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.

References

[1] Shvydun, S. (2025). Zoo of Centralities: Encyclopedia of Node Metrics in Complex Networks. arXiv: 2511.05122 https://doi.org/10.48550/arXiv.2511.05122
[2] Tulu, M. M., Hou, R., & Younas, T. (2018). Identifying influential nodes based on community structure to speed up the dissemination of information in complex network. IEEE access, 6, 7390-7401. doi: 10.1109/ACCESS.2018.2794324.