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.