On the Asymptotic Connectivity of Random Networks under the Random Connection Model

TL;DR

This paper analyzes the asymptotic behavior of isolated nodes in a random network with probabilistic connections, providing conditions for almost sure connectivity as node density increases, extending geometric graph theory to more realistic models.

Contribution

It introduces a detailed asymptotic analysis of isolated nodes in a random connection model, expanding connectivity results beyond the traditional unit disk model.

Findings

Derived the asymptotic distribution of isolated nodes using Chen-Stein technique.

Established a necessary condition for the network to be asymptotically almost surely connected.

Analyzed the boundary effects on network connectivity as node density tends to infinity.

Abstract

Consider a network where all nodes are distributed on a unit square following a Poisson distribution with known density $\rho$ and a pair of nodes separated by an Euclidean distance $x$ are directly connected with probability $g(\frac{x}{r_{\rho}})$, where $g:[0,\infty)\rightarrow[0,1]$ satisfies three conditions: rotational invariance, non-increasing monotonicity and integral boundedness, $r_{\rho}=\sqrt{\frac{\log\rho+b}{C\rho}}$, $C=\int_{\Re^{2}}g(\Vert \boldsymbol{x}\Vert)d\boldsymbol{x}$ and $b$ is a constant, independent of the event that another pair of nodes are directly connected. In this paper, we analyze the asymptotic distribution of the number of isolated nodes in the above network using the Chen-Stein technique and the impact of the boundary effect on the number of isolated nodes as $\rho\rightarrow\infty$. On that basis we derive a necessary condition for the above network to be asymptotically almost surely connected. These results form an important link in expanding recent results on the connectivity of the random geometric graphs from the commonly used unit disk model to the more generic and more practical random connection model.