Global and local structure (GLS) centrality
The
global and local structure (GLS) centrality
evaluates node importance by integrating both local and global network structures [2]. The global influence of node \(i\) is defined as
\begin{equation*}
f_g(i) = d_i \sum_{j \in \mathcal{N}(i)} α^{|\mathcal{N}(i) \cap \mathcal{N}(j)|},
\end{equation*}
where \(d_i\) is the degree of node \(i\), \(\mathcal{N}(i)\) is the set of neighbors of node \(i\), and \(α = 1.1\) is a constant.
The local influence of node \(i\) accounts for the normalized degree of its neighbors and the inverse average degree of each neighbor's neighbors:
\begin{equation*}
f_l(i) = \sum_{j \in \mathcal{N}(i)} \frac{d_j}{N-1} \cdot \frac{d_j}{\sum_{l \in \mathcal{N}(j)} d_l}.
\end{equation*}
The GLS centrality of node \(i\) combines its global and local influence as
\begin{equation*}
c_{GLS}(i) = f_g(i) \cdot f_l(i)
= \frac{d_i}{N-1} \sum_{j \in \mathcal{N}(i)} \frac{d_j^2 α^{|\mathcal{N}(i) \cap \mathcal{N}(j)|}}{\sum_{l \in \mathcal{N}(j)} d_l}.
\end{equation*}
Hence, the GLS centrality combines the weighted contributions of immediate neighbors, accounting for their degrees, with the overlap between the neighborhoods of connected nodes, thereby integrating local connectivity and semi-global structural information.