More is less: Connectivity in fractal regions

TL;DR

This paper reveals that in fractal-boundary regions, increasing node density can decrease the probability of full network connectivity, due to the complex boundary geometry affecting line of sight connections.

Contribution

It introduces a model linking fractal boundary dimensions to network connectivity decay and discusses strategies to mitigate this effect in practical scenarios.

Findings

Connectivity probability decays as a stretched exponential with fractal dimension.

Increasing nodes can reduce network connectivity in fractal regions.

Mitigation strategies can improve network reliability in complex terrains.

Abstract

Ad-hoc networks are often deployed in regions with complicated boundaries. We show that if the boundary is modeled as a fractal, a network requiring line of sight connections has the counterintuitive property that increasing the number of nodes decreases the full connection probability. We characterise this decay as a stretched exponential involving the fractal dimension of the boundary, and discuss mitigation strategies. Applications of this study include the analysis and design of sensor networks operating in rugged terrain (e.g. railway cuttings), mm-wave networks in industrial settings and vehicle-to-vehicle/vehicle-to-infrastructure networks in urban environments.