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This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

Connectivity is a fundamental issue in research on wireless sensor networks. However, unreliable and asymmetric links have a great impact on the global quality of connectivity (QoC). By assuming the deployment of nodes a homogeneous Poisson point process and eliminating the border effect, this paper derives an explicit expression of node non-isolation probability as the upper bound of one-connectivity, based on an analytical link model which incorporates important parameters such as path loss exponent, shadowing variance of channel, modulation, encoding method etc. The derivation has built a bridge over the local link property and the global network connectivity, which makes it clear to see how various parameter impact the QoC. Numerical results obtained further confirm the analysis and can be used as reference for practical design and simulation of wireless ad hoc and sensor networks. Besides, we find giant component size a good relaxed measure of connectivity in some applications that do not require full connectivity.

Considerable studies have discussed the issues such as the capacity [

Pioneering works [

The rest of the paper is organized as follows. Section 2 provides a short overview of both wireless link model and network connectivity issue. The detailed analysis of node isolation probability under an analytical link model is presented in Section 3. In Section 4 we do extensive simulations to evaluate the analysis in Section 3 and also discuss the impact of various parameters on the global QoC. Finally we conclude the paper in Section 5 and point out the future direction.

A lot of existing works are based on the Boolean disk model, as illustrated in

The third link model discussed above is supported by many experimental studies and significantly affects networks behavior. Woo

Several recent studies have proposed communication models based on empirical data and analyzed related phenomena for more accurate evaluation of upper-layer protocols. Woo

Most important to our work is the analysis of transitional region in [

One of the first papers on connectivity in wireless multi-hop networks was [

Noisy links are introduced in [

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. Miorand ^{∞} queue model (In ^{∞} 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 [

Suppose the sensor nodes are scattered from the air and the process resembles Poisson arrival in a service queue. Thus we use a homogeneous Poisson point process with intensity λ to model the spatial distribution of the nodes. This process is defined as follows [

Consider a measure space (

If _{1}, _{2}, … are disjoint sets then _{1}), _{2}), … are independent random variables:

If

We consider an arbitrary subarea

A graph is said to be

Our focus will first be on one-connectivity. Higher orders of connectivity will be discussed in the future work. We are interested in the critical density

Since _{0}. Hence

In the following, we calculate the mean node degree _{0}. Let _{0} can be calculated by integrating

The solution of the equation is

Depending on different radio link model, the link probability

Empirical studies have shown that the lognormal shadowing model provides more accurate multipath channel models than other models like Nakagami and Rayleigh models [_{r}_{t}^{2}, and _{0}) is the power decay for the reference distance _{0}.

_{i}

For a modulation method _{M}, while _{M} is a function of SNR _{M} depends on modulation, and more generally on transmitting and receiving techniques (diversity, equalization, etc.). Here we take non-coherent FSK for example, which is the default modulation scheme in Mica2 Mote. Expressions for _{M} is:
_{n}^{2}, where

According to

If the link between two nodes has a high probability (>_{h}_{h}_{l}

If the link between two nodes has a high probability (>_{h}_{l}_{h}

According to definition 3 and definition 4, the region between _{l}_{h}

With consideration of different pass loss and shadow fading along different direction, we draw contour plot of the reception rate for a node located at the center of the region.

To calculate _{0} and _{1} represent the receiver's range of threshold.

Similarly, _{0}, _{1}] is relatively narrow compared with [

_{h}_{h}

In the following, we will use simulations to verify the theoretical analysis.

In this section, various simulations are conducted to validate the analytical results and show how different parameters impact the network connectivity.

For different applications, the requirement of QoC can be different. For example, a surveillance sensor system with one sink node requires the network to be one-connected so as to collect data from any node in the network. However to detect or track an intrusion [^{2/3}) a.s.; for

To determine the probability of giant component and one-connectivity of a wireless _{i}_{i}_{1}≥λ_{2}≥…≥λ_{N-1}≥0=λ_{N} is called the Laplacian spectrum of a graph

If the graph

In all simulation cases, as the node density increases, the network graph becomes denser and the transition from low connectivity to nearly full connected or appearance of giant component is quite sharp over a small range of ^{2} according to homogeneous Poisson process, the node density must be higher than 0.007 to form giant component and higher than 0.0125 to reach one-connectivity. The density threshold is an energy- efficient point of operation, in that to the left of this threshold the network is disconnected with high probability, and to the right of this threshold, additional energy expenditure results in a negligible increase in the high probability of connectivity.

In all simulation cases, the non-isolation probability serves as the upper bound for probability of one-connectivity. Generally, the difference between the two probabilities is non-negligible. However as node density increases, the two probabilities converge to 1. This result agrees with inequality

It should be noted that, the analytical performance of non-isolation probability closely matches the real non-isolation probability in different settings except when

Comparing

The impact of area size ‖

In summary, a higher transmission power, larger shadowing variance and more complicated encoding method can improve the network connectivity. However, it is energy inefficient to sacrifice the transmission power for better QoC unconditionally; higher shadowing variance ^{2} can increase the chance of link asymmetry and typically comes with a higher path loss exponent

In this paper, we have presented an analytical procedure of calculating the node non-isolation probability based on a generalized radio link model in wireless sensor networks. The non-isolation probability is the upper bound of one-connectivity probability. We use a combination of analytical and simulation-based methods to study the impact of various parameters on the connectivity behavior. The results can be applied for practical design and simulation in wireless

Several issues remain for future work in this area. First, this paper only focus on the connectivity in the spatial domain, it is also interesting to study the temporal dynamics of wireless connectivity. For more challenging dynamic environment, statistics of time-varying links are required; second, the giant component size needs further analytical study. It is important if an explicit expression could be derived for the giant component probability or largest component size; third, this paper assumes the node distribution follows homogeneous Poisson point process. However, in many real-world scenarios, the nodes are in an inhomogeneous distribution. Bettstetter

This work is supported by Natural Science Foundation of China under grants No. 60773181, No. 60873223; the National High-Tech Research and Development Plan of China under Grant No. 2007AA041406, 2006AA01Z218; Shanghai Science and Technology R&D Program under Grant No. 07DZ15012.

Different kinds link models.

Analytical PRR to distance, obtained from _{t}

Contour plot for different link models. (a) Boolean disk model

The procedure of linear approximation for Ψ(

Simulated results for different settings. _{t}

Critical density for giant component, one-connectivity and non-isolation under different settings, comparison of analytical and simulation results.

^{-2}) simulation |
^{-2}) simulation |
^{-2}) simulation |
^{-2}) analysis | |
---|---|---|---|---|

(a) | 7.00 · 10^{-3} |
1.25 · 10^{-2} |
1.22 · 10^{-3} |
1.25 · 10^{-3} |

(b) | 1.15 · 10^{-3} |
2.27 · 10^{-2} |
2.21 · 10^{-3} |
2.15 · 10^{-3} |

(c) | 4.50 · 10^{-3} |
1.07 · 10^{-2} |
1.07 · 10^{-3} |
6.00 · 10^{-3} |

(d) | 4.80 · 10^{-3} |
8.75 · 10^{-3} |
8.50 · 10^{-3} |
8.00 · 10^{-3} |

(e) | 5.00 · 10^{-3} |
9.00 · 10^{-3} |
8.70 · 10^{-3} |
8.70 · 10^{-3} |

(f) | 7.50 · 10^{-3} |
1.33 · 10^{-2} |
1.30 · 10^{-3} |
1.32 · 10^{-3} |

(g) | 7.00 · 10^{-3} |
1.40 · 10^{-2} |
1.35 · 10^{-3} |
1.32 · 10^{-3} |