Connectivity and giant component in random distance graphs

TL;DR

This paper introduces a new random graph model based on metric spaces and distance-dependent edge probabilities, analyzing the conditions for the emergence of a giant component and connectivity, especially in lattice structures.

Contribution

It proposes a novel metric space-based random graph model and determines thresholds for connectivity and giant component emergence, comparing it with Waxman graphs.

Findings

Thresholds for giant component emergence identified

Connectivity thresholds established in lattice models

Comparison with Waxman graph model provided

Abstract

Various different random graph models have been proposed in which the vertices of the graph are seen as members of a metric space, and edges between vertices are determined as a function of the distance between the corresponding metric space elements. We here propose a model $G=G(X, f)$, in which $(X, d)$ is a metric space, $V(G)=X$, and $\mathbb{P}(u\sim v) = f(d(u, v))$, where $f$ is a decreasing function on the set of possible distances in $X$. We consider the case that $X$ is the $n\times n \times \dots\times n$ integer lattice in dimension $r$, with $d$ the $\ell_1$ metric, and $f(d) = \frac{1}{n^\beta d}$, and determine a threshold for the emergence of the giant component and connectivity in this model. We compare this model with a traditional Waxman graph. Further, we discuss expected degrees of nodes in detail for dimension 2.