The eccentricity centrality [2], also referred to as Harary graph centrality [3], measures how close a node is to the farthest node in a connected graph. For a node \(i\) in a connected graph \(G\), the eccentricity centrality, denoted by \(c_{Eccentricity}(i)\), is defined as the reciprocal of the maximum shortest-path distance from \(i\) to any other node:
\begin{equation*}
c_{Eccentricity}(i) = \frac{1}{\max_{j \in \mathcal{N}} d_{ij}},
\end{equation*}
where \(d_{ij}\) represents the shortest-path distance between nodes \(i\) and \(j\). Nodes with the highest eccentricity centrality are considered the most central, as they are closest, on average, to the farthest nodes in the network.

References

[1] Shvydun, S. (2025). Zoo of Centralities: Encyclopedia of Node Metrics in Complex Networks. arXiv: 2511.05122 https://doi.org/10.48550/arXiv.2511.05122
[2] Hage, P., & Harary, F. (1995). Eccentricity and centrality in networks. Social networks, 17(1), 57-63. doi: 10.1016/0378-8733(94)00248-9.
[3] Brandes, U. (2005). Network analysis: methodological foundations (Vol. 3418). Springer Science & Business Media. doi: 10.1007/b106453.