# Multi-Objective Optimization of the Robustness of Complex Networks Based on the Mixture of Weighted Surrogates

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Related Work

#### 2.1. Network Controllability Robustness

#### 2.2. Surrogate Models

#### 2.2.1. Radial Basis Function (RBF) Networks

#### 2.2.2. Least Square (LS) Method

#### 2.2.3. Inverse Distance Weighting (IDW) Interpolation Method

#### 2.2.4. Kriging Interpolation Method

#### 2.3. New Contributions of the Proposed Algorithm

## 3. Algorithm Framework

Algorithm 1: Multi-objective evolutionary algorithm optimization. | ||

Input: | ||

Initialize network population P; | ||

Iterations t = 0; | ||

Max iterations MaxGen; | ||

output: | ||

The non-dominated solution set and network structure; | ||

Calculate the network controllability robustness and initialize the surrogate model; | ||

While t < MaxGen: | ||

Conduct the crossover operator on ${P}_{t}$ to generate ${Q}_{t}$; | ||

Conduct the mutation operator on ${Q}_{t}$; | ||

Select better individuals from ${P}_{t}$ and ${Q}_{t}$ to ${P}_{t+1}$; | ||

Conduct the local search operator on ${P}_{t+1}$; | ||

Update the surrogate model; | ||

Update EP with ${P}_{t+1}$; | ||

t = t + 1; | ||

end while |

## 4. Selection and Mixture of Surrogate Models

#### 4.1. Dempster–Shafer Theory Weighting Method

#### 4.2. Single Selection of Surrogate Models

Algorithm 2: The process of calculating the weights of D-S theory. | ||

Input: | ||

The number of surrogate: N; | ||

The number of complex networks: G; | ||

output: | ||

Weights of different surrogate models in complex networks; | ||

Randomly initialize the complex network and set iterator t = 0; | ||

Calculate the true network controllability robustness; | ||

While t < N: | ||

Training surrogate models with controllability robustness at population; | ||

The trained surrogate model is used to evaluate the controllability robustness of | ||

the network; | ||

Calculate the correlation coefficient between the true value and the predicted | ||

value ${m}_{t}^{CC}$; | ||

Root mean square error ${m}_{t}^{RMSE}$ and maximal absolute errors ${m}_{t}^{MAE}$; | ||

Several feature attribute values have been calculated in the previous step, and | ||

D-S theory | ||

is used to calculate the weights under these feature attribute values; | ||

t = t + 1; | ||

end while | ||

Output the weight of each surrogate model; |

#### 4.3. Mixture of Weighted Surrogate Models

#### 4.4. Adaptively Updating Surrogate Models

Algorithm 3: Updated pseudo-code for the surrogate model. | ||

Input: | ||

The number of surrogates: N; | ||

Maximum number of iterations: MaxGen; | ||

Surrogate model update probability: update_rate; | ||

output: | ||

Output the two surrogate models with the largest weights; | ||

Randomly initialize the complex network and set iterator t = 0; | ||

Using the obtained controllability robustness at the population to train the surrogate model; | ||

Evaluate the network controllability robustness on a trained surrogate model | ||

using the evaluated complex network structure; | ||

Calculate CC, RMSE, and MAE between the true value and the evaluated value; | ||

The D-S theory is used to calculate the weights under three feature attribute values; | ||

While t < MaxGen: | ||

if random.random() < update_rate: | ||

D-S theory computational process is used to assign weights to the | ||

surrogate model; | ||

Select the top N surrogate models based on weights; | ||

Retrain the surrogate model using the obtained non-dominated solution; | ||

t = t + 1; | ||

Output the surrogate model with the highest weights and save the weights; | ||

end while |

## 5. Experimental Results

#### 5.1. Experimental Results of Single Selection and Mixture Weighted Surrogate Models

#### 5.2. Experimental Results of Adaptively Updating Surrogate Models

## 6. Discussions

## 7. Conclusions and Future Work

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 5.**$\mathrm{MOEA}\_\mathrm{Two}$ and $\mathrm{MOEA}\_\mathrm{Two}\_\mathrm{Adapt}$ under four types of complex network.

**Figure 6.**HV values of $\mathrm{MOEA}\_\mathrm{Two}$ and $\mathrm{MOEA}\_\mathrm{Two}\_\mathrm{Adapt}$ in four types of network.

