Relative entropy
Relative entropy
is a hybrid measure that integrates multiple centrality indices to evaluate node importance using linear programming and Kullback-Leibler divergence [2]. Suppose there are \(m\) centrality measures, each represented as a discrete distribution \(u_j = (u_{j1}, \dots, u_{jN})\) for \(j=1,\dots,m\), where \(u_{ji} \ge 0\) and \(\sum_{i=1}^N u_{ji} = 1\).
The relative entropy of the nodes, denoted by \(w = (w_1, \dots, w_N)\), is obtained by solving the linear programming problem
\begin{equation*}
\min_{w_1, \dots, w_N} \sum_{j=1}^m \sum_{i=1}^N w_i \log_2 \frac{w_i}{u_{ji}}, \quad
\text{subject to } \sum_{i=1}^N w_i = 1, \quad w_i > 0 \ \forall i.
\end{equation*}
The solution for node \(i\) is given by the normalized geometric mean of its centrality values:
\begin{equation*}
w_i = \frac{\prod_{j=1}^m (u_{ji})^{1/m}}{\sum_{k=1}^N \prod_{j=1}^m (u_{jk})^{1/m}}.
\end{equation*}
Chen et al. [2] apply relative entropy using \(m=4\) centrality measures: degree, closeness, betweenness, and Burt's constraint coefficient. By computing the normalized geometric mean of these measures for each node, the approach produces a single composite score \(w_i\) that reflects the node’s overall importance across multiple centrality perspectives.