M-centrality
The
M-centrality
evaluates the influence of a node by combining local information from its neighborhood with global information about its position in the network [2].
The centrality of node \(i\) is defined as
\begin{equation*}
c_{M}(i) = μ \, k_s(i) + (1-μ) \, \Delta D(i),
\end{equation*}
where \(k_s(i)\) is the \(k\)-shell centrality of node \(i\), representing its global importance, and \(\Delta D(i)\) captures local degree variation:
\begin{equation*}
\Delta D(i) = \sum_{j \in \mathcal{N}(i)} d_i \left| \frac{d_j - d_i}{\sum_{l \in \mathcal{N}(i)} d_l} \right|,
\end{equation*}
with \(\mathcal{N}(i)\) denoting the set of neighbors of node \(i\), and \(d_j\) the degree of neighbor \(j\).
The parameter \(μ \in [0,1]\) balances the contributions of the global (\(k\)-shell) and local (degree variation) measures.
Ibnoulouafi et al. [2] suggest setting \(μ\) based on the relative entropies of the two distributions:
\begin{equation*}
μ = \frac{1 - E_1}{2 - E_1 - E_2},
\end{equation*}
where \(E_1\) and \(E_2\) are the entropies of the \(k\)-shell centrality and \(\Delta D(i)\) distributions, respectively.