# Multi-Criteria Seed Selection for Targeting Multi-Attribute Nodes in Complex Networks

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Methodology

#### 3.1. Multi-Attribute Nature of the Targeted Nodes

#### 3.2. Multi-Attribute Seed Selection

#### 3.3. MCDA Foundations of the Proposed Approach and the Research Method Justification

#### 3.4. Multi-Criteria Seed Selection for Multi-Attribute Nodes Targeting

## 4. Empirical Study

#### 4.1. Real-Life Usage Example

#### 4.1.1. Target 1: Male Aged 0–29

#### 4.1.2. Target 2: Female Aged 30–59

#### 4.1.3. Real-Life Example Discussion

#### 4.2. Setup of the Comprehensive Experiment

- Betweenness—1687.295;
- Degree—3.994;
- Closeness—0.0002310899;
- Eigen Centrality—0.03661858.

- young, i.e., aged 0–49, $64.62\%$ of the population;
- mid-aged, i.e., aged 50–69, $25.34\%$ of the population;
- elderly, i.e., aged 70 and above, $10.04\%$ of the population.

#### 4.3. Criteria for Seed Selection

#### 4.4. Scenario 1: Single Criterion

#### 4.5. Scenario 2: Two Criteria

#### 4.6. Scenario 3: Four Criteria

#### 4.7. Sensitivity Analysis

#### 4.8. Full Range Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Appendix A

## References

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**Figure 1.**Conceptual framework of the proposed approach. Marks A–E provide anchors to be referred in the main text of the paper.

**Figure 2.**Visual presentation of two real-life usage scenarios for targeting male aged 0–29 (target 1) or female aged 30–59 (target 2). The table contains: values of the sex and age attributes, information on targeted nodes for both scenarios, and the rankings of nodes for seeding.

**Figure 3.**Visual comparison of ranks of nodes obtained in rankings for various scenarios: (

**A**) scenarios 1 and 2; (

**B**) scenarios 1 and 3; (

**C**) scenarios 2 and 3.

**Figure 4.**Sensitivity analysis on the subset of 63 network vertices. The charts represent how changes in a single criterion (1–8) affect the score obtained by the analysed vertices, when the weights of the other criteria are set to 1 (

**A**), 25 (

**B**), 50 (

**C**) or 75 (

**D**).

**Figure 5.**Sensitivity analysis on the subset of 63 network vertices. The charts represent how changes in a single criterion (1–8) affect the ranks obtained by the analysed vertices, when the weights of the other criteria are set to 1 (

**A**), 25 (

**B**), 50 (

**C**) or 75 (

**D**).

No | Criterion | Preference |
---|---|---|

C1 | Degree | max |

C2 | Sex (Match/Mismatch) | min |

C3 | Degree Male | max |

C4 | Degree Female | max |

C5 | Age (Match/Mismatch) | min |

C6 | Degree Young | max |

C7 | Degree Mid-Aged | max |

C8 | Degree Elderly | max |

**Table 2.**Seeds selected for Scenario 1, ordered by their rank and CCi score obtained in the applied TOPSIS method.

Rank | Vertex | Score | Rank | Vertex | Score | Rank | Vertex | Score | Rank | Vertex | Score | Rank | Vertex | Score |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 3 | 0.9975 | 11 | 49 | 0.4000 | 21 | 29 | 0.2800 | 31 | 18 | 0.2400 | 41 | 151 | 0.1800 |

2 | 4 | 0.6800 | 12 | 6 | 0.4000 | 22 | 170 | 0.2800 | 32 | 153 | 0.2400 | 42 | 97 | 0.1800 |

3 | 2 | 0.6000 | 13 | 11 | 0.3800 | 23 | 47 | 0.2800 | 33 | 57 | 0.2200 | 43 | 65 | 0.1800 |

4 | 12 | 0.5400 | 14 | 16 | 0.3400 | 24 | 21 | 0.2600 | 34 | 10 | 0.2200 | 44 | 59 | 0.1800 |

