On the Mixing Time of Geographical Threshold Graphs

TL;DR

This paper analyzes the mixing times of geographical threshold graphs, a generalization of random geometric graphs, showing they have similar mixing properties near the connectivity threshold under certain weight distribution conditions.

Contribution

It extends known mixing time results from random geometric graphs to the more general geographical threshold graphs, accounting for heterogeneous degree distributions.

Findings

Mixing time bounds match those of RGGs under specific weight decay conditions.

Connectivity threshold for GTGs is comparable to RGGs, around (log n / n)^{1/d}.

Degree heterogeneity in GTGs does not significantly affect mixing times near the threshold.

Abstract

We study the mixing time of random graphs in the $d$-dimensional toric unit cube $[0,1]^d$ generated by the geographical threshold graph (GTG) model, a generalization of random geometric graphs (RGG). In a GTG, nodes are distributed in a Euclidean space, and edges are assigned according to a threshold function involving the distance between nodes as well as randomly chosen node weights, drawn from some distribution. The connectivity threshold for GTGs is comparable to that of RGGs, essentially corresponding to a connectivity radius of $r=(\log n/n)^{1/d}$. However, the degree distributions at this threshold are quite different: in an RGG the degrees are essentially uniform, while RGGs have heterogeneous degrees that depend upon the weight distribution. Herein, we study the mixing times of random walks on $d$-dimensional GTGs near the connectivity threshold for $d \geq 2$. If the weight distribution function decays with $\mathbb{P}[W \geq x] = O(1/x^{d+\nu})$ for an arbitrarily small constant $\nu>0$ then the mixing time of GTG is $\mixbound$. This matches the known mixing bounds for the $d$-dimensional RGG.