Isotropic random geometric networks in two dimensions with a penetrable cavity

TL;DR

This paper introduces the isotropic random geometric graph (IRGG), a new model with circular boundaries and a penetrable cavity, providing analytical insights into its topological properties relevant for wireless networks.

Contribution

The paper develops and analyzes the IRGG model, highlighting the impact of a penetrable cavity on network properties, which was not previously studied in standard RGGs.

Findings

Analytical expressions for degree, connectivity, clustering, and shortest path.

Good agreement between theoretical results and simulations.

Potential applications in wireless communication networks with node-free regions.

Abstract

In this work, a novel model of the random geometric graph (RGG), namely the isotropic random geometric graph (IRGG) has been developed and its topological properties in two dimensions have been studied in details. The defining characteristics of RGG and IRGG are the same --- two nodes are connected by an edge if their distance is less than a fixed value, called the connection radius. However, IRGGs have two major differences from regular RGGs. Firstly, the shape of their boundaries --- which is circular. It brings very little changes in final results but gives a significant advantage in analytical calculations of the network properties. Secondly, it opens up the possibility of an empty concentric region inside the network. The empty region contains no nodes but allows the communicating edges between the nodes to pass through it. This second difference causes significant alterations in physically relevant network properties such as average degree, connectivity, clustering coefficient and average shortest path. Analytical expressions for most of these features have been provided. These results agree well with those obtained from simulations. Apart from the applicability of the model due to its symmetry and simplicity, the scope of incorporating a penetrable cavity makes it suitable for potential applications in wireless communication networks that often have a node-free region.