Gateway coefficient
The
gateway coefficient
quantifies a node’s role in connecting different network modules by combining information about community structure and nodal centrality [2]. It extends the participation coefficient by introducing a weight that reflects the “importance” of the node’s intermodular connections.
Assume that the network \(G\) has a community structure consisting of \(K\) communities \(C_1, \ldots, C_K\). The gateway coefficient of node \(i\) is defined as
\begin{equation*}
c_{\mathrm{gateway}}(i) = 1 - \sum_{s=1}^{K}
\left( \frac{d_{is}}{d_i} \right)^2
\left( 1 - \overline{d}_{iS} \cdot \overline{c}_{iS} \right)^2,
\end{equation*}
where \(d_{is}\) is the number of links from node \(i\) to nodes in community \(C_s\), and \(d_i\) is the total degree of node \(i\). The term \(\overline{d}_{iS} = \frac{d_{iS}}{\sum_{j \in C_s} d_{jS}}\) represents the fraction of all connections in community \(C_s\) that belong to node \(i\), and
\[
\overline{c}_{iS} = \frac{\sum_{j \in \mathcal{N}(i,s)} c(j)}{\max_s \sum_{j \in C_s} c(j)}
\]
accounts for the average centrality \(c(j)\) of the neighbors \(\mathcal{N}(i,s)\) of node \(i\) within community \(C_s\). Ruiz Vargas and Wahl [2] define \(c(j)\) as either the degree or betweenness centrality of node \(j\), depending on the application.