Weighted h-index
The
weighted h-index
is an extension of the classical h-index to weighted networks [2]. First, the weight \(w_{ij}\) of the edge \((i,j)\) is defined as
\begin{equation*}
w_{ij} = d_i d_j,
\end{equation*}
where \(d_i\) and \(d_j\) are the degrees of nodes \(i\) and \(j\).
Next, each neighbor \(j\) of node \(i\) is conceptually cloned \(d_j\) times. Each cloned neighbor \(j_c\) is assigned a virtual edge weight \(w_{ij_c} = w_{ij}\), effectively repeating the original edge weight \(d_j\) times.
The weighted h-index \(c_{wh}(i)\) of node \(i\) is then calculated as
\begin{equation*}
c_{wh}(i) = H(w_{ij_1,1}, \dots, w_{ij_1,d_{j_1}}, \dots, w_{ij_{d_i},1}, \dots, w_{ij_{d_i},d_{j_{d_i}}}),
\end{equation*}
where \(H\) is the
h
-index operator, which returns the largest integer \(h\) such that there are at least \(h\) elements in the set with value no less than \(h\). In other words, the weighted
h
-index reflects both the degrees of a node’s neighbors and the multiplicity of their connections, capturing a more nuanced measure of local influence.