5 | 5 | 0.5200 | 15 | 26 | 0.3400 | 25 | 14 | 0.2600 | 35 | 40 | 0.2200 | 45 | 101 | 0.1800 |

6 | 24 | 0.4400 | 16 | 7 | 0.3400 | 26 | 45 | 0.2600 | 36 | 238 | 0.2200 | 46 | 36 | 0.1600 |

7 | 1 | 0.4400 | 17 | 113 | 0.3400 | 27 | 103 | 0.2600 | 37 | 56 | 0.2000 | 47 | 116 | 0.1600 |

8 | 30 | 0.4200 | 18 | 135 | 0.2800 | 28 | 82 | 0.2600 | 38 | 172 | 0.2000 | 48 | 37 | 0.1600 |

9 | 185 | 0.4200 | 19 | 17 | 0.2800 | 29 | 9 | 0.2400 | 39 | 20 | 0.1801 | 49 | 93 | 0.1600 |

10 | 19 | 0.4000 | 20 | 53 | 0.2800 | 30 | 42 | 0.2400 | 40 | 143 | 0.1800 | 50 | 55 | 0.1600 |

**Table 3.**Seeds selected for Scenario 2, ordered by their rank and CCi score obtained in the applied TOPSIS method.

Rank | Vertex | Score | Rank | Vertex | Score | Rank | Vertex | Score | Rank | Vertex | Score | Rank | Vertex | Score |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 3 | 0.9980 | 11 | 30 | 0.4075 | 21 | 20 | 0.3645 | 31 | 116 | 0.3073 | 41 | 34 | 0.2560 |

2 | 4 | 0.8142 | 12 | 9 | 0.4048 | 22 | 18 | 0.3606 | 32 | 26 | 0.3045 | 42 | 93 | 0.2476 |

3 | 2 | 0.7554 | 13 | 19 | 0.4036 | 23 | 7 | 0.3482 | 33 | 29 | 0.3045 | 43 | 464 | 0.2476 |

4 | 5 | 0.5836 | 14 | 11 | 0.3936 | 24 | 170 | 0.3482 | 34 | 152 | 0.3044 | 44 | 14 | 0.2445 |

5 | 12 | 0.5392 | 15 | 113 | 0.3857 | 25 | 153 | 0.3442 | 35 | 174 | 0.2913 | 45 | 48 | 0.2445 |

6 | 24 | 0.5178 | 16 | 17 | 0.3857 | 26 | 185 | 0.3260 | 36 | 82 | 0.2900 | 46 | 56 | 0.2354 |

7 | 6 | 0.4741 | 17 | 42 | 0.3856 | 27 | 53 | 0.3260 | 37 | 10 | 0.2840 | 47 | 69 | 0.2341 |

8 | 1 | 0.4452 | 18 | 21 | 0.3708 | 28 | 172 | 0.3250 | 38 | 238 | 0.2839 | 48 | 33 | 0.2341 |

9 | 135 | 0.4296 | 19 | 57 | 0.3658 | 29 | 16 | 0.3135 | 39 | 195 | 0.2839 | 49 | 97 | 0.2325 |

10 | 49 | 0.4164 | 20 | 143 | 0.3658 | 30 | 47 | 0.3135 | 40 | 122 | 0.2589 | 50 | 295 | 0.2325 |

**Table 4.**Seeds selected for Scenario 3, ordered by their rank and CCi score obtained in the applied TOPSIS method.

Rank | Vertex | Score | Rank | Vertex | Score | Rank | Vertex | Score | Rank | Vertex | Score | Rank | Vertex | Score |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 3 | 0.9069 | 11 | 9 | 0.4120 | 21 | 20 | 0.3750 | 31 | 29 | 0.3197 | 41 | 122 | 0.2782 |

