Centroid centrality
The
centroid centrality
evaluates how close a node is to all other nodes in a network from a game-theoretical perspective [2, 3]. It considers pairwise comparisons between nodes to identify those that are, on average, more centrally located.
For a pair of nodes \(i\) and \(j\), let \(γ_i(j)\) denote the number of nodes that are closer (in terms of shortest-path distance) to \(i\) than to \(j\):
\[
γ_i(j) = \left| \left\{ v \in \mathcal{N} : d_{iv} < d_{jv} \right\} \right|,
\]
where \(d_{iv}\) is the shortest-path distance between nodes \(i\) and \(v\).
The centroid centrality of node \(i\) is then defined as
\begin{equation*}
c_{\text{centroid}}(i) = \min_{j \in \mathcal{N} \setminus \{i\}} f(i,j),
\end{equation*}
where
\begin{equation*}
f(i,j) = γ_i(j) - γ_j(i).
\end{equation*}
Intuitively, centroid centrality quantifies the positional advantage of node \(i\) within the network. Nodes with high centroid values are closer, on average, to a larger portion of the network than competing nodes, making them strategically well-positioned or ``centrally dominant'' within the network structure.