In a connected graph, the eccentricity centrality of a graph vertex 
is the reciprocal of its graph
eccentricity. Thus, for a vertex of eccentricity 
,
 |
Eccentricity centrality emphasizes worst-case distance from a vertex to the rest of its connected component, and so gives large values to vertices whose farthest vertices are relatively close (Hage and Harary 1995). It is useful in center-periphery questions where worst-case reachability, rather than average distance, is the relevant constraint, such as choosing central locations in transportation or infrastructure networks. Like closeness centrality, it is distance-based, but it depends on the farthest reachable vertex rather than on an average distance.
Eccentricity centrality is implemented in the Wolfram Language as EccentricityCentrality[g],
and precomputed values for many named graphs can be obtained using GraphData[graph,
"EccentricityCentralities"]. For a disconnected
graph, both use the eccentricity of 
within its connected component,
with vertices in singleton components assigned eccentricity centrality 0. This differs
from GraphData[graph,
"Eccentricities"], which uses infinite eccentricity for every
vertex of a disconnected graph, so taking reciprocals of that property gives the
centralities only in special cases such as edgeless graphs.
