# On Connectivity of Wireless Sensor Networks with Directional Antennas

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## Abstract

**:**

## 1. Introduction

- We establish a general framework to analyze the network connectivity with various existing directional antenna models and our proposed iris model. In particular, we investigate both the local connectivity and the overall connectivity of WSNs in the presence of channel randomness. More specifically, the local connectivity mainly concerns the probability of the node isolation of a node, while the overall connectivity evaluates the probability that there exists at least one path for each node pair in the network from the viewpoint of the entire network.
- We conduct extensive simulations to validate the analytical framework and evaluate the accuracy of the existing antenna models and our proposed model. Our simulation results match the analytical results, indicating that the analytical framework is quite accurate and effective. Besides, our proposed iris model provides a relatively better approximation to realistic antennas than the keyhole model and the sector model on average.
- We find that the network connectivity heavily depends on different antenna models and different channel conditions. We demonstrate that the channel randomness (such as the path loss and the shadow fading) has significant impacts on the network connectivity. For example, the path loss effect is always detrimental to the network connectivity, and the shadow fading effect is somewhat beneficial to the connectivity.

## 2. Related Works

## 3. Antenna Models

#### 3.1. Isotropic Antenna

#### 3.2. Directional Antennas

- The radiation beam (lobe) is a clear peak in the radiation intensity surrounded by regions of weaker radiation intensity.
- The Half Power Beam Width (HPBW) is the angular width between the half-power (−3 dB) points of the lobe.
- The main beam represents the radiation lobe with the maximum antenna gain.
- The side or back lobes represent the lobes in any directions other than the direction of the main beam.
- The nulling capability is the capability of a directional antenna employing nulls to counteract unwanted interference in some undesired directions.

#### 3.2.1. Uniform Circular Array

#### 3.2.2. Uniform Linear Array

#### 3.3. Existing Simplified Models of Directional Antennas

#### 3.4. Iris Antenna Model

**Definition**

**1.**

## 4. Channel Models

## 5. Local Connectivity

#### 5.1. Probability of Isolation

**Definition**

**2.**

#### 5.2. Empirical Results of Local Connectivity

^{2}. To minimize the impacts of the border effect, we use the subarea approach [2], in which we only consider the nodes within an inner square of area ${l}^{\prime}\times {l}^{\prime}$ m

^{2}(${l}^{\prime}$ must be sufficiently smaller than l). For example, for the network using the sector model, we consider l = 12,000 m, and ${l}^{\prime}$ = 1000 m. Besides, each value of the probability of node isolation is obtained by averaging over a large number of random topologies (e.g., 5000). Note that we fixed the threshold attenuation ${\beta}_{0}=50\phantom{\rule{0.166667em}{0ex}}\text{dB}$ in all simulations. Table 4 lists the detailed parameters in simulations.

#### 5.2.1. Comparisons of the Probability of Node Isolation with UCA Antennas, Keyhole, Sector and Iris-UCA Models

#### 5.2.2. Comparisons of the Probability of Node Isolation with the ULA Antenna and Iris-ULA Model

## 6. Overall Connectivity

#### 6.1. One-Connectivity

**Definition**

**3.**

#### 6.2. Empirical Results of One-Connectivity

#### 6.2.1. Comparisons of One-Connectivity with UCA Antenna, Keyhole, Sector and Iris-UCA Models

#### 6.2.2. Comparisons of One-Connectivity with ULA Antenna and Iris-ULA Model

## 7. Discussion and Future Directions

## 8. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 6.**Directional antenna models. (

**a**) Realistic vs. keyhole; (

**b**) Realistic vs. sector; (

**c**) Realistic vs. iris.

**Figure 8.**Probability of node isolation $\mathbb{P}(\text{iso})$ with UCA antenna, keyhole, sector and iris-UCA models, where curves are analytical results and markers are simulation results. (

**a**) $\alpha =2.5$, $\sigma =4$; (

**b**) $\alpha =2.5$, $\sigma =8$; (

**c**) $\alpha =4$, $\sigma =8$.

**Figure 9.**Probability of node isolation $\mathbb{P}(\text{iso})$ with the ULA antenna and iris-ULA model, where curves are analytical results and markers are simulation results. (

