# The Effects of Topology on Throughput Capacity of Large Scale Wireless Networks
## 1 Introduction
Hence, it is necessary to develop the throughput capacity of three-dimensional wireless networks. Actually, there are some studies that have been done on throughput and routing protocol of three-dimensional wireless networks. [PERSON] et al. [13] and [PERSON] [14] investigated the throughput capacity and routing protocol of three-dimensional wireless networks, respectively. [PERSON] et al. [15] analyzed the throughput of three-dimensional wireless networks via constructing a backbone routing scheme. The connectivity of three-dimensional wireless networks is analyzed in [16]. For the large scale three-dimensional wireless network, [PERSON] and [PERSON] [17] derived the transport capacity of three-dimensional arbitrary ad hoc networks and the throughput capacity of three-dimensional random ad hoc networks using both Physical Model and Protocol Model. After that, [PERSON] et al. [18] obtained an achievable throughput capacity of three-dimensional wireless networks via percolation theory. Later, [PERSON] et al. [19] investigated the throughput capacity of three-dimensional regular ad hoc networks by employing a generalized physical model. Nevertheless, all of the research mentioned above focused on the homogeneous three-dimensional wireless network. Study of throughput capacity on heterogeneous three-dimensional wireless networks is still not involved, since the inhomogeneity would influence the through capacity significantly. In this paper, we explore the achievable throughput capacity of three-dimensional heterogeneous wireless ad hoc networks via percolation theory [20; 21]. Due to the heterogeneity of network topology, we establish a spheroidal percolation model and some \"information pipes\" to analyze throughput capacity. Specifically, we consider that there are \(n\) nodes distributed in a three-dimensional cube with edge \(L=\Theta(n^{\frac{\gamma}{3}})\), and the network volume \(A=\Theta(n^{\gamma})\), where \(0\leq\gamma\leq 1\) is the parameter of network size, if \(\gamma=0\), the network is a dense network, \(\gamma=1\), and the network is an extended network. The distribution of the nodes is according to the Shot Noise Cox Process (SNCP), which is an inhomogeneous Poisson node process. According to the node's distribution, we deploy \(m\) clusters in the network randomly and uniformly, where \(m=\Theta(n^{v})\), \(0\leq v\leq 1\). Due to the inhomogeneity of network topology, we divide the transmission into two parts: the intra-cluster transmission and the inter-cluster transmission. For the first part, we establish a spheroidal percolation model and propose a routing strategy with five phases. On the basis of spheroidal percolation model, an achievable rate of \(\Omega\left(1/(n^{1-v}((1-v)\log n)^{2})^{\frac{1}{3}}\right)\) in the intra-cluster transmission is achieved. While for the inter-cluster transmission, since the node density among the clusters is much lower, following the idea of homogeneous three-dimensional wireless networks, we establish some \"information pipes\" crossing each cluster. Using these \"information pipes\", nodes located in different clusters can successfully communicate with each other. By a proposed routing scheme, we can get an achievable rate of \(\Omega\left(n^{\frac{2\gamma}{3}-1}\underline{\Phi}^{\frac{3}{3}}\right)\) over the inter-cluster transmission, where \(\underline{\Phi}\) is the minimum node density. If the traffic of network is also heterogeneous, i.e., a node would be inclined to select a destination located in the same cluster, then we conclude that the per-node rate of three-dimensional heterogeneous wireless ad hoc networks falls into an interval of \(\left[\Omega\left(n^{\frac{2\gamma}{3}-1}\underline{\Phi}^{\frac{3}{3}}\right),\Omega\left(1/(n^{1-v}((1-v)\log n)^{2})^{\frac{1}{3}}\right)\right]\). The contributions of this paper can be summarized as follows:  
* We firstly study the impact of inhomogeneity on the throughput capacity three-dimensional wireless networks.