# Constrained Binary Optimization Approach for Pinned Node Selection in Pinning Control of Complex Dynamical Networks

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## Abstract

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## 1. Introduction

- We present an approach for finding the optimal pinned node selection for pinning control, aimed at stabilizing the complex network state to zero, given the total number of pinned nodes.
- We conduct a comparative study of state-of-the-art binary optimization algorithms to compute the optimal pinned nodes. The algorithms considered for comparison include the genetic algorithm (GA), arithmetic optimization algorithm (AOA), hybrid algorithm of particle swarm optimization and grey wolf optimizer (HPSOGWO), rat swarm optimization (RSO), grasshopper optimization algorithm GOA, firefly algorithm FA, and its discrete version (DFA).

## 2. Problem Formulation

#### 2.1. Complex Network and Pinning Control

#### 2.2. Objective Function Formulation

## 3. Algorithms Used and Their Modifications

#### 3.1. Genetic Algorithms

Algorithm 1: Parent selection algorithm. |

Algorithm 2: Crossover. |

Algorithm 3: Mutation. |

Algorithm 4: Binary-constrained mutation. |

#### 3.2. Arithmetic Optimization Algorithm

Algorithm 5: Binary-constrained transform. |

#### 3.3. Optimization Hybrid Grey Wolf Optimization

#### 3.4. Grasshopper Optimization Algorithm

#### 3.5. Rat Swarm Optimization Algorithm

#### 3.6. Firefly Algorithm

#### 3.7. Summary

## 4. Results

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AOA | arithmetic optimization algorithm |

BinAOA | binary arithmetic optimization algorithm |

BGOA | binary grasshopper optimization algorithm |

BGWOPSO | binary hybrid grey wolf optimization-particle swarm optimization |

BRSO | binary rat swarm optimization |

DE | differential evolution |

DFA | discrete firefly algorithm |

FA | firefly algorithm |

GA | genetic algorithm |

GOA | grasshopper optimization algorithm |

GWO | grey wolf optimization |

HPSOGWO | hybrid algorithm of particle swarm optimization and grey wolf optimizer |

MOA | math optimizer accelerated |

MOP | math optimization probability |

NBGOA | New binary grasshopper optimization algorithm |

PSO | particle swarm optimization |

RSO | rat swarm optimization |

WHO | whale optimization algorithm |

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**Figure 4.**Stabilization errors for each algorithm for every iteration in the experiment, with 5 different systems.

Algorithm | Modification |
---|---|

GA | Binary-Constrained Mutation |

AOA | Binary-Constrained Transform |

HPSOGWO | Binary-Constrained Transform |

GOA | Binary-Constrained Transform |

RSO | Binary-Constrained Transform |

FA | Binary-Constrained Transform |

DFA | None |

Algorithm | Parameters |
---|---|

GA | $M=20,{p}_{m}=0.7$ |

AOA | $M=20,\alpha =5,\mu =0.5,\mathsf{\epsilon}=0.000001,Max=0.2,Min=0.05$ |

HPSOGWO | $M=20,{c}_{1}={c}_{2}={c}_{3}=0.5$ |

GOA | $M=20,\beta =0.5,l=1.5$ |

RSO | $M=20,$ |

FA | $M=10,\gamma =0.4,\alpha =0.01$ |

DFA | $M=10,\gamma =0.4,\alpha =0.01$ |

Selected Nodes (𝚤) | Stabilization Error |
---|---|

1,3,4,6,7,10,32,33,48,50 | 0.0105133261 |

3,4,13,18,19,20,24,37,44,50 | 0.0107734755 |

1,9,19,20,21,25,34,45,46,48 | 0.0108550197 |

1,6,11,16,24,28,30,31,44,50 | 0.0142530794 |

1,3,19,28,32,36,38,42,48,50 | 0.0150178204 |

Selected Nodes (𝚤) | Stabilization Error |
---|---|

3,7,33,48,19,39,8,11,16,23 | 0.0012275025 |

3,7,33,48,19,39,8,11,16,41 | 0.0023177425 |

3,7,33,48,19,39,8,11,23,41 | 0.0031266226 |

3,7,33,48,19,39,8,11,23,41 | 0.0013908266 |

3,7,33,48,19,39,11,16,23,41 | 0.0004301837 |

Algorithm | Selected Nodes (𝚤) | Stabilization Error |
---|---|---|

GA | 3,14,22,33,34,35,37,38,43,48 | 0.0001228105 |

AOA | 1,3,4,6,33,38,39,43,48,50 | 0.0037335172 |

HPSOGWO | 1,3,16,18,20,29,33,34,48,50 | $7.0052533573\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ |

