Asymptotics of Closeness Centralities of Graphs

Santiago Frias, Adriana Galindo Silva, Bryan Romero and Darren A. Narayan ()
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Santiago Frias: Department of Mathematics, Michigan State University, East Lansing, MI 48825, USA
Adriana Galindo Silva: Mathematics and Statistics Department, Sonoma State University, Rohnert Park, CA 94928, USA
Bryan Romero: Department of Mathematics, University of Massachusetts Boston, Boston, MA 02125, USA
Darren A. Narayan: School of Mathematics and Statistics, Rochester Institute of Technology, Rochester, NY 14623, USA

Mathematics, 2025, vol. 13, issue 23, 1-24

Abstract: Given a connected graph G with n vertices, the distance between two vertices is the number of edges in a shortest path connecting them. The sum of the distances in a graph G from a vertex v to all other vertices is denoted by S D G ( v ) . The closeness centrality of a vertex in a graph was defined by Bavelas to be C C ( v ) = n − 1 S D G ( v ) and the closeness centrality of G is C C ( G ) = ∑ v ∈ G n − 1 S D G ( v ) . We consider the asymptotic limit of C C ( G ) as the number of vertices tends to infinity and provide an elegant and insightful proof of a 2025 result by Britz, Hu, Islam, and Tang, lim n → ∞ C C ( P n ) = π , using uniform convergence and Riemann sums. We applied the same technique for the union of a cycle C m and path P n and the union of a path and a complete graph. We prove that of all graphs, paths have the minimum closeness centrality. Next we show for any c ∈ [ π , ∞ ) , there exists a sequence of graphs { G n } such that lim n → ∞ C C ( G n ) = c . In addition, we investigate the mean distance of a graph, l ¯ ( G ) = 1 n ( n − 1 ) ∑ v ∈ V ( G ) S D ( v ) and the normalized closeness centrality, C ¯ C ( G ) = 1 n C C ( G ) . We verify a conjecture of Britz, Hu, Islam, and Tang that the set of products { l ¯ ( G ) C ¯ C ( G ) : G is finite and connected } is dense in [ 1 , 2 ) .

Keywords: graph distance; closeness centrality; asymptotics (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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