Coleman-Theil disorder index
The
Coleman-Theil disorder index
, also known as the hierarchy index, quantifies the extent to which a node's aggregate Burt's constraint is concentrated on a single contact [2]. High values indicate that most of the constraint arises from a single relationship, whereas low values reflect a more even distribution of constraint across multiple contacts.
For node \(i\), the Coleman-Theil disorder index \(c_{\mathrm{CTDI}}(i)\) is defined as
\begin{equation*}
c_{\mathrm{CTDI}}(i) = \frac{\sum_{j \in \mathcal{N}(i)} \tilde{c}_{ij} \ln(\tilde{c}_{ij})}{N_i \ln(N_i)},
\end{equation*}
where \(\mathcal{N}(i)\) is the set of neighbors of node \(i\), \(N_i = |\mathcal{N}(i)|\) is the number of neighbors and \(\tilde{c}_{ij}\) represents the relative contribution of contact \(j\) to the total constraint of node \(i\), defined as
\[
\tilde{c}_{ij} = \frac{c_{ij}}{\frac{1}{N_i} \sum_{k \in \mathcal{N}(i)} c_{ik}}.
\]
The term \(c_{ij}\) denotes the Burt's constraint imposed by contact \(j\) and is given by
\[
c_{ij} = \left( p_{ij} + \sum_{k \in \mathcal{N}(i) \setminus \{j\}} p_{ik} p_{kj} \right)^2,
\]
where \(p_{ij} = \frac{w_{ij}}{\sum_{k \in \mathcal{N}(i)} w_{ik}}\) is the proportion of node \(i\)'s connections invested in contact \(j\), and \(w_{ij}\) represents the weight of the edge between nodes \(i\) and \(j\) (equal to 1 for unweighted networks).
The Coleman-Theil disorder index attains its minimum value of 0 when constraint is equally distributed among all neighbors, and reaches its maximum of 1 when all constraint is concentrated on a single neighbor.