Effective size, efficiency and constraint of Burt (1992) for every node of a valued or directed network, with the option of treating the alters of the same category as redundant (Everett and Borgatti, 2026).
Arguments
- A
A square matrix, binary or valued
- B
An optional matrix of the membership of the nodes (rows) in the categories (columns), or a vector with the category of each node
- beta
The strength of the tie added between two alters of the same category
- ego_network
Whether the measures are computed within the ego network of each node (TRUE) or within the whole network
Value
This function returns a data frame with the number of alters, the effective size, the efficiency and the constraint of each node.
Details
Burt (1992) measures the ties of a node \(i\) with its alters \(j\) through the proportion of the ties of \(i\) that go to \(j\), \(p_{ij} = (a_{ij} + a_{ji}) / \sum_k (a_{ik} + a_{ki})\), so a directed tie counts in both directions. The effective size is \(\sum_j (1 - \sum_q p_{iq} m_{jq})\), where \(m_{jq}\) is the tie of \(j\) with \(q\) divided by the strongest tie of \(j\); the efficiency is the effective size divided by the number of alters; and the constraint is \(\sum_j (p_{ij} + \sum_q p_{iq} p_{qj})^2\).
With ego_network = TRUE (default, as in UCINET) the proportions are computed within the ego network of
each node, and with ego_network = FALSE within the whole network, as in igraph::constraint(). For
binary undirected networks and ego networks, the effective size is the one of redundancy() and the
constraint the one of eb_constraint().
When B is given, two alters of the same category are treated as partly redundant even when they are not
tied: each missing tie between two alters is given the value \(\beta \sum_k B_{xk} B_{yk}\), the product of
their memberships, before the measures are computed (Everett and Borgatti, 2026: Eq. 6). With a partition and
beta = 1 two alters of the same category count as tied; beta = 0 gives the original measures, and
the values in between set how much the category matters.
References
Burt, R.S., 1992. Structural Holes: the Social Structure of Competition. Harvard University Press, Cambridge.
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
Examples
data(campnet)
structural_holes(campnet$network)
#> alters effective_size efficiency constraint
#> HOLLY 5 3.857143 0.7714286 0.4363719
#> BRAZEY 3 1.000000 0.3333333 1.0238889
#> CAROL 3 2.000000 0.6666667 0.8044444
#> PAM 5 3.875000 0.7750000 0.5163194
#> PAT 4 3.571429 0.8928571 0.3900227
#> JENNIE 3 2.333333 0.7777778 0.6111111
#> PAULINE 5 3.857143 0.7714286 0.5365079
#> ANN 3 1.600000 0.5333333 0.9333333
#> MICHAEL 5 3.071429 0.6142857 0.6010603
#> BILL 3 1.000000 0.3333333 1.0800000
#> LEE 3 1.666667 0.5555556 0.8227778
#> DON 4 2.142857 0.5357143 0.7018141
#> JOHN 3 2.333333 0.7777778 0.7283951
#> HARRY 4 1.750000 0.4375000 0.7903321
#> GERY 4 2.900000 0.7250000 0.6491111
#> STEVE 5 3.062500 0.6125000 0.6148247
#> BERT 4 2.214286 0.5535714 0.7183263
#> RUSS 4 2.785714 0.6964286 0.6275510
# Alters of the same gender are partly redundant
structural_holes(campnet$network, B = campnet$attributes$gender, beta = 0.5)
#> alters effective_size efficiency constraint
#> HOLLY 5 3.500000 0.7000000 0.5185714
#> BRAZEY 3 1.000000 0.3333333 1.0238889
#> CAROL 3 1.500000 0.5000000 0.9023480
#> PAM 5 2.000000 0.4000000 0.6359449
#> PAT 4 2.285714 0.5714286 0.6843537
#> JENNIE 3 1.666667 0.5555556 0.8227778
#> PAULINE 5 3.142857 0.6285714 0.5869581
#> ANN 3 1.300000 0.4333333 0.9467222
#> MICHAEL 5 2.047619 0.4095238 0.6829291
#> BILL 3 1.000000 0.3333333 1.0800000
#> LEE 3 1.666667 0.5555556 0.8227778
#> DON 4 2.071429 0.5178571 0.7123272
#> JOHN 3 2.333333 0.7777778 0.7283951
#> HARRY 4 1.666667 0.4166667 0.8059043
#> GERY 4 1.483333 0.3708333 0.7961985
#> STEVE 5 2.437500 0.4875000 0.6535387
#> BERT 4 1.928571 0.4821429 0.7505675
#> RUSS 4 1.797619 0.4494048 0.7328385