Brokerage roles of Gould and Fernandez (1989), for categories that can overlap (Everett and Borgatti, 2026).
Value
This function returns a matrix with the score of each node (rows) in each role, and the total.
Details
A node \(b\) brokers in a path \(a \to b \to c\) when \(a\) has no tie to \(c\). The role depends on
the categories of the three nodes: coordinator (all in the same category), gatekeeper (\(b\)
and \(c\) in the same category, \(a\) in another), representative (\(a\) and \(b\) in the same
category, \(c\) in another), consultant (also called itinerant; \(a\) and \(c\) in the same
category, \(b\) in another) and liaison (the three in different categories).
With overlapping categories, each path counts partly for each role (Everett and Borgatti, 2026: Eq. 5). The coordinator part is \(\sum_j B_{aj} B_{bj} B_{cj}\), the joint membership of the three nodes in each category; the gatekeeper part is \(\sum_j B_{bj} B_{cj} (1 - B_{aj})\); the representative part \(\sum_j B_{aj} B_{bj} (1 - B_{cj})\); the consultant part \(\sum_j B_{aj} B_{cj} (1 - B_{bj})\); and the liaison part is the rest. When the categories are a partition, each path counts for a single role, as in Gould and Fernandez (1989). The parts of each path add up to one, so the total of each node is the number of paths it brokers.
A symmetric matrix is treated as a directed network with ties in both directions, so each path is counted from both ends.
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
Gould, R. V. and Fernandez, R. M. (1989). Structures of mediation: A formal approach to brokerage in transaction networks. Sociological Methodology, 19, 89–126. doi:10.2307/270949
Examples
A <- matrix(c(
0, 1, 0, 0,
0, 0, 1, 1,
0, 0, 0, 1,
1, 0, 0, 0
), byrow = TRUE, ncol = 4)
rownames(A) <- colnames(A) <- c("a", "b", "c", "d")
brokerage_roles(A, c("x", "x", "y", "y"))
#> coordinator gatekeeper representative consultant liaison total
#> a 0 1 0 0 0 1
#> b 0 0 2 0 0 2
#> c 0 0 0 0 0 0
#> d 0 0 1 1 0 2
B <- matrix(c(
10, 0,
5, 5,
0, 8,
2, 6
), byrow = TRUE, ncol = 2)
brokerage_roles(A, B)
#> coordinator gatekeeper representative consultant liaison total
#> a 0.125 0.375 0.125 0.375 0 1
#> b 0.125 0.875 0.875 0.125 0 2
#> c 0.000 0.000 0.000 0.000 0 0
#> d 0.125 0.375 1.125 0.375 0 2