Average shortest path centrality (AC)
The
average shortest path centrality
(AC), also known as the relative change in average shortest path (RASP), quantifies node influence by measuring the relative change in the network's average shortest path (ASP) after removing a node [2].
The centrality of node \(i\) is defined as
\begin{equation*}
c_{AC}(i) = \frac{ASP(G_i) - ASP(G)}{ASP(G)},
\end{equation*}
where \(G\) is the original network, and \(G_i\) is the subgraph obtained by removing node \(i\) and all its adjacent links. The average shortest path of a graph \(G\) is
\begin{equation*}
ASP(G) = \frac{\sum_{i \neq j} d_{ij}}{N(N-1)},
\end{equation*}
where \(d_{ij}\) is the shortest path length between nodes \(i\) and \(j\) if they are connected; otherwise, \(d_{ij}\) is set equal to the diameter of \(G\).
Hence, AC centrality quantifies the importance of a node in preserving efficient communication paths within the network.