Entropy-based gravity model (EGM)
The
entropy-based gravity model
(EGM) is a variant of the local gravity model in which a node’s mass is determined by an entropy-based hybrid centrality measure [2]. Specifically, the mass \( m(i) \) of node \( i \) is defined as a weighted linear combination of \( m \) normalized centrality measures:
\begin{equation*}
m(i) = \sum_{j=1}^m w_j c_j(i),
\end{equation*}
where \( c_j(i) \) denotes the normalized value of the \( j\)-th centrality index, and \( w_j \) represents its corresponding weight. The weights \( w_j \) are determined using the entropy weight method, which quantifies the amount of information each centrality measure contributes. Specifically,
\begin{equation*}
w_j = \frac{1 - S_j}{m - \sum_{k=1}^m S_k},
\end{equation*}
where
\begin{equation*}
S_j = -\frac{1}{\ln N} \sum_{i=1}^N
\frac{c_j(i)}{\sum_{k=1}^N c_j(k)}
\ln \left( \frac{c_j(i)}{\sum_{k=1}^N c_j(k)} \right),
\end{equation*}
and \( S_j \) represents the entropy value of the \( j\)-th centrality measure across all \( N \) nodes.
Using the entropy-weighted mass, the EGM centrality of node \( i \) is given by
\begin{equation*}
c_{\text{egm}}(i) = \sum_{j \in \mathcal{N}^{(\leq l)}(i)} \frac{m(i)\, m(j)}{d_{ij}^2},
\end{equation*}
where \( d_{ij} \) is the shortest path distance between nodes \( i \) and \( j \), and \( \mathcal{N}^{(\leq l)}(i) \) denotes the set of nodes whose shortest-path distance from \(i\) is less than or equal to \(l\) (typically \( l = 2 \)).
The performance of the entropy-based gravity model depends on the selection of centrality measures included in the hybrid formulation. Yan et al. [2] compared nine different combinations of centrality measures and demonstrated that the combination incorporating the h-index (Lobby index), closeness centrality, betweenness centrality and PageRank yields the best performance. The effectiveness of this approach was validated on six real-world networks through simulations of the Susceptible-Infected-Recovered (SIR) spreading process.