Flow coefficient
The
flow coefficient
is a measure of local centrality introduced by Honey et al. [2] to quantify the capacity of a node to mediate information flow among its immediate neighbors. For a node \(i\), the flow coefficient \(c_{\mathrm{fc}}(i)\) is defined as the fraction of all possible two-step paths between pairs of its neighbors that actually pass through node \(i\):
\begin{equation*}
c_{\mathrm{fc}}(i) =
\frac{\sum_{j \neq k \in \mathcal{N}(i)} (A^2)_{jk}}
{|\mathcal{N}(i)| \, (|\mathcal{N}(i)| - 1)},
\end{equation*}
where \(\mathcal{N}(i)\) denotes the set of neighbors of node \(i\) and \(A\) is the adjacency matrix of the network. The term \((A^2)_{jk}\) counts the number of paths of length two between nodes \(j\) and \(k\).
High values of \(c_{\mathrm{fc}}(i)\) indicate that node \(i\) plays an important role in facilitating information flow among its neighbors, whereas low values suggest that the node’s neighbors are more directly connected to one another.