Community-based mediator (CbM)
The
community-based mediator
(CbM) is a community-aware centrality measure that evaluates a node’s influence based on the entropy of its connections across different communities [2].
Let the network \(G\) consist of \(K\) communities \(C_1, C_2, \dots, C_K\).
The CbM centrality \(c_{CbM}(i)\) of node \(i\) is defined as
\begin{equation*}
c_{CbM}(i) = H_i \frac{d_i}{2L},
\end{equation*}
where \(L\) is the total number of edges in \(G\), \(d_i\) is the degree of node \(i\), and \(H_i\) is the entropy associated with the distribution of its connections among the communities:
\begin{equation*}
H_i = -\sum_{k=1}^{K} \frac{d_i(C_k)}{d_i} \log \frac{d_i(C_k)}{d_i}.
\end{equation*}
Here, \(d_i(C_k) = \sum_{j \in C_k} a_{ij}\) denotes the number of links from node \(i\) to nodes in community \(C_k\).
Nodes that distribute their links more evenly across different communities have higher entropy \(H_i\) and thus higher \(c_{CbM}(i)\), reflecting their stronger mediating role in inter-community connectivity.