Multi-characteristics gravity model (MCGM)
The multi-characteristics gravity model (MCGM) is a variant of the local gravity model designed to identify influential spreaders in complex networks. In this model, a node’s mass is determined by a combination of three structural features: degree, \(k\)-shell index and eigenvector centrality [2]. Let \(\mathcal{N}^{(\leq l)}(i)\) denote the set of nodes whose shortest-path distance from \(i\) is less than or equal to \(l\). The centrality \(c_{\text{MCGM}}(i)\) of node \(i\) is then defined as\begin{equation*}c_{\text{MCGM}}(i) = \sum_{j \in \mathcal{N}^{(\leq l)}(i)} \frac{\left( \frac{d_i}{d_{\max}} + \frac{α\, k_s(i)}{ks_{\max}} + \frac{ev(i)}{ev_{\max}} \right)\left( \frac{d_j}{d_{\max}} + \frac{α\, k_s(j)}{ks_{\max}} + \frac{ev(j)}{ev_{\max}} \right)}{d_{ij}^2},\end{equation*}where \(d_{ij}\) is the shortest-path distance between nodes \(i\) and \(j\); \(d_i\), \(k_s(i)\), and \(ev(i)\) denote the degree, \(k\)-shell index and eigenvector centrality of node \(i\), respectively. The terms \(d_{\max}\), \(ks_{\max}\) and \(ev_{\max}\) represent the corresponding maximum values across all nodes in the network.The coefficient \(α\) adjusts the relative influence of the \(k\)-shell index and is computed as\begin{equation*}α = \frac{\max \left( \frac{d_{\text{mid}}}{d_{\max}}, \frac{ev_{\text{mid}}}{ev_{\max}} \right)}{\frac{ks_{\text{mid}}}{ks_{\max}}},\end{equation*}where \(d_{\text{mid}}\), \(ks_{\text{mid}}\), and \(ev_{\text{mid}}\) denote the median values of the degree, \(k\)-shell index and eigenvector centrality, respectively. Li and Huang [2] consider \(l = 2\) as the truncated radius.