Effective distance gravity model (EffG)
The effective distance gravity model (EffG) is a variant of the gravity model that incorporates both static and dynamic interactions between nodes by utilizing the concept of effective distance [2]. The effective distance \(D_{j|i}\) from node \(i\) to node \(j\), which are directly connected, was introduced by Brockmann and Helbing [3] and is defined as\begin{equation*}D_{j|i} = 1 - \log_2\!\left(\frac{a_{ij}}{d_i}\right),\end{equation*}where \(a_{ij}\) is the element of the adjacency matrix representing the connection between nodes \(i\) and \(j\), and \(d_i\) denotes the degree of node \(i\). The effective distance is not necessarily symmetric, even in undirected networks, because nodes may have different degrees. The effective shortest path distance \(d̃_{ij}\) between nodes \(i\) and \(j\) is computed as the length of the shortest path in a weighted graph, where each direct link \((i,j)\) is assigned a weight equal to \(D_{j|i}\).The EffG centrality of node \(i\), denoted by \(c_{\text{EffG}}(i)\), is then given by\begin{equation*}c_{\text{EffG}}(i) = \sum_{j \neq i} \frac{d_i\,d_j}{d̃_{ij}^2}.\end{equation*}