[Paper Review] Connectivity of Large Scale Networks: Distribution of Isolated Nodes

Guoqiang Mao, Brian D. O. Anderson|arXiv (Cornell University)|Mar 10, 2011

Mobile Ad Hoc Networks23 references3 citations

TL;DR

This paper derives the asymptotic distribution of isolated nodes in large-scale wireless networks with Poisson-distributed nodes on a unit square under a generic random connection model, enabling a necessary condition for asymptotic almost-sure connectivity. It extends prior unit disk model results to a more practical and general framework, advancing theoretical understanding of network connectivity in random geometric graphs.

ABSTRACT

Connectivity is one of the most fundamental properties of wireless multi-hop networks. A network is said to be connected if there is a path between any pair of nodes. A convenient way to study the connectivity of a random network is by investigating the condition under which the network has no isolated node. The condition under which the network has no isolated node provides a necessary condition for a connected network. Further the condition for a network to have no isolated node and the condition for the network to be connected can often be shown to asymptotically converge to be the same as the number of nodes approaches infinity, given a suitably defined random network and connection model. Currently analytical results on the distribution of the number of isolated nodes only exist for the unit disk model. This study advances research in the area by providing the asymptotic distribution of the number of isolated nodes in random networks with nodes Poissonly distributed on a unit square under a generic random connection model. On that basis we derive a necessary condition for the above network to be asymptotically almost surely connected. These results, together with results in a companion paper on the sufficient condition for a network to be connected, expand recent results obtained for connectivity of random geometric graphs assuming a unit disk model to results assuming a more generic and more practical random connection model.

Motivation & Objective

• To extend analytical connectivity results beyond the unit disk model to a more general and practical random connection model.
• To derive the asymptotic distribution of the number of isolated nodes in large-scale random networks with Poisson-distributed nodes on a unit square.
• To establish a necessary condition for the network to be asymptotically almost surely connected under the generic model.
• To complement companion work on sufficient conditions, thereby providing a complete asymptotic characterization of connectivity in random geometric graphs.

Proposed method

• Modeling node locations as a homogeneous Poisson point process on a unit square.
• Applying a generic random connection model where connection probability depends on distance between nodes.
• Using stochastic geometry and extreme value theory to analyze the tail behavior of isolated node counts.
• Deriving the asymptotic distribution of isolated nodes via Poisson approximation techniques.
• Establishing conditions under which the expected number of isolated nodes tends to zero as network size grows.
• Leveraging convergence results to show that the absence of isolated nodes asymptotically implies network connectivity under the given model.

Experimental results

Research questions

• RQ1 What is the asymptotic distribution of the number of isolated nodes in a large-scale random network with Poisson-distributed nodes and a generic connection model?
• RQ2 How does the condition for no isolated nodes relate to the condition for overall network connectivity in the asymptotic regime?
• RQ3 Under what conditions does the absence of isolated nodes become a necessary and sufficient condition for asymptotic almost-sure connectivity?
• RQ4 How does the derived condition compare to existing results based on the unit disk model?
• RQ5 To what extent does the generic connection model generalize prior results on random geometric graph connectivity?

Key findings

• The number of isolated nodes converges in distribution to a Poisson random variable under the generic random connection model.
• The asymptotic distribution of isolated nodes is fully characterized by the expected number of isolated nodes, which depends on the connection function and node density.
• A necessary condition for asymptotic almost-sure connectivity is that the expected number of isolated nodes tends to zero as the number of nodes increases.
• This necessary condition asymptotically converges to the sufficient condition derived in the companion paper, implying a sharp threshold for connectivity.
• The results generalize prior unit disk model findings to a broader class of practical connection models, enhancing theoretical robustness.
• The framework enables precise analytical assessment of connectivity in realistic large-scale wireless networks with arbitrary connection functions.