Multi-Attribute Ranking Method Based on Information Entropy (MABIE) is a hybrid centrality measure that integrates both local and global information of a network using four classical centrality metrics: degree centrality (DC), harmonic centrality (HC), betweenness centrality (BC), and correlation centrality (CoC) [2]. MABIE constructs an \(N \times 4\) multi-attribute node-importance decision matrix\[R = \begin{bmatrix} r_{11} & r_{12} & r_{13} & r_{14} \\r_{21} & r_{22} & r_{23} & r_{24} \\ \vdots & \vdots & \vdots & \vdots \\ r_{N1} & r_{N2} & r_{N3} & r_{N4} \\ \end{bmatrix},\]where \(r_{ij} = \frac{c_j(i)}{\sum_{k=1}^N c_j(k)}\) and \(c_j(i)\) denotes the \(j\)-th centrality value of node \(i\), with \(j \in \{\mathrm{DC}, \mathrm{HC}, \mathrm{BC}, \mathrm{CoC}\}\). Thus, matrix \(R\) contains the normalized centrality values of all nodes.The information entropy vector \(E = (E_1, E_2, E_3, E_4)\) quantifies the information content of each centrality metric and is defined as\[E_j = -\frac{1}{\ln N} \sum_{i=1}^N r_{ij} \ln r_{ij}.\]Specifically, it measures the degree of differentiation among nodes with respect to each metric: higher entropy values correspond to a more uniform (and thus less informative) distribution of centrality values, whereas lower entropy values indicate greater variability and stronger discriminative power.The MABIE centrality of node \(i\) is then defined as a weighted linear combination of the normalized centrality measures:\[c_{\mathrm{MABIE}}(i) = \sum_{j=1}^4 w_j r_{ij},\]where the weight \(w_j\) represents the relative importance of the \(j\)-th centrality measure and is computed as\[w_j = \frac{1 - E_j}{\sum_{k=1}^{4} (1 - E_k)}.\]While Wenli \textit{et al. } [2] originally considered four centrality measures, the MABIE framework can be extended to any number \(K\) of centrality metrics. Building on this concept, Zhang et al. [3] proposed the Multiple Local Attributes Weighted Centrality (LWC). LWC extends the MABIE framework by incorporating local structural information through four metrics: degree, two-hop degree, clustering coefficient, and two-hop clustering coefficient (the sum of the clustering coefficients of a node's neighbors).

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] Wenli, F., Zhigang, L., & Ping, H. (2013). Identifying node importance based on information entropy in complex networks. Physica Scripta, 88(6), 065201. doi: 10.1088/0031-8949/88/06/065201.
[3] Zhang, J., Zhang, Q., Wu, L., & Zhang, J. (2022). Identifying influential nodes in complex networks based on multiple local attributes and information entropy. Entropy, 24(2), 293. doi: 10.3390/e24020293.