Connectivity of Confined Dense Networks: Boundary Effects and Scaling Laws

TL;DR

This paper analyzes how boundary effects influence the connectivity of dense networks within confined geometries, deriving scaling laws and applying cluster expansion methods across various MIMO link models.

Contribution

It introduces a cluster expansion approach to study boundary effects on network connectivity and derives scaling laws for different MIMO configurations in confined spaces.

Findings

Boundary effects significantly impact network connectivity.

Derived scaling laws mitigate boundary effects in dense networks.

Numerical verification confirms analytical predictions.

Abstract

In this paper, we study the probability that a dense network confined within a given geometry is fully connected. We employ a cluster expansion approach often used in statistical physics to analyze the effects that the boundaries of the geometry have on connectivity. To maximize practicality and applicability, we adopt four important point-to-point link models based on outage probability in our analysis: single-input single-output (SISO), single-input multiple-output (SIMO), multiple-input single-output (MISO), and multiple-input multiple-output (MIMO). Furthermore, we derive diversity and power scaling laws that dictate how boundary effects can be mitigated (to leading order) in confined dense networks for each of these models. Finally, in order to demonstrate the versatility of our theory, we analyze boundary effects for dense networks comprising MIMO point-to-point links confined within a right prism, a polyhedron that accurately models many geometries that can be found in practice. We provide numerical results for this example, which verify our analytical results.