# MOPIO: A Multi-Objective Pigeon-Inspired Optimization Algorithm for Community Detection

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## Abstract

**:**

## 1. Introduction

## 2. Related Works

#### 2.1. Community Detection Problem

#### 2.2. Multi-Objective Optimization

#### 2.3. Basic PIO

## 3. Method

#### 3.1. Pigeon Representation and Initialization

#### 3.2. Fitness Function

#### 3.3. Pareto Sorting Scheme

#### 3.4. Search Strategy

#### 3.4.1. Optimal Solution Selecting Strategy

#### 3.4.2. Crossover and Mutation

#### 3.5. Leader Selection Operation

## 4. Results and Discussion

#### 4.1. Evaluation Metrics

#### 4.2. Experimental Results

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**True structure of the Zachary’s karate club network and results detected by five methods. (

**a**) Benchmark network; (

**b**) MOPIO; (

**c**) MOPSO; (

**d**) MOGA-Net; (

**e**) FN; (

**f**) Meme-Net.

**Figure 4.**True structure of American College Football network and results detected by five methods. (

**a**) Benchmark network; (

**b**) MOPIO; (

**c**) MOPSO; (

**d**) MOGA-Net; (

**e**) FN; (

**f**) Meme-Net.

**Figure 5.**True structure of FB50 and results detected by five methods. (

**a**) Benchmark network; (

**b**) MOPIO; (

**c**) MOPSO; (

**d**) MOGA-Net; (

**e**) FN; (

**f**) Meme-Net.

**Figure 6.**True structure of Krebs’ books on US politics network and results detected by five methods. (

**a**) Benchmark network; (

**b**) MOPIO; (

**c**) MOPSO; (

**d**) MOGA-Net; (

**e**) FN; (

**f**) Meme-Net.

N | Population Size | 100 |
---|---|---|

I | The number of MOPIO iteration | 50 |

pc | crossover probability | 0.8 |

pm | mutation probability | 0.4 |

p | preferred ratio | 0.25 |

Network | Node | Edge | Community |
---|---|---|---|

Zachary’s karate club | 34 | 78 | 2 |

FB50 | 50 | 404 | 4 |

American College Football | 115 | 613 | 12 |

Krebs’ books on US politics | 105 | 441 | 3 |

Dataset | Metrics | MOPIO | MOPSO | MOGA-Net | FN | Meme-Net |
---|---|---|---|---|---|---|

Karate | NMI (max) | 1.000 | 0.930 | 0.707 | 0.692 | 1.000 |

NMI (avg) | 0.860 | 0.556 | 0.628 | 0.692 | 0.501 | |

ARI (max) | 1.000 | 0.882 | 0.416 | 0.680 | 1.000 | |

ARI (avg) | 0.856 | 0.467 | 0.415 | 0.680 | 0.477 | |

Football | NMI (max) | 0.816 | 0.399 | 0.800 | 0.726 | 0.887 |

NMI (avg) | 0.754 | 0.122 | 0.762 | 0.726 | 0.795 | |

ARI (max) | 0.670 | 0.113 | 0.629 | 0.491 | 0.744 | |

ARI (avg) | 0.573 | 0.045 | 0.485 | 0.491 | 0.581 | |

Fb50 | NMI (max) | 1.000 | 0.902 | 0.938 | 0.938 | 1.000 |

NMI (avg) | 1.000 | 0.794 | 0.938 | 0.938 | 0.997 | |

ARI (max) | 1.000 | 0.814 | 0.954 | 0.954 | 1.000 | |

ARI (avg) | 1.000 | 0.580 | 0.954 | 0.954 | 0.998 | |

Polbooks | NMI (max) | 0.606 | 0.456 | 0.564 | 0.516 | 0.574 |

NMI (avg) | 0.494 | 0.163 | 0.524 | 0.516 | 0.427 | |

ARI (max) | 0.709 | 0.248 | 0.665 | 0.609 | 0.675 | |

ARI (avg) | 0.559 | 0.080 | 0.579 | 0.609 | 0.434 |

Dataset | Metrics | MOPIO | MOPSO | MOGA-Net | FN | Meme-Net |
---|---|---|---|---|---|---|

Karate | P | 0.624 | 0.631 | 0.229 | 0.370 | 0.402 |

R | 0.576 | 0.612 | 0.172 | 0.185 | 0.453 | |

F | 0.596 | 0.615 | 0.196 | 0.247 | 0.410 | |

Football | P | 0.164 | 0.028 | 0.110 | 0.097 | 0.101 |

R | 0.182 | 0.108 | 0.150 | 0.165 | 0.142 | |

F | 0.160 | 0.040 | 0.118 | 0.119 | 0.108 | |

Fb50 | P | 1.000 | 0.462 | 0.625 | 0.625 | 0.981 |

R | 1.000 | 0.538 | 0.750 | 0.750 | 0.988 | |

F | 1.000 | 0.492 | 0.667 | 0.667 | 0.983 | |

Polbooks | P | 0.095 | 0.209 | 0.052 | 0.056 | 0.123 |

R | 0.080 | 0.359 | 0.054 | 0.024 | 0.104 | |

F | 0.074 | 0.237 | 0.033 | 0.033 | 0.085 |

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**MDPI and ACS Style**

Shang, J.; Li, Y.; Sun, Y.; Li, F.; Zhang, Y.; Liu, J.-X.
MOPIO: A Multi-Objective Pigeon-Inspired Optimization Algorithm for Community Detection. *Symmetry* **2021**, *13*, 49.
https://doi.org/10.3390/sym13010049

**AMA Style**

Shang J, Li Y, Sun Y, Li F, Zhang Y, Liu J-X.
MOPIO: A Multi-Objective Pigeon-Inspired Optimization Algorithm for Community Detection. *Symmetry*. 2021; 13(1):49.
https://doi.org/10.3390/sym13010049

**Chicago/Turabian Style**

Shang, Junliang, Yiting Li, Yan Sun, Feng Li, Yuanyuan Zhang, and Jin-Xing Liu.
2021. "MOPIO: A Multi-Objective Pigeon-Inspired Optimization Algorithm for Community Detection" *Symmetry* 13, no. 1: 49.
https://doi.org/10.3390/sym13010049