Towards a Better Understanding of Large Scale Network Models

TL;DR

This paper critically examines the differences between infinite, dense, and extended network models in wireless multi-hop networks, highlighting the limitations of applying infinite network results to finite models through case studies and asymptotic analysis.

Contribution

It demonstrates the subtle differences between network models and emphasizes the need for careful application of infinite network results to dense and extended networks.

Findings

Differences between infinite and finite network models are significant for connectivity analysis.

Boundary effects are negligible for the number of isolated nodes in large networks.

Asymptotic results on network components and isolated nodes are derived for dense and extended models.

Abstract

Connectivity and capacity are two fundamental properties of wireless multi-hop networks. The scalability of these properties has been a primary concern for which asymptotic analysis is a useful tool. Three related but logically distinct network models are often considered in asymptotic analyses, viz. the dense network model, the extended network model and the infinite network model, which consider respectively a network deployed in a fixed finite area with a sufficiently large node density, a network deployed in a sufficiently large area with a fixed node density, and a network deployed in $\Re^{2}$ with a sufficiently large node density. The infinite network model originated from continuum percolation theory and asymptotic results obtained from the infinite network model have often been applied to the dense and extended networks. In this paper, through two case studies related to network connectivity on the expected number of isolated nodes and on the vanishing of components of finite order k>1 respectively, we demonstrate some subtle but important differences between the infinite network model and the dense and extended network models. Therefore extra scrutiny has to be used in order for the results obtained from the infinite network model to be applicable to the dense and extended network models. Asymptotic results are also obtained on the expected number of isolated nodes, the vanishingly small impact of the boundary effect on the number of isolated nodes and the vanishing of components of finite order k>1 in the dense and extended network models using a generic random connection model.