# Construction and Optimization of Biconnected and Wide-Coverage Topology Based on Node Mobility

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## Abstract

**:**

## 1. Introduction

## 2. Background and Nodes Distribution Model

#### 2.1. Background

#### 2.2. Network Model and Coverage Model

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

#### 2.3. Objective Optimization Model

## 3. Critical Node Detection Algorithm and NNR Algorithm

#### 3.1. Critical Node Detection Algorithm

#### 3.2. NNR

- Node ${v}_{i}$ and the neighbor node exchange the information about their positions. Assume that the counter-clockwise direction is positive, the two nodes ${v}_{l}$ and ${v}_{r},$ which are closest to node ${v}_{i}$ on the azimuth, are selected, namely ${\theta}_{r}<{\theta}_{i}<{\theta}_{l}$, and the azimuth of any other nodes ${v}_{j}$ is in accordance with ${\theta}_{j}<{\theta}_{r}$ or ${\theta}_{j}>{\theta}_{l}$.
- Calculate the angle that only involves the node ${v}_{i}$, then calculate the angular bisector of this angle, and the azimuth of the angular bisector is the azimuth ${\theta}_{i}\left(t+1\right)$ at the next time of the node. If the nodes ${v}_{r}$ and ${v}_{l}$ meet ${\theta}_{r}\left(t\right)<{\theta}_{i}\left(t\right)<{\theta}_{l}\left(t\right)$, ${\theta}_{i}\left(t+1\right)$ is the average value of the azimuth ${\theta}_{l}\left(t\right)$ and ${\theta}_{r}\left(t\right)$. If all neighbor nodes of ${v}_{i}$ are only at one side of the node, namely ${\theta}_{i}\left(t\right)<{\theta}_{j}\left(t\right)$ or ${\theta}_{i}\left(t\right)>{\theta}_{j}\left(t\right)$, and ${v}_{j}\in N\left({v}_{i}\right)$, the virtual node ${v}_{vir}$ is constructed at the other side of the node ${v}_{i}$ and ${\theta}_{i}\left(t+1\right)$ is the average value of the azimuth between ${v}_{l}$ (or ${v}_{r}$) and ${v}_{vir}$, as shown in Formula (4). Moreover, the azimuth of the virtual node ${v}_{vir}$ is ${v}_{virr}=$ max{$\alpha ,\mathrm{min}\left\{{\theta}_{j}|{v}_{j}\in V\right\}-\Delta \theta $} or ${v}_{virl}=\mathrm{min}\left\{\beta ,\mathrm{max}\left\{{\theta}_{j}|{v}_{j}\in V\right\}+\Delta \theta \right\}$, among which $\Delta \theta $ is the control constant.$${\theta}_{i}\left(t+1\right)=\{\begin{array}{c}\frac{{\theta}_{virr}+{\theta}_{l}\left(t\right)}{2},if{v}_{i}closesttoboundary\alpha \\ \frac{{\theta}_{r}\left(t\right)+{\theta}_{l}\left(t\right)}{2},ifthereexists{v}_{r}and{v}_{l}\\ \frac{{\theta}_{r}\left(t\right)+{\theta}_{virl}}{2},if{v}_{i}closesttoboundary\beta \end{array}$$
- Move the position ${\theta}_{i}\left(t\right)$ of node ${v}_{i}$ to the angular bisector ${\theta}_{i}\left(t+1\right)$, repeat the step 1 and 2 until ${\theta}_{i}\left(t\right)$ and ${\theta}_{i}\left(t+1\right)$ coincide.

## 4. The Optimization Method of the Topological Structure

#### 4.1. The Limit Value for Moving

**Proposition**

**1.**

**Proof.**

#### 4.2. Building up the Biconnected Topology and Enhancing the Coverage Based on Mobility

#### 4.2.1. Constructing the Biconnected Topology

#### 4.2.2. Enhancing Coverage

**Proposition**

**2.**

**Proof.**

#### 4.2.3. Remove Redundant Connections

#### 4.2.4. Termination of the Algorithm

- Obtain the position coordinate $\mathrm{I}\left({x}_{i},{y}_{i}\right)$ of the local node by GPS and obtain the position coordinates of its neighbors by communication.
- Calculate the actual destination coordinate ${\mathrm{I}}^{\prime}\left({x}_{i}^{\prime},{y}_{i}^{\prime}\right)$ and move to the destination.
- Repeat Step 1 and Step 2 until ${\mathrm{I}}^{\prime}\left({x}_{i}^{\prime},{y}_{i}^{\prime}\right)\approx \mathrm{I}\left({x}_{i},{y}_{i}\right),$ then the algorithm is terminated.

