## 2 Related Work
### Connectivity of Wireless Networks: State of the Art  
One of the first papers on connectivity in wireless multi-hop networks was [20], which investigated how far a node's message percolates for Poisson distributed nodes on an infinitely large area. More recently, [PERSON] and [PERSON] [5] performed a fundamental study on the connectivity of uniform distributed nodes in the asymptotic case. They have shown that when the covered area of each node equals to (log\(N\)+c(\(N\)))/\(N\), where \(N\) is the number of nodes in the unit disk and \(\lim\limits_{N\rightarrow\infty}\text{inf }c(N)=+\infty\), the resulting wireless network is asymptotically connected with probability one if \(c(N)\rightarrow+\infty\). Afterwards, various methods and theories have been applied to the study of connectivity. The model of continuum percolation with the Poisson Boolean model is commonly used to study the phase transition phenomenon in wireless connectivity [14, 20-22]. Considering nodes with identical range distributed in an infinite two-dimensional plane according to Poisson point process, there is critical node density, above which there will be an infinite cluster almost surely (it is commonly abbreviated to a.s. and means the event happens with probability 1 if the plane is infinite), below which there will be bounded clusters a.s. Noisy links are introduced in [23] to study the impact of interference on the connectivity of ad hoc networks using percolation theory. It is found that there is a critical noise coefficient value above which the network is made of disconnected clusters. When the noise coefficient is small enough, the asymptotic connectivity can be achieved. Indubitably, whether two nodes can communicate with each other at any given distance and any moment also depends on the interference condition, which is caused by simultaneous communication between other nodes in the range. Due to interference, communication between two connected nodes may drop to lower bit rate or even become impossible at certain time. However, in our study we regard this case as a medium access issue instead of connectivity problem. In other words, when the link failure is caused by interference or collision, we say that the network MAC is not effective and the link capacity is reduced, instead of saying that the connectivity between two nodes is of lower degree. In fact, a simple collision free MAC protocol, such as TDMA scheme, can help achieve a better connectivity [23]. Although geometric random graph and percolation theory applied to the study of wireless connectivity shows great effectiveness, their basis is a deterministic radio link model, which reduces their charm in real environment. To date, several papers have investigated the connectivity under realistic radio channel model. Fading and shadowing effects are considered. [PERSON] _et al_. [24] analyze connectivity issues in one-dimensional ad hoc networks using queueing theoretical approach. They use _GI/G/\({}^{\infty}\)_ queue model (In _GI/G/\({}^{\infty}\)_ queue, the arrival process follows general independent distribution and the service process is general random process, which corresponds to general transmission range distribution and general node placement statistics.) to study broadcast percolation problem with general node placement with fading channel. In [9], it is found that when shadowing gets more severe, the link probability at short distances reduces, while increases at large distances. Long-distance connectivity probability will affect the global network connectivity and the routing performance, similar to the small world [25] networks extended with a few long links. It seems that shadowing fading improves global connectivity. However in our study, we find that the effect has been counteracted with the increase in asymmetric links and thus lead to limited improvement of global connectivity.