Krackhardt Hierarchy Measures

Krackhardt's graph-theoretic dimensions provide four indices that together measure how closely a directed network approximates a pure hierarchy — formally, an out-tree. The dimensions are connectedness (is everyone linked?), hierarchy (are ties asymmetric, i.e., non-reciprocated?), efficiency (are there no redundant ties?), and least upper bound (does every pair share a common superior?). Each is scaled from 0 to 1, and a network scoring 1 on all four is a perfect hierarchy.