Participation coefficient
The
participation coefficient
is a community-based measure that quantifies how evenly a node's links are distributed among different communities (modules) [2]. Consider a graph \(G\) with a community structure consisting of \(K\) communities \(C_1, \dots, C_K\). For example, [3] applied the participation coefficient to networks whose communities were detected using the Louvain algorithm. The participation coefficient quantifies how a node's links are distributed within its own module compared to other modules:
\begin{equation*}
c_{\mathrm{Part.coeff}}(i) = 1 - \sum_{s=1}^{K} \left( \frac{d_{is}}{d_i} \right)^2
= 1 - \sum_{s=1}^{K} \left( \frac{\sum_{j \in C_s \setminus \{i\}} a_{ij}}{d_i} \right)^2,
\end{equation*}
where \(d_{is}\) is the number of links from node \(i\) to nodes in module \(C_s\), and \(d_i\) is the degree of node \(i\).
The participation coefficient \(c_{\mathrm{Part.coeff}}(i)\) approaches 1 when a node's links are uniformly distributed across all modules, and is close to 0 when all links are confined within its own module. While the participation coefficient was originally intended to describe the roles of nodes within community-structured networks, it has also proven useful for identifying highly connected hub nodes in real-world networks [4].