## 5. Simulation

#### 5.1. Feasibility Analysis

#### 5.2. Performance Comparison

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Illustration of jamming node distribution. The square marks denote the Radars, and the bold dots denote the jamming nodes. (

**a**) At the initial moment, a number of jamming nodes are deployed randomly in the sector region near the enemy radar; (

**b**) denotes that the jamming nodes are deployed reasonably after network topology optimization.

**Figure 2.**Illustration of coverage by jamming nodes. The square marks denote the radars, and the bold dots denote the jamming nodes. The sector area is divided into $m$ subregions, and the angle of the subregion is equal to the width of the main lobe of the radar beam. (

**a**) The sector area is completely covered where the shaded area represents covered area. (

**b**) The sector area is not completely covered.

**Figure 3.**Illustration of the process of nodes moving by the nearest neighbor rule (NNR). The circle marks denote the initial positions, and the plus signs denote the positions at next iteration. (

**a**) Take node ${v}_{i}$ as example, calculate the angle which only involves the node ${v}_{i}$, then calculate the angular bisector of this angle. The target position of node ${v}_{i}$ is located on the angular bisector. (

**b**) The process of nodes moving from the initial position to the target position. (

**c**) Repeatable process to update the positions of nodes.

**Figure 4.**Motion planning in Scene 1. The solid circles denote the current positions of nodes. The empty circles with a solid line denote the destination that the nodes can really reach. The empty circles with a dashed line denote the intentional destination.

**Figure 8.**Topology sketch of single critical node. The red circle denotes the critical node ${v}_{c}$, and the blue circles denote the neighbors.

**Figure 9.**Simulation results under three scenes. (

**a-1**), the initial topology with 5 nodes. (

**a-2**), the final topology constructed by the Localized Topology Optimized Method (LTOM). (

**a-3**), the final topology constructed by NNR. (

**b-1**), the initial topology with 20 nodes. (

**b-2**), the final topology constructed by LTOM. (

**b-3**), the final topology constructed by NNR. (

**c-1**), a node fails during the operation of the algorithm. (

**c-2**), the final topology constructed by LTOM. (

**c-3**), the final topology constructed by NNR.

n | Number of the Jamming Nodes |
---|---|

${r}_{c}$ | Communication radius of the nodes |

$\left({x}_{i},{y}_{i}\right)$ | The coordinates of node ${v}_{i}$ in rectangular coordinates |

${\theta}_{i}$ | The angle between the line linking the node ${v}_{i}$ with the radar and the X-axis, called azimuth |

${R}_{i}$ | The distance from the node to the radar |

$\left[\alpha ,\beta \right]$ | The combat zone is the sector area, and $\left[\alpha ,\beta \right]$ is the angle range of the combat zone, see Figure 1 |

$\left[{R}_{1},{R}_{2}\right]$ | The radial range of the combat zone, see Figure 1 |

$P\left({v}_{i},{v}_{j}\right)$ | The sets of disjoint paths between the node ${v}_{i}and{v}_{j}$ |

${n}_{p}\left({v}_{i},{v}_{j}\right)$ | The number of the edges on the shortest path ${P}_{1}\left({v}_{i},{v}_{j}\right)$ |

$N\left({v}_{i}\right)$ | The sets of neighbor nodes for the node ${v}_{i}$ |

$\u25b3{\theta}_{r}$ | The beam width of the radar main lobe, which is also the interval angle for dividing the subregions |

$m$ | Divide the combat zone into $m$ subregions |

${S}_{i}$ | The subregion where the node ${v}_{i}$ is located |

$\u25b3{d}_{i}\left(t\right)$ | The moving distance of node ${v}_{i}$ at the $t$ iteration |

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**MDPI and ACS Style**

Zhao, P.; Wang, J.; Kong, L.
Construction and Optimization of Biconnected and Wide-Coverage Topology Based on Node Mobility. *Symmetry* **2020**, *12*, 791.
https://doi.org/10.3390/sym12050791

**AMA Style**

Zhao P, Wang J, Kong L.
Construction and Optimization of Biconnected and Wide-Coverage Topology Based on Node Mobility. *Symmetry*. 2020; 12(5):791.
https://doi.org/10.3390/sym12050791

**Chicago/Turabian Style**

Zhao, Peng, Jianzhong Wang, and Lingren Kong.
2020. "Construction and Optimization of Biconnected and Wide-Coverage Topology Based on Node Mobility" *Symmetry* 12, no. 5: 791.
https://doi.org/10.3390/sym12050791