**a**) $\alpha =2.5$, $\sigma =4$; (

**b**) $\alpha =2.5$, $\sigma =8$; (

**c**) $\alpha =4$, $\sigma =8$.

**Figure 10.**One-connectivity $\mathbb{P}$(1-con) of UCA antenna, keyhole, sector and iris-UCA models. (

**a**) $\alpha =2.5$, $\sigma =4$; (

**b**) $\alpha =3$, $\sigma =4$.

**Figure 11.**One-connectivity $\mathbb{P}$(1-con) of the ULA antenna and iris-ULA model. (

**a**) $\alpha =2.5$, $\sigma =4$; (

**b**) $\alpha =3$, $\sigma =4$.

Features | Keyhole Model | Sector Model | Iris Model (This Paper) |
---|---|---|---|

Main beam | Yes | Yes | Yes |

Side/back lobes | Yes | No | Yes |

Nulling capability | No | Yes | Yes |

More than one main beam | No | No | Yes |

**Table 2.**$\mathbb{E}[{({G}_{r}{G}_{t})}^{\frac{2}{\alpha}}]$ of the UCA antenna, keyhole, sector and iris-UCA with deviations compared with the UCA antenna.

Path Loss α | Antenna Models | |||
---|---|---|---|---|

UCA | Keyhole | Sector | Iris-UCA | |

2 | 1.61 | 2.33 (+44.34%) | 22.60 (+1302.28%) | 1.55 (−3.67%) |

2.25 | 1.32 | 1.92 (+45.31%) | 9.42 (+612.86%) | 1.16 (−12.28%) |

2.5 | 1.15 | 1.68 (+46.59%) | 4.68 (+307.40%) | 0.93 (−19.15%) |

2.75 | 1.04 | 1.53 (+47.72%) | 2.64 (+154.39%) | 0.78 (−24.74%) |

3 | 0.96 | 1.43 (+48.55%) | 1.64 (+70.14%) | 0.68 (−29.39%) |

3.25 | 0.91 | 1.36 (+49.06%) | 1.09 (+20.16%) | 0.61 (−33.31%) |

3.5 | 0.87 | 1.30 (+49.28%) | 0.77 (−11.33%) | 0.55 (−36.67%) |

3.75 | 0.84 | 1.26 (+49.23%) | 0.57 (−32.18%) | 0.51 (−39.56%) |

4 | 0.82 | 1.23 (+48.98%) | 0.44 (−46.54%) | 0.48 (−42.08%) |

Mean absolute deviation | N/A | 47.67% | 284.14% | 26.76% |

**Table 3.**$\mathbb{E}[{({G}_{r}{G}_{t})}^{\frac{2}{\alpha}}]$ of the ULA antenna and iris-ULA antenna with deviations compared with the ULA antenna.

Path Loss α | Antenna Models | |
---|---|---|

Realistic ULA | Iris-ULA | |

2 | 6.07 | 6.26 (+3.23%) |

2.25 | 4.13 | 4.04 (−2.10%) |

2.5 | 3.08 | 2.87 (−6.64%) |

2.75 | 2.44 | 2.19 (−10.58%) |

3 | 2.04 | 1.75 (−14.04%) |

3.25 | 1.76 | 1.46 (−17.12%) |

3.5 | 1.56 | 1.25 (−19.87%) |

3.75 | 1.42 | 1.10 (−22.36%) |

4 | 1.31 | 0.99 (−24.61%) |

Mean absolute deviation | N/A | 13.39% |

Parameters | Values |
---|---|

Number of topologies | 5000 |

Attenuation threshold ${\beta}_{0}$ | 50 dB |

Path loss exponent α | 2.5, 4 |

Standard deviation of shadow effect σ | 4, 8 |

**Table 5.**Critical node density ${\rho}_{c}$ with UCA antenna, keyhole, sector and iris-UCA models when $\sigma =4$.

α | A (m^{2}) | Antenna Models | |||
---|---|---|---|---|---|

Realistic UCA | Keyhole | Sector | Iris-UCA | ||

2 | 10^{6} | 8.75 × 10^{−6} | 5.67 × 10^{−6} (−35.20%) | 3.18 × 10^{−7} (−96.37%) | 9.14 × 10^{−6} (+4.46%) |

2.5 | 10^{6} | 2.10 × 10^{−4} | 1.37 × 10^{−4} (−34.76%) | 4.34 × 10^{−5} (−79.33%) | 2.66 × 10^{−4} (+26.67%) |

3 | 2.5 × 10^{5} | 1.32 × 10^{−3} | 8.58 × 10^{−4} (−35.00%) | 7.38 × 10^{−4} (−44.09%) | 1.90 × 10^{−3} (+43.94%) |

3.5 | 2.5 × 10^{5} | 5.20 × 10^{−3} | 3.35 × 10^{−3} (−35.50%) | 5.93 × 10^{−3} (+14.04%) | 8.56 × 10^{−4} (+64.57%) |

