An extremal problem: How small scale-free graph can be

TL;DR

This paper analytically estimates the tight lower bound of diameter for scale-free graphs by constructing a candidate model with ultra-small diameter and proving its optimality among all such graphs.

Contribution

It introduces a constructive method to determine the minimal possible diameter of scale-free graphs and proves its optimality, providing a rigorous lower bound.

Findings

Constructed a candidate scale-free graph with ultra-small diameter.

Proved the candidate graph's diameter is the smallest among all scale-free graphs.

Established a tight lower bound for the diameter of scale-free graphs.

Abstract

The bloom of complex network study, in particular, with respect to scale-free ones, is considerably triggering the research of scale-free graph itself. Therefore, a great number of interesting results have been reported in the past, including bounds of diameter. In this paper, we focus mainly on a problem of how to analytically estimate the lower bound of diameter of scale-free graph, i.e., how small scale-free graph can be. Unlike some pre-existing methods for determining the lower bound of diameter, we make use of a constructive manner in which one candidate model $\mathcal{G^*} (\mathcal{V^*}, \mathcal{E^*})$ with ultra-small diameter can be generated. In addition, with a rigorous proof, we certainly demonstrate that the diameter of graph $\mathcal{G^{*}}(\mathcal{V^{*}},\mathcal{E^{*}})$ must be the smallest in comparison with that of any scale-free graph. This should be regarded as the tight lower bound.