Capacity of Three-Dimensional Erasure Networks

TL;DR

This paper characterizes the capacity scaling laws of large-scale 3D erasure networks, showing that 3D space allows higher throughput than 2D due to increased geographic diversity and percolation highways.

Contribution

It introduces a 3D erasure network model, analyzes its capacity scaling laws, and proposes a routing protocol that achieves near-optimal throughput scaling.

Findings

Achieves throughput scaling of n^{min{1-λ,1-μ,1-ν}} in 3D networks.

Demonstrates 3D networks can achieve n^{2/3} throughput, outperforming 2D networks.

Provides cut-set bounds confirming the order-optimality of the proposed scheme.

Abstract

In this paper, we introduce a large-scale three-dimensional (3D) erasure network, where $n$ wireless nodes are randomly distributed in a cuboid of $n^{\lambda}\times n^{\mu}\times n^{\nu}$ with $\lambda+\mu+\nu=1$ for $\lambda,\mu,\nu>0$, and completely characterize its capacity scaling laws. Two fundamental path-loss attenuation models (i.e., exponential and polynomial power-law models) are used to suitably model an erasure probability for packet transmission. Then, under the two erasure models, we introduce a routing protocol using percolation highway in 3D space, and then analyze its achievable throughput scaling laws. It is shown that, under the two erasure models, the aggregate throughput scaling $n^{\min\{1-\lambda,1-\mu,1-\nu\}}$ can be achieved in the 3D erasure network. This implies that the aggregate throughput scaling $n^{2/3}$ can be achieved in 3D cubic erasure networks while $\sqrt{n}$ can be achieved in two-dimensional (2D) square erasure networks. The gain comes from the fact that, compared to 2D space, more geographic diversity can be exploited via 3D space, which means that generating more simultaneous percolation highways is possible. In addition, cut-set upper bounds on the capacity scaling are derived to verify that the achievable scheme based on the 3D percolation highway is order-optimal within a polylogarithmic factor under certain practical operating regimes on the decay parameters.