The mixed core, semi-local degree, and weighted entropy (MCSDWE) method combines the MCDWE and MCSDE approaches to produce a fully weighted, semi-local centrality measure [2]. The centrality of node \(i\) is defined as\[c_{MCSDWE}(i) = α k_s(i) + β \, c_{\mathrm{LR}}(i) + γ H(i),\]where \(k_s(i)\) is the \(k\)-shell index, \(c_{\mathrm{LR}}(i)\) is the LocalRank centrality of node \(i\) [3], and \(α, β, γ\) are weights controlling the contributions of each component (Sheikhahmadi and Nematbakhsh [2] suggest \(α = β = γ = 1\)). The weighted entropy \(H(i)\) is given by\[H(i) = \sum_{k=0}^{k_{\max}} \frac{p_k(i) \log_2 p_k(i)}{k_{\max} - |\{r : \exists j \in \mathcal{N}(i), k_s(j) = r\}| + 1},\]where \(p_k(i)\) is the fraction of neighbors of node \(i\) in the \(k\)th shell and \(k_{\max}\) denotes the maximum \(k\)-shell index in the network. MCSDWE integrates local, semi-local, and hierarchical neighborhood information, providing a comprehensive and nuanced assessment of node influence in complex networks.

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] Sheikhahmadi, A., & Nematbakhsh, M. A. (2017). Identification of multi-spreader users in social networks for viral marketing. Journal of Information Science, 43(3), 412-423. doi: 10.1177/0165551516644171.
[3] Chen, D., Lü, L., Shang, M. S., Zhang, Y. C., & Zhou, T. (2012). Identifying influential nodes in complex networks. Physica a: Statistical mechanics and its applications, 391(4), 1777-1787. doi: 10.1016/j.physa.2011.09.017.