4 | 2.5 × 10^{5} | 1.40 × 10−2 | 9.10 × 10^{−3} (−35.15%) | 2.76 × 10−2 (+97.01%) | 2.54 × 10−2 (+80.67%) |

**Table 6.**Critical node density ${\rho}_{c}$ with UCA antenna, keyhole, sector and iris-UCA models when $\sigma =8$.

α | A (m^{2}) | Antenna Models | |||
---|---|---|---|---|---|

Realistic UCA | Keyhole | Sector | Iris-UCA | ||

2 | 10^{6} | 1.90 × 10^{−6} | 1.20 × 10^{−5} (−36.84%) | 2.48 × 10^{−8} (−98.69%) | 1.99 × 10^{−6} (+4.74%) |

2.5 | 10^{6} | 8.46 × 10^{−5} | 5.50 × 10^{−5} (−34.99%) | 1.71 × 10^{−5} (−79.79%) | 1.07 × 10^{−4} (+26.48%) |

3 | 2.5 × 10^{5} | 7.06 × 10^{−4} | 4.57 × 10^{−4} (−35.27%) | 3.92 × 10^{−4} (−44.48%) | 1.00 × 10^{−3} (+41.64%) |

3.5 | 2.5 × 10^{5} | 3.30 × 10^{−3} | 2.12 × 10^{−3} (−35.61%) | 3.77 × 10^{−3} (+14.10%) | 5.44 × 10^{−3} (+64.86%) |

4 | 2.5 × 10^{5} | 9.93 × 10^{−3} | 6.43 × 10^{−3} (−35.22%) | 1.96 × 10^{−2} (+97.31%) | 1.80 × 10^{−2} (+81.23%) |

**Table 7.**Critical node density ${\rho}_{c}$ with the ULA antenna and the iris-ULA model when $\sigma =4$.

α | A (m^{2}) | Antenna Models | |
---|---|---|---|

Realistic ULA | Iris-ULA Model | ||

2 | 10^{6} | 1.78 × 10^{−6} | 1.71 × 10^{−6} (−3.86%) |

2.5 | 10^{6} | 6.98 × 10^{−5} | 7.54 × 10^{−5} (+8.05%) |

3 | 2.5 × 10^{5} | 5.75 × 10^{−4} | 6.81 × 10^{−4} (+18.39%) |

3.5 | 2.5 × 10^{5} | 2.74 × 10^{−3} | 3.50 × 10^{−3} (+27.53%) |

4 | 2.5 × 10^{5} | 8.48 × 10^{−3} | 1.15 × 10^{−3} (+35.97%) |

**Table 8.**Critical node density ${\rho}_{c}$ with the ULA antenna and the iris-ULA model when $\sigma =8$.

α | A (m^{2}) | Antenna Models | |
---|---|---|---|

Realistic ULA | Iris-ULA | ||

2 | 10^{6} | 3.38 × 10^{−6} | 3.23 × 10^{−6} (−4.44%) |

2.5 | 10^{6} | 2.77 × 10^{−5} | 2.99 × 10^{−5} (+7.94%) |

3 | 2.5 × 10^{5} | 3.05 × 10^{−4} | 3.62 × 10^{−4} (+18.69%) |

3.5 | 2.5 × 10^{5} | 1.70 × 10^{−3} | 2.70 × 10^{−3} (+29.41%) |

4 | 2.5 × 10^{5} | 6.00 × 10^{−3} | 8.20 × 10^{−3} (+36.67%) |

Parameters | Values |
---|---|

Number of topologies | 5000 |

Attenuation threshold ${\beta}_{0}$ | 50 dB |

Path loss exponent α | 2.5, 3 |

Standard deviation of shadow effect σ | 4 |

© 2017 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Wang, Q.; Dai, H.-N.; Zheng, Z.; Imran, M.; Vasilakos, A.V.
On Connectivity of Wireless Sensor Networks with Directional Antennas. *Sensors* **2017**, *17*, 134.
https://doi.org/10.3390/s17010134

**AMA Style**

Wang Q, Dai H-N, Zheng Z, Imran M, Vasilakos AV.
On Connectivity of Wireless Sensor Networks with Directional Antennas. *Sensors*. 2017; 17(1):134.
https://doi.org/10.3390/s17010134

**Chicago/Turabian Style**

Wang, Qiu, Hong-Ning Dai, Zibin Zheng, Muhammad Imran, and Athanasios V. Vasilakos.
2017. "On Connectivity of Wireless Sensor Networks with Directional Antennas" *Sensors* 17, no. 1: 134.
https://doi.org/10.3390/s17010134