Weighted community betweenness (WCB) centrality
Weighted community betweenness
(WCB) centrality is a centrality measure that integrates a node’s betweenness centrality at both the global network level and within its local community [2]. Let the graph \(G\) have a community structure consisting of \(K\) non-overlapping communities \(C_1, \dots, C_K\). This measure highlights the importance of nodes based on their contributions to both global connectivity and local community structure.
The centrality \(c_{\textsc{WCB}}(i)\) of node \(i\) is defined as
\begin{equation*}
c_{\textsc{WCB}}(i) = (1-μ_{C_l})\, c_b(i,C_l) + μ_{C_l}\, c_b(i,G),
\end{equation*}
where \(C_l\) is the community to which node \(i\) belongs, \(c_b(i,C_l)\) is the local betweenness centrality of node \(i\) within \(C_l\), and \(c_b(i,G)\) is the global betweenness centrality of node \(i\) in the entire graph \(G\). The weighting factor \(μ_{C_l}\) quantifies the relative importance of global connectivity by reflecting the proportion of inter-community links, and is calculated as
\begin{equation*}
μ_{C_l} = \frac{\sum_{i \in C_l} \sum_{j \in C_l} a_{ij}}{\sum_{i=1}^N \sum_{j=1}^N a_{ij}},
\end{equation*}
where \(a_{ij}\) are the elements of the adjacency matrix of \(G\).