The diameter of Inhomogeneous random graphs

TL;DR

This paper investigates the diameter of inhomogeneous random graphs generated by kernels on metric spaces, providing bounds that generalize known results from Erdős-Rényi graphs.

Contribution

It extends diameter analysis to inhomogeneous graphs with kernels on metric spaces, offering bounds based on expansion factors and kernel behavior.

Findings

Derived upper and lower diameter bounds for inhomogeneous random graphs

Generalized Erdős-Rényi diameter results to more complex kernels

Identified constants influencing graph diameter based on kernel properties

Abstract

In this paper we study the diameter of Inhomogeneous random graphs $G(n,\kappa,p)$ that are induced by irreducible kernels $\kappa$. The kernels we consider act on separable metric spaces and are almost everywhere continuous. We generalize results known for the Erd\H{o}s-R\'enyi model $G(n,p)$ for several ranges of $p$. We find upper and lower bounds for the diameter of $G(n,\kappa,p)$ in terms of the expansion factor and two explicit constants that depend on the behavior of the kernel over partitions of the metric space.