Entropy-based ranking measure (ERM)
The
Entropy-Based Ranking Measure
(ERM) is a centrality metric that quantifies the influence of a node based on the degrees of its first- and second-order neighbors [2]. Let \(d_i^{(1)}\) denote the total degree of the neighbors of node \(i\), defined as
\begin{equation*}
d_i^{(1)} = \sum_{j \in \mathcal{N}(i)} d_j,
\end{equation*}
where \(d_j\) is the degree of neighbor \(j\). Similarly, let \(d_i^{(2)}\) be the total degree of the neighbors of node \(i\)'s neighbors:
\begin{equation*}
d_i^{(2)} = \sum_{j \in \mathcal{N}(i)} d_j^{(1)}.
\end{equation*}
The ERM centrality of node \(i\) is defined as
\begin{equation*}
c_{\mathrm{ERM}}(i) = \sum_{j \in \mathcal{N}(i)} \sum_{k \in \mathcal{N}(j)} EC(i),
\end{equation*}
where \(EC(i)\) represents
the entropy centrality
of node \(i\), given by
\begin{equation*}
EC(i) = E_1(i) + λ_i E_2(i)
= -\sum_{j \in \mathcal{N}(i)} \frac{d_j}{d_i^{(1)}} \log{\frac{d_j}{d_i^{(1)}}}
+ λ_i \left(-\sum_{j \in \mathcal{N}(i)} \frac{d_j^{(1)}}{d_i^{(2)}} \log{\frac{d_j^{(1)}}{d_i^{(2)}}} \right).
\end{equation*}
Here, \(E_1(i)\) and \(E_2(i)\) denote the entropy of the degrees of the first- and second-order neighbors of node \(i\), respectively, and \(λ_i \in [0,1]\) is a tunable parameter that balances their contributions. Following [2], \(λ_i\) can be set as
\begin{equation*}
λ_i = \frac{d_i^{(2)}}{\max_k d_k^{(2)}},
\end{equation*}
so that nodes with larger second-order neighborhoods give proportionally more weight to \(E_2(i)\).