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).