Burt’s constraint
Burt's constraint
, also called network constraint, was proposed in [2, 3] to quantify the extent to which a node's connections constrain its brokerage opportunities in a network. The constraint of a node \(i\) is high if its neighbors \(\mathcal{N}(i)\) communicate heavily with one another (dense network) or if they share information indirectly through a central contact (hierarchical network).
Mathematically, the strength \(p_{ij}\) of a link \((i,j)\) is defined as the proportion of \(i\)'s time or energy invested in contact \(j\):
\begin{equation*} \label{eq_burt_energy}
p_{ij} = \frac{a_{ij}+a_{ji}}{\sum_{k \in \mathcal{N}(i)} (a_{ik}+a_{ki})}.
\end{equation*}
Burt's constraint of node \(i\) is then
\begin{equation*}
c_{\mathrm{Burt}}(i) = \sum_{j \in \mathcal{N}(i)} \left( p_{ij} + \sum_{k \in \mathcal{N}(i) \setminus \{j\}} p_{ik} p_{kj} \right)^2.
\end{equation*}
For unweighted undirected networks, this simplifies to
\begin{equation*}
c_{\mathrm{Burt}}(i) = \sum_{j \in \mathcal{N}(i)} \left( \frac{1}{|\mathcal{N}(i)|} + \sum_{k \in \mathcal{N}(i) \setminus \{j\}} \frac{1}{|\mathcal{N}(i)||\mathcal{N}(k)|} \right)^2,
\end{equation*}
where \(|\mathcal{N}(i)|\) denotes the degree of node \(i\).