Heatmap centrality
Heatmap centrality captures both local and global network information by comparing the farness of a node with the average farness of its neighbors [2]. The centrality of node \(i\), denoted \(c_{\text{heatmap}}(i)\), is defined as\begin{equation*}c_{\text{heatmap}}(i) = \sum_{j=1}^{N} d_{ij} - \frac{\sum_{j=1}^{N} \left( a_{ij} \sum_{k=1}^{N} d_{jk} \right)}{\sum_{j=1}^{N} a_{ij}},\end{equation*}where \(d_{ij}\) is the shortest-path distance between nodes \(i\) and \(j\), so that \(\sum_{j=1}^{N} d_{ij}\) represents the farness of node \(i\). Intuitively, this measure identifies “hot spot” nodes within their local neighborhoods: a node whose farness is smaller than the average farness of its neighbors is considered more influential in the network.