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
\(\tilde{d}_{ij}\) 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}{\tilde{d}_{ij}^2}.
\end{equation*}