Bounds on the closeness centrality of a graph

TL;DR

This paper establishes fundamental bounds on the normalized closeness centrality of connected graphs, explores explicit formulas for specific graph families, and conjectures the density of these values within a certain interval.

Contribution

It introduces new tight bounds on closeness centrality and its product with mean distance, providing explicit formulas for certain graph families and proposing a conjecture on the density of these values.

Findings

Bounds on $ar{l}ar{ ext{C}}_C$ between 1 and 2.

Explicit formulas for paths and cycles.

Asymptotic values for specific graph families.

Abstract

We present new values and bounds on the (normalised) closeness centrality $\bar{\mathsf{C}}_C$ of connected graphs and on its product $\bar{l}\bar{\mathsf{C}}_C$ with the mean distance $\bar{l}$ of these graphs. Our main result presents the fundamental bounds $1\leq \bar{l}\bar{\mathsf{C}}_C<2$. The lower bound is tight and the upper bound is asymptotically tight. Combining the lower bound with known upper bounds on the mean distance, we find ten new lower bounds for the closeness centrality of graphs. We also present explicit expressions for $\bar{\mathsf{C}}_C$ and $\bar{l}\bar{\mathsf{C}}_C$ for specific families of graphs. Elegantly and perhaps surprisingly, the asymptotic values $n\bar{\mathsf{C}}_C\big(P_n\big)$ and of $n\bar{\mathsf{C}}_C\big(L_n\big)$ both equal $\pi$, and the asymptotic limits of $\bar{l}\bar{\mathsf{C}}_C$ for these families of graphs are both equal to $\pi/3$. We conjecture that the set of values $\bar{l}\bar{\mathsf{C}}_C$ for all connected graphs is dense in the interval $[1,2)$.