Topological Properties of an Exponential Random Geometric Graph Process

TL;DR

This paper analyzes a one-dimensional exponential AR(1) random geometric graph process, deriving transition probabilities, stationary distributions, and topological properties relevant for mobile wireless network modeling.

Contribution

It introduces a novel AR(1) process-based model for geometric graphs and provides analytical results on connectivity, degree distributions, and topological bounds.

Findings

Derived transition probability matrix and stationary distribution.

Characterized hitting time for disconnectivity.

Established bounds and laws for connectivity and nearest neighbor distances.

Abstract

In this paper, we consider a one-dimensional random geometric graph process with the inter-nodal gaps evolving according to an exponential AR(1) process, which may serve as a mobile wireless network model. The transition probability matrix and stationary distribution are derived for the Markov chains in terms of network connectivity and the number of components. We characterize an algorithm for the hitting time regarding disconnectivity. In addition, we also study topological properties for static snapshots. We obtain the degree distributions as well as asymptotic precise bounds and strong law of large numbers for connectivity threshold distance and the largest nearest neighbor distance amongst others. Both closed form results and limit theorems are provided.