Surrogates | SF | ER | ||||
---|---|---|---|---|---|---|

CC | RMSE | MAE | CC | RMSE | MAE | |

RBF | 0.25 | 0.252 | 0.252 | 0.25 | 0.279 | 0.285 |

LS | 0.25 | 0.245 | 0.241 | 0.25 | 0.212 | 0.202 |

IDW | 0.25 | 0.252 | 0.253 | 0.25 | 0.267 | 0.265 |

Kriging | 0.25 | 0.251 | 0.253 | 0.25 | 0.242 | 0.248 |

Surrogates | SW | RR | ||||

CC | RMSE | MAE | CC | RMSE | MAE | |

RBF | 0.25 | 0.250 | 0.248 | 0.25 | 0.251 | 0.247 |

LS | 0.25 | 0.250 | 0.247 | 0.25 | 0.249 | 0.250 |

IDW | 0.25 | 0.251 | 0.255 | 0.25 | 0.250 | 0.252 |

Kriging | 0.25 | 0.249 | 0.246 | 0.25 | 0.250 | 0.251 |

Surrogates | SF | ER | SW | RR |
---|---|---|---|---|

RBF | 0.26 | 0.31 | 0.26 | 0.14 |

LS | 0.23 | 0.19 | 0.24 | 0.13 |

IDW | 0.27 | 0.27 | 0.27 | 0.37 |

Kriging | 0.24 | 0.23 | 0.23 | 0.36 |

Surrogates | SF | ER | SW | RR |
---|---|---|---|---|

RBF | 0.51 | 0.47 | 0.51 | 0.59 |

LS | 0.49 | 0.53 | 0.49 | - |

IDW | - | - | - | - |

Kriging | - | - | - | 0.41 |

Surrogates | SF | ER | SW | RR |
---|---|---|---|---|

RBF | 0.35 | 0.33 | 0.35 | 0.43 |

LS | 0.34 | 0.38 | 0.24 | 0.16 |

IDW | - | - | 0.31 | - |

Kriging | 0.31 | 0.31 | - | 0.41 |

Networks | Method | HV | Run_Time |
---|---|---|---|

${\mathrm{MOEA}}_{0}$ | 0.1430 | 209.34 | |

$\mathrm{SP}\_\mathrm{RV}\_{\mathrm{MOEA}}_{\mathrm{Net}}$ | 0.1690 | 95.62 | |

SF | $\mathrm{MOEA}\_\mathrm{One}$ | 0.1660 | 57.26 |

$\mathrm{MOEA}\_\mathrm{Two}$ | 0.1929 | 58.65 | |

$\mathrm{MOEA}\_\mathrm{Three}$ | 0.1672 | 107.09 | |

${\mathrm{MOEA}}_{0}$ | 0.1404 | 257.95 | |

$\mathrm{SP}\_\mathrm{RV}\_{\mathrm{MOEA}}_{\mathrm{Net}}$ | 0.1698 | 104.21 | |

ER | $\mathrm{MOEA}\_\mathrm{One}$ | 0.1559 | 70.85 |

$\mathrm{MOEA}\_\mathrm{Two}$ | 0.1721 | 72.34 | |

$\mathrm{MOEA}\_\mathrm{Three}$ | 0.1646 | 124.36 | |

${\mathrm{MOEA}}_{0}$ | 0.2149 | 198.63 | |

$\mathrm{SP}\_\mathrm{RV}\_{\mathrm{MOEA}}_{\mathrm{Net}}$ | 0.2345 | 101.36 | |

SW | $\mathrm{MOEA}\_\mathrm{One}$ | 0.2362 | 30.10 |

$\mathrm{MOEA}\_\mathrm{Two}$ | 0.1721 | 36.54 | |

$\mathrm{MOEA}\_\mathrm{Three}$ | 0.2342 | 102.32 | |

${\mathrm{MOEA}}_{0}$ | 0.1030 | 234.36 | |

$\mathrm{SP}\_\mathrm{RV}\_{\mathrm{MOEA}}_{\mathrm{Net}}$ | 0.1149 | 126.31 | |

RR | $\mathrm{MOEA}\_\mathrm{One}$ | 0.1390 | 42.13 |

$\mathrm{MOEA}\_\mathrm{Two}$ | 0.1671 | 45.57 | |

$\mathrm{MOEA}\_\mathrm{Three}$ | 0.1200 | 102.46 |

**Table 6.**Running time (hours) and HV values for $\mathrm{MOEA}\_\mathrm{Two}\_\mathrm{Adapt}$ and $\mathrm{MOEA}\_\mathrm{Two}$ for four types of complex network.

Networks | Method | HV | Run_time |
---|---|---|---|

SF | MOEA_Two | 0.1929 | 58.65 |

MOEA_Two_Adapt | 0.1939 | 178.59 | |

ER | MOEA_Two | 0.1721 | 72.34 |

MOEA_Two_Adapt | 0.1766 | 180.77 | |

SW | MOEA_Two | 0.2362 | 36.54 |

MOEA_Two_Adapt | 0.2439 | 176.78 | |

RR | MOEA_Two | 0.1671 | 45.57 |

MOEA_Two_Adapt | 0.1765 | 137.52 |

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**MDPI and ACS Style**

Nie, J.; Yu, Z.; Li, J.
Multi-Objective Optimization of the Robustness of Complex Networks Based on the Mixture of Weighted Surrogates. *Axioms* **2023**, *12*, 404.
https://doi.org/10.3390/axioms12040404

**AMA Style**

Nie J, Yu Z, Li J.
Multi-Objective Optimization of the Robustness of Complex Networks Based on the Mixture of Weighted Surrogates. *Axioms*. 2023; 12(4):404.
https://doi.org/10.3390/axioms12040404

**Chicago/Turabian Style**

Nie, Junfeng, Zhuoran Yu, and Junli Li.
2023. "Multi-Objective Optimization of the Robustness of Complex Networks Based on the Mixture of Weighted Surrogates" *Axioms* 12, no. 4: 404.
https://doi.org/10.3390/axioms12040404