GOA | 3,10,18,20,22,28,30,33,37,48 | $1.2757280761\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ |

RSO | 3,5,10,12,13,19,24,33,37,48 | $1.5527264707\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ |

FA | 1,3,19,20,26,31,37,42,44,48 | $4.681870615\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ |

DFA | 3,16,18,20,29,31,33,44,47,48 | 2.0350233065$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ |

System | Nodes$\left(\U0001d6a4\right)$ |
---|---|

Lorenz | 1,6,11,16,21,26,31,36,41,46 |

Chen | 2,7,12,17,22,27,32,37,42,47 |

Lü | 3,8,13,18,23,28,33,38,43,48 |

Qi | 4,9,14,19,24,29,34,39,44,49 |

Chua | 5,10,15,20,25,30,35,40,45,50 |

Selected Nodes (𝚤) | Stabilization Error |
---|---|

1,3,14,26,29,34,36,42,47,50 | 0.0107664230 |

3,6,10,11,16,24,26,33,37,44 | 0.0693599189 |

1,2,3,7,24,27,31,36,42,50 | 0.1874461797 |

3,4,8,19,20,23,29,35,38,44 | 0.2147937241 |

2,7,8,9,13,21,25,31,34,50 | 0.2685484577 |

Selected Nodes (𝚤) | Stabilization Error |
---|---|

3,7,33,48,19,39,8,11,16,23 | N/A |

3,7,33,48,19,39,8,11,16,41 | 22.495094562 |

3,7,33,48,19,39,8,11,23,41 | 25.991754941 |

3,7,33,48,19,39,8,11,23,41 | N/A |

3,7,33,48,19,39,11,16,23,41 | 5.0256082799 |

Algorithm | Selected Nodes (𝚤) | Stabilization Error |
---|---|---|

GA | 9,19,27,29,30,34,35,37,46,48 | 0.0018476030 |

AOA | 1,3,5,6,12,19,35,39,48,50 | 0.0001338463 |

HPSOGWO | 10,11,12,17,19,21,26,33,35,47 | 0.0045804490 |

GOA | 1,5,9,10,15,22,24,32,35,44 | 0.0099802442 |

RSO | 5,9,15,22,23,26,32,34,44,47 | 0.0051811245 |

FA | 1,3,19,20,26,31,37,42,44,48 | 0.0008024141 |

DFA | 5,7,14,19,26,29,31,34,35,46 | 0.0180924252 |

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**MDPI and ACS Style**

Alanis, A.Y.; Hernandez-Barragan, J.; Ríos-Rivera, D.; Sanchez, O.D.; Martinez-Soltero, G.
Constrained Binary Optimization Approach for Pinned Node Selection in Pinning Control of Complex Dynamical Networks. *Axioms* **2023**, *12*, 1088.
https://doi.org/10.3390/axioms12121088

**AMA Style**

Alanis AY, Hernandez-Barragan J, Ríos-Rivera D, Sanchez OD, Martinez-Soltero G.
Constrained Binary Optimization Approach for Pinned Node Selection in Pinning Control of Complex Dynamical Networks. *Axioms*. 2023; 12(12):1088.
https://doi.org/10.3390/axioms12121088

**Chicago/Turabian Style**

Alanis, Alma Y., Jesus Hernandez-Barragan, Daniel Ríos-Rivera, Oscar D. Sanchez, and Gabriel Martinez-Soltero.
2023. "Constrained Binary Optimization Approach for Pinned Node Selection in Pinning Control of Complex Dynamical Networks" *Axioms* 12, no. 12: 1088.
https://doi.org/10.3390/axioms12121088