_homophily(A, B)
#>   a   b   c   d 
#> 0.5 0.0 0.5 0.0 
alter_homophily(A, B, method = "yule", similarity = "minimum")
#>    a    b    c    d 
#>  0.0   NA -0.8  0.5