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.
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.
References
Burt, R. S. (1991). STRUCTURE, version 4.2, Reference Manual, Columbia University, pages 181-183. These pages reproduce the hierarchy definition attributed to equation 2.9 in Burt (1992), Structural Holes: The Social Structure of Competition, Harvard University Press.
Examples
centrality_coleman_theil(igraph::make_star(5, mode = "undirected"))
#> 1 2 3 4 5
#> 0 1 1 1 1