These function provide different measures of the degree to which nodes fill structural holes, as outlined in Burt (1992). Burt's theory holds that while those nodes embedded in dense clusters of close connections are likely exposed to the same or similar ideas and information, those who fill structural holes between two otherwise disconnected groups can gain some comparative advantage from that position.

node_bridges(object)

node_redundancy(object)

node_effsize(object)

node_efficiency(object)

node_constraint(object)

node_hierarchy(object)

## Arguments

object

An object of a migraph-consistent class:

• matrix (adjacency or incidence) from {base} R

• edgelist, a data frame from {base} R or tibble from {tibble}

• igraph, from the {igraph} package

• network, from the {network} package

• tbl_graph, from the {tidygraph} package

## Details

A number of different ways of measuring these structural holes are available. Note that we use Borgatti's reformulation for unweighted networks in node_redundancy() and node_effsize(). Redundancy is thus $$\frac{2t}{n}$$, where $$t$$ is the sum of ties and $$n$$ the sum of nodes in each node's neighbourhood, and effective size is calculated as $$n - \frac{2t}{n}$$. Node efficiency is the node's effective size divided by its degree.

## Functions

• node_bridges(): Returns the sum of bridges to which each node is adjacent.

• node_redundancy(): Returns a measure of the redundancy of each nodes' contacts.

• node_effsize(): Returns nodes' effective size

• node_efficiency(): Returns nodes' efficiency

• node_constraint(): Returns nodes' constraint scores for one-mode networks according to Burt (1992) and for two-mode networks according to Hollway et al (2020).

• node_hierarchy(): Returns nodes' exposure to hierarchy, where only one or two contacts are the source of closure

## References

Burt, Ronald S. 1992. Structural Holes: The Social Structure of Competition. Cambridge, MA: Harvard University Press.

Borgatti, Steven. 1997. “Structural Holes: Unpacking Burt’s Redundancy MeasuresConnections 20(1):35-38.

Hollway, James, Jean-Frédéric Morin, and Joost Pauwelyn. 2020. "Structural conditions for novelty: the introduction of new environmental clauses to the trade regime complex." International Environmental Agreements: Politics, Law and Economics 20 (1): 61–83. doi:10.1007/s10784-019-09464-5 .

Other measures: centralisation, centrality, closure, cohesion(), diversity, features, tie_centrality

## Examples

node_bridges(ison_adolescents)
#>   Betty   Sue Alice  Jane  Dale   Pam Carol  Tina
#> 1     1     1     0     0     0     1     2     1
node_bridges(ison_southern_women)
#>   EVELYN LAURA THERESA BRENDA CHARLOTTE FRANCES ELEANOR PEARL  RUTH VERNE  MYRA
#> 1      0     0       0      0         0       0       0     0     0     0     0
#> # ... with 7 more from this nodeset in the vector.
#>      E1    E2    E3    E4    E5    E6    E7    E8    E9   E10   E11   E12   E13
#> 1     0     0     0     0     0     0     0     0     0     0     0     0     0
#> # ... with 1 more from this nodeset in the vector.
#>   Betty   Sue Alice  Jane  Dale   Pam Carol  Tina
#> 1     6   2.5   2.5     4  3.33     4     2     2
node_redundancy(ison_southern_women)
#>   EVELYN LAURA THERESA BRENDA CHARLOTTE FRANCES ELEANOR PEARL  RUTH VERNE  MYRA
#> 1   12.5  12.9    14.2   13.1        12      16      18  20.7    20  19.5  17.5
#> # ... with 7 more from this nodeset in the vector.
#>      E1    E2    E3    E4    E5    E6    E7    E8    E9   E10   E11   E12   E13
#> 1  12.7  13.3  10.7  11.5   9.5  10.2    10     9     8    10   7.5  9.43    12
#> # ... with 1 more from this nodeset in the vector.
#>   Betty   Sue Alice  Jane   Dale   Pam Carol  Tina
#> 1    -5   1.5   1.5    -2 -0.333    -1     0    -1
node_effsize(ison_southern_women)
#>   EVELYN LAURA THERESA BRENDA CHARLOTTE FRANCES ELEANOR PEARL  RUTH VERNE  MYRA
#> 1   -4.5 -5.86   -6.25  -6.14        -8     -12     -14 -17.7   -16 -15.5 -13.5
#> # ... with 7 more from this nodeset in the vector.
#>      E1    E2    E3    E4    E5    E6    E7    E8    E9   E10   E11   E12   E13
#> 1 -9.67 -10.3 -4.67  -7.5  -1.5 -2.25     0     5     4    -4  -3.5 -2.43    -8
#> # ... with 1 more from this nodeset in the vector.
#>      V1    V2    V3    V4     V5     V6    V7    V8
#> 1    -5 0.375 0.375    -1 -0.111 -0.333     0    -1
node_efficiency(ison_southern_women)
#>       V1     V2     V3     V4    V5    V6    V7    V8    V9   V10   V11   V12
#> 1 -0.562 -0.837 -0.781 -0.878    -2    -3  -3.5 -5.89    -4 -3.88 -3.38 -1.28
#> # ... with 6 more from this nodeset in the vector.
#>      V1    V2     V3    V4     V5     V6    V7    V8    V9    V10    V11    V12
#> 1 -3.22 -3.44 -0.778 -1.88 -0.188 -0.281     0 0.357 0.333 -0.667 -0.875 -0.347
#> # ... with 2 more from this nodeset in the vector.
node_constraint(ison_southern_women)
#>   EVELYN LAURA THERESA BRENDA CHARLOTTE FRANCES ELEANOR PEARL  RUTH VERNE  MYRA
#> 1  0.276 0.306   0.294  0.317     0.384   0.412   0.415 0.461 0.398 0.407 0.434
#> # ... with 7 more from this nodeset in the vector.
#>      E1    E2    E3    E4    E5    E6    E7    E8    E9   E10   E11   E12   E13
#> 1 0.476 0.476 0.338 0.394 0.275 0.241 0.192 0.170 0.161 0.327 0.308 0.292 0.432
#> # ... with 1 more from this nodeset in the vector.