Coleman-Theil Hierarchy Index

The Coleman-Theil index (Burt 1992) measures how concentrated Burt's dyadic constraint is across the $d_i$ contacts of node $i$. Let $p_{ij}$ be the share of the mutual tie strength of $i$ invested in $j$, $c_{ij} = (p_{ij} + \sum_q p_{iq} p_{qj})^2$ the constraint, and $r_{ij}$ the constraint divided by its mean over the contacts. Then $$H_i = \frac{\sum_{j \in N(i)} r_{ij} \log r_{ij}}{d_i \log d_i}.$$

Usage

centrality_coleman_theil(x, ...)

Arguments

x: Network input accepted by centrality.
...: Further arguments to centrality. The measure uses weighted (use edge weights, default TRUE).

Value

A named numeric vector with one score per node, in input node order.

Details

The mutual tie strength of a pair is the sum of the weights in both directions, so direction is combined and loops are removed. Edge weights must be finite and nonnegative, and weighted = FALSE gives every edge weight one before the directions are combined. The index lies between zero, for equal constraints, and one, for constraint concentrated on one contact. Following Burt's STRUCTURE 4.2 manual (pages 181-183), an isolated node scores zero and a node with one contact scores one. The organizational and oligopoly multipliers of STRUCTURE are fixed at one. A weight range beyond double precision raises an error.

References

Burt, R. S. (1992). Structural Holes: The Social Structure of Competition. Harvard University Press. doi:10.4159/9780674029095 .

Examples

centrality_coleman_theil(regulation_net)
#>    Explore       Plan    Monitor      Adapt    Reflect    Discuss Synthesize 
#> 0.07165311 0.10844104 0.15612256 0.06285124 0.17918268 0.05549900 0.39363179 
#>   Evaluate     Create      Share 
#> 0.12236245 0.14733940 0.05273089