Multi-attribute ranking method based on information entropy (MABIE)
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
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).