On Characterizing the Local Pooling Factor of Greedy Maximal Scheduling in Random Graphs

TL;DR

This paper characterizes the local pooling factor of greedy maximal scheduling in large random networks, providing bounds based on network density and connectivity, with implications for optimality and network design.

Contribution

It offers rigorous bounds on the local pooling factor for Erdős-Rényi and random geometric graphs, linking network structure to scheduling optimality in large networks.

Findings

LoP factor between 1/2 and 2/3 in dense graphs

GMS is optimal in sparse graphs with high probability

Connectivity and cycle subgraphs influence LoP bounds

Abstract

The study of the optimality of low-complexity greedy scheduling techniques in wireless communications networks is a very complex problem. The Local Pooling (LoP) factor provides a single-parameter means of expressing the achievable capacity region (and optimality) of one such scheme, greedy maximal scheduling (GMS). The exact LoP factor for an arbitrary network graph is generally difficult to obtain, but may be evaluated or bounded based on the network graph's particular structure. In this paper, we provide rigorous characterizations of the LoP factor in large networks modeled as Erd\H{o}s-R\'enyi (ER) and random geometric (RG) graphs under the primary interference model. We employ threshold functions to establish critical values for either the edge probability or communication radius to yield useful bounds on the range and expectation of the LoP factor as the network grows large. For sufficiently dense random graphs, we find that the LoP factor is between 1/2 and 2/3, while sufficiently sparse random graphs permit GMS optimality (the LoP factor is 1) with high probability. We then place LoP within a larger context of commonly studied random graph properties centered around connectedness. We observe that edge densities permitting connectivity generally admit cycle subgraphs which forms the basis for the LoP factor upper bound of 2/3. We conclude with simulations to explore the regime of small networks, which suggest the probability that an ER or RG graph satisfies LoP and is connected decays quickly in network size.