centrality_coleman_theil: Coleman-Theil hierarchy index

In cograph: Analysis and Visualization of Complex Networks

Coleman-Theil hierarchy index

Description

Measures concentration of Burt's dyadic constraints over a node's contacts. Let mutual tie strength be z_ij+z_ji, and p_ij its proportion of all mutual strength incident to i. With organizational weights fixed at one, define

c_{ij}=(p_{ij}+\sum_q p_{iq}p_{qj})^2,\quad r_{ij}=c_{ij}/\operatorname{mean}_{k\in N(i)}c_{ik}.

The index is \sum_{j\in N(i)}r_{ij}\log(r_{ij})/(d_i\log(d_i)). Contacts are distinct nodes with positive mutual strength. Investment proportions use the full supplied graph, including alters' outside ties.

Usage

centrality_coleman_theil(x, ...)

Arguments

x	Network input accepted by centrality.
...	Additional arguments to centrality.

Details

Follows Burt's STRUCTURE 4.2 manual (pages 181-183): isolates score zero and nodes with one contact score one. The general formula is undefined in these two cases; these are the author's explicit conventions. JUNG's documented implementation instead returns NaN for isolates. Values range from zero for equal constraints to one for complete concentration. Input organizational/oligopoly multipliers from STRUCTURE are not implemented; they are fixed at one, as in the Zoo's formula.

Finite nonnegative weights are supported. Zero-weight ties are absent, loops are removed, and remaining parallel edges sum after generic simplification. Directed ties are combined by summing both directions; weighted=FALSE assigns unit weight to each retained edge before combining them, so reciprocity can affect mutual investment. Generic mode, shortest-path inversion and cutoff do not affect the result. Empty input returns no scores. Components are independent before global normalization.

The default output is already the unit-interval hierarchy index. normalized=TRUE additionally divides by the largest node score; an all-zero vector remains zero. Dense native arithmetic costs O(n cubed) time and O(n squared) memory. Global weight scaling precedes mutual sums. Unrepresentable positive weight or investment ranges raise an error; tiny squared constraints may underflow and use the zero-log-zero limit. Relative deviations of local constraints within 16 machine epsilons are treated as uniform; a series stabilizes the entropy near uniformity.

Value

Named numeric vector in input node order.

References

Burt, R. S. (1992). Structural Holes: The Social Structure of Competition. Harvard University Press. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.4159/9780674029095")}.

Examples

centrality_coleman_theil(igraph::make_star(5, mode = "undirected"))

cograph documentation built on Sept. 30, 2026, 5:08 p.m.