Everett and Borgatti specification of the constraint measure for binary, directed and valued matrices
Value
This function returns term 1, 2 and 3, the normalization and the maximum value of the specification of Everett and Borgatti (2020), and the constraint of Burt (1992).
Details
The constraint of Burt (1992) is computed in the ego network, from the proportion \(p_{ij}\) of the ties of each node \(i\) that go to \(j\), and is split in the three terms of Everett and Borgatti (2020: Eq. 2): $$\sum_j p_{ij}^2 + 2 \sum_j p_{ij} \sum_q p_{iq} p_{qj} + \sum_j \left(\sum_q p_{iq} p_{qj}\right)^2$$ For binary undirected networks, the first term is one over the number of alters \(N\).
Burt (1992) uses the ties in both directions, \(p_{ij} \propto a_{ij} + a_{ji}\), so the constraint of a directed network is that of the undirected valued network \(A + A^T\), in which a reciprocated tie counts twice and an unreciprocated tie once (Everett and Borgatti, 2020: 53).
The normalization is \((c - 1/N) / (c_{max} - 1/N)\), where \(1/N\) is the minimum and \(c_{max}\) the maximum constraint of an ego with \(N\) alters. The maximum is reached in a complete ego network or in a shadow ego network, in which one alter is tied to all the others and there are no other ties among alters (Everett and Borgatti, 2020: Eq. 4, 6, 7, 8 and 9). For valued networks the maximum depends on the smallest (\(m\)) and the largest (\(M\)) value of the ties in the ego network, and binary networks are the case \(m = M = 1\). The maximum is a conjecture of Everett and Borgatti, checked by enumerating ego networks. An ego with a single alter has a normalized constraint of one.
References
Burt, R.S., 1992. Structural Holes: the Social Structure of Competition. Harvard University Press, Cambridge.
Everett, M.G. and Borgatti, S., 2020. Unpacking Burt's constraint measure. Social Networks 62, pp. 50-57. doi:10.1016/j.socnet.2020.02.001
Examples
A <- matrix(c(
0, 1, 1, 0, 0, 1,
1, 0, 1, 0, 0, 1,
1, 1, 0, 0, 0, 1,
0, 0, 0, 0, 1, 1,
0, 0, 0, 1, 0, 1,
1, 1, 1, 1, 1, 0
), ncol = 6, byrow = TRUE)
rownames(A) <- letters[1:nrow(A)]
colnames(A) <- letters[1:ncol(A)]
eb_constraint(A, ego = "f")
#> $results
#> term1 term2 term3 constraint normalization
#> f 0.2 0.24 0.073 0.513 0.699
#>
#> $maximum
#> f
#> 0.648
#>
# Directed network: f -> a is not reciprocated
D <- A
D["a", "f"] <- 0
eb_constraint(D, ego = "f", digraph = TRUE)
#> $results
#> term1 term2 term3 constraint normalization
#> f 0.21 0.237 0.075 0.522 0.517
#>
#> $maximum
#> f
#> 0.823
#>
# Valued network
W <- A
W["f", "a"] <- W["a", "f"] <- 3
eb_constraint(W, ego = "f", weighted = TRUE)
#> $results
#> term1 term2 term3 constraint normalization
#> f 0.265 0.199 0.055 0.519 0.429
#>
#> $maximum
#> f
#> 0.944
#>