# Throughput Capacity of Selfish Wireless _Ad Hoc_ Networks with General Node Density
## 1 Introduction  
Wireless _ad hoc_ networks (WANETs) are an emerging networking technology, which are widely used in environmental monitoring, emergency communication, and military applications, _etc._ The unique feature of such networks is formed by the huge number of nodes. Each node communicates over a wireless channel without any centralized control. One of the problems in WANETs is the routing protocol. Much work has been done on this issue; for example, [PERSON] _et al._[1] studied the routing protocol in Machine-to-Machine communication network, and the case of vehicular applications was investigated by [PERSON] _et al._[2]. The routing protocol in mobile WANETs was analyzed by [PERSON] _et al._[3]. The other problem is the network capacity. [PERSON] and [PERSON] [4] gave an outstanding result on the scaling law of wireless network capacity. They considered \(n\) nodes randomly located in a unit network area, each node randomly selected a destination and derived a capacity upper bound of \(O\left(\frac{1}{\sqrt{n}}\right)\) and a lower bound of \(\Theta\left(\frac{1}{\sqrt{n\log n}}\right)\), respectively. (Given two functions \(f(n)\) and \(g(n)\): \(f(n)=o(g(n))\) means \(\lim_{n\to\infty}f(n)/g(n)=0\); \(f(n)=O(g(n))\) means \(\lim_{n\to\infty}f(n)/g(n)=c<\infty\); if \(g(n)=O(f(n))\), \(f(n)=\Omega\big{(}g(n)\big{)}\)_w.h.p._ ; if both \(f(n)=\Omega\big{(}g(n)\big{)}\) and \(f(n)=O(g(n))\), \(f(n)=\Theta(g(n))\); \(f(n)=\widehat{\Theta}(g(n))\) means \(f(n)=\Theta\big{(}g(n)\big{)}\) when logarithmic terms are ignored.) The results showed that the per-node rate decreases as the number of nodes increases. This pessimistic result is a milestone work of wireless network capacity proceeding. Motivated by [4], [PERSON] _et al._[5] exploited percolation theory and variable transmission radius to construct a highway system in the network. Based on highways systems, a per-node rate of \(\Omega\left(\frac{1}{\sqrt{n}}\right)\) was achieved. The result closed the capacity gap in [PERSON]'s work [4]. Since mobility plays an important role in wireless networks, [PERSON] _et al._[6] found that mobility can increase the throughput of networks. They demonstrated that a per-node rate of \(\Theta(1)\) can be achieved while the transmissiondelay was up to \(\Theta(n)\). Due to the results on network capacity, many researchers became devoted to solving the problem of capacity and delay; most of these works can be generalized into two categories: (1) Increase the number of simultaneous transmission; (2) Decrease the count of hops from source nodes to destination nodes. For the first case, some advanced technologies were employed, such as directional antennas (DA) [7; 8], multi-packet reception (MPR) [9] and Multi-input Multi-output (MIMO) [10]. With regard to the second case, infrastructure or base station and nodes mobility were explored; for example, in [11; 12], the authors added a base station into the networks. By constructing an optimized transmission scheme, the distance between source and destination can be decreased. Mobility can increase the throughput capacity to \(\Theta(1)\), but at the cost of increasing the delay to \(\Theta(n)\). Therefore, the work on the tradeoff between throughput and delay was elaborated in [13; 14; 15; 16]. However, most of the previous works on network capacity of WANETs assumed that node was cooperative [17; 18; 19]. That is, whenever a node receives a request to relay traffic, it will forward the packets on no condition. This ignores the nodes' viewpoint. In addition, previous works either focused on dense or extended networks; that is, the node density is \(n\) or \(1\). They do not consider the impact of node density on the throughput. Therefore, the question remained: what is the throughput capacity of the network if the node exposed selfishness and the node density is general? In this work, we study the throughput of wireless _ad hoc_ networks with selfish nodes under general node density.