All spatial random graphs with weak long-range effects have chemical distance comparable to Euclidean distance

TL;DR

This paper establishes a sufficient condition for linear lower bounds on chemical distances in spatial random graphs, emphasizing the scarcity of long edges and weak large-distance correlations, applicable to translation invariant graphs.

Contribution

It introduces a general criterion linking edge scarcity and correlation decay to chemical distance bounds in spatial graphs, extending previous methods.

Findings

Chemical distances are linearly bounded below under certain conditions.

The conditions involve scarcity of long edges and weak correlations.

The proof uses a renormalisation scheme from Berger (2004).

Abstract

This note provides a sufficient condition for linear lower bounds on chemical distances (compared to the Euclidean distance) in general spatial random graphs. The condition is based on the scarceness of long edges in the graph and weak correlations at large distances and is valid for all translation invariant and locally finite graphs that fulfil these conditions. The proof is based on a renormalisation scheme introduced by Berger [arXiv: 0409021 (2004)].