# The Effects of Topology on Throughput Capacity of Large Scale Wireless Networks
## 1 Introduction  
Wireless ad hoc networks consist of a group of nodes. Each node communicates with each other over a wireless channel with the help of relay nodes. The lack of any centralized control and possible node mobility cause many issues in the network. The main problem is that of network capacity. Network capacity has been a hot issue in the past few years, and tremendous efforts have been made related to the capacity of wireless networks. [PERSON] and [PERSON] [1] started the research on the capacity of large scale wireless networks and provided ground-breaking results on the scaling law of wireless network capacity. They considered a unit area network consisting of \(n\) nodes uniformly distributed. By the rigorous derivation, a per-node transport capacity \(\Theta\left(\frac{1}{\sqrt{n}}\right)\) (Given two functions \(f(n)\) and \(g(n)\): \(f(n)=o(g(n))\) means \(\lim_{n\rightarrow\infty}\frac{f(n)}{g(n)}=0\); \(f(n)=O(g(n))\) means \(\lim_{n\rightarrow\infty}\frac{f(n)}{g(n)}=c<\infty\); if \(g(n)=O(f(n))\), \(f(n)=\Omega(g(n))\) with high probability (w.h.p. ); if both \(f(n)=\Omega(g(n))\) and \(f(n)=O(g(n))\), \(f(n)=\Theta(g(n))\); \(f(n)=\widetilde{\Theta}(g(n))\) means \(f(n)=\Theta(g(n))\) when logarithmic term is ignored.) was obtained in arbitrary ad hoc networks. For the random ad hoc networks, an achievable per-node throughput capacity was \(\Theta\left(\frac{1}{\sqrt{n\log n}}\right)\). These results indicated that the per-node rate decreased when the number of nodes increased. After that, [PERSON] et al. [2] exploited percolation theory to wireless ad hoc network and showed that a per-node rate \(\Theta\left(\frac{1}{\sqrt{n}}\right)\) was achievable, which improved [PERSON]'s [1] by an order of \(\Theta(\sqrt{\log n})\). Motivated by their works, studieson network capacity of some specific networks such as cognitive network [3] hybrid network [4] and mobile wireless Ad hoc networks [5] were conducted. In the seminal works of network capacity [1; 2], the authors only considered the uniformly network topology, and each node randomly and uniformly selected a destination. They did not consider the impact of other factors, such as traffic patterns, transmission models and node attributes. [PERSON] et al. [6] analyzed the impact of social interaction on the throughput capacity of wireless network. Later, [PERSON] et al. [7] considered that the node is selfish and derived the throughput capacity of wireless networks under various extents of selfish via percolation theory. Since the results of throughput capacity were pessimistic, [PERSON] [8] proposed a strategy to improve the throughput capacity by physical layer caching. Recently, [PERSON] [9] introduced infrastructure into the network, and the throughput was increased dramatically if the amount of infrastructure was larger than a threshold. Except for the above works which focused on homogeneous wireless ad hoc networks, many researchers turned their attention to the heterogeneous networks. In particular, [PERSON] et al. [10] considered the nodes distributed according to the Shot Noise Cox Process (SNCP) [11] and obtained the upper bounds of throughput capacity using a combination of geometric and percolation arguments. Later, they further optimized [10] by employing a layered scheduling and complicated routing schemes [12]. However, most works are only conducted over the planar networks. With the development of network technology, wireless networks are expected to extend from two-dimensional space to three-dimensional space, connecting all kinds of objects such as sensors, computers, mobile phones, etc. The future three-dimensional wireless networks will be a fusion of the digital world and the physical world and bring together everything from individuals to objects, from data to services, etc. For example, in modern battlefields, three-dimensional wireless networks need to be deployed to connect various military units together, like aircrafts, troops, and fleets.