Asymptotics of Closeness Centralities of Graphs

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 SDG (v). The closeness centrality of a vertex in a graph was defined by Bavelas to be CC (v) =n−1SDG (v) and the closeness centrality of G is CC (G) =∑v∈Gn−1SDG (v). We consider the asymptotic limit of CC (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, limn→∞CC (Pn) =π, using uniform convergence and Riemann sums. We applied the same technique for the union of a cycle Cm and path Pn 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 Gn such that limn→∞CC (Gn) =c. In addition, we investigate the mean distance of a graph, l¯ (G) =1n (n−1) ∑v∈V (G) SD (v) and the normalized closeness centrality, C¯C (G) =1nCC (G). We verify a conjecture of Britz, Hu, Islam, and Tang that the set of products l¯ (G) C¯C (G): Gisfiniteandconnected is dense in [1, 2).