2 | 4 | 0.7842 | 12 | 11 | 0.4023 | 22 | 153 | 0.3561 | 32 | 185 | 0.3197 | 42 | 34 | 0.2731 |

3 | 2 | 0.7191 | 13 | 30 | 0.3985 | 23 | 170 | 0.3535 | 33 | 116 | 0.3148 | 43 | 33 | 0.2717 |

4 | 5 | 0.5821 | 14 | 19 | 0.3950 | 24 | 18 | 0.3534 | 34 | 152 | 0.3125 | 44 | 93 | 0.2679 |

5 | 24 | 0.5291 | 15 | 143 | 0.3862 | 25 | 7 | 0.3412 | 35 | 174 | 0.3067 | 45 | 14 | 0.2660 |

6 | 12 | 0.5248 | 16 | 21 | 0.3810 | 26 | 53 | 0.3326 | 36 | 195 | 0.2934 | 46 | 130 | 0.2577 |

7 | 6 | 0.4782 | 17 | 113 | 0.3775 | 27 | 172 | 0.3315 | 37 | 82 | 0.2846 | 47 | 69 | 0.2566 |

8 | 1 | 0.4508 | 18 | 17 | 0.3774 | 28 | 16 | 0.3279 | 38 | 464 | 0.2822 | 48 | 97 | 0.2543 |

9 | 49 | 0.4236 | 19 | 42 | 0.3774 | 29 | 47 | 0.3278 | 39 | 10 | 0.2788 | 49 | 74 | 0.2474 |

10 | 135 | 0.4198 | 20 | 57 | 0.3757 | 30 | 26 | 0.3197 | 40 | 238 | 0.2788 | 50 | 104 | 0.2474 |

(A) RANKS | Scenario 1 | Scenario 2 | Scenario 3 | (B) SCORE | Scenario 1 | Scenario 2 | Scenario 3 |
---|---|---|---|---|---|---|---|

Scenario 1 | x | 0.7510 | 0.7099 | Scenario 1 | x | 0.9022 | 0.8186 |

Scenario 2 | 0.7510 | x | 0.7308 | Scenario 2 | 0.9022 | x | 0.8933 |

Scenario 3 | 0.7099 | 0.7308 | x | Scenario 3 | 0.8186 | 0.8933 | x |

Scenario | Preferences | Avg. Last Iter. | Inf. Nodes | Coverage | Targeted Inf. Nodes | Targeted Coverage |
---|---|---|---|---|---|---|

1 | 100-1-1-1-1-1-1-1 | 8.60 | 433.60 | 0.4336 | 50.50 | 0.3885 |

2 | 1-1-1-100-1-1-100-1 | 9.10 | 435.60 | 0.4356 | 52.00 | 0.4000 |

3 | 1-100-1-100-100-1-100-1 | 8.70 | 435.00 | 0.4350 | 52.70 | 0.4054 |

Average Last Iteration | Average Coverage | Average Targeted Coverage | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Δ | S1 | S2 | S3 | Δ | S1 | S2 | S3 | Δ | S1 | S2 | S3 |

S1 | x | $-0.5$ | $-0.1$ | S1 | x | $-0.0020$ | $-0.0014$ | S1 | x | $-0.0115$ | $-0.0169$ |

S2 | 0.5 | x | 0.4 | S2 | 0.0020 | x | 0.0006 | S2 | 0.0115 | x | $-0.0054$ |

S3 | 0.1 | $-0.4$ | x | S3 | 0.0014 | $-0.0006$ | x | S3 | 0.0169 | 0.0054 | x |

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**MDPI and ACS Style**

Karczmarczyk, A.; Jankowski, J.; Wątrobski, J.
Multi-Criteria Seed Selection for Targeting Multi-Attribute Nodes in Complex Networks. *Symmetry* **2021**, *13*, 731.
https://doi.org/10.3390/sym13040731

**AMA Style**

Karczmarczyk A, Jankowski J, Wątrobski J.
Multi-Criteria Seed Selection for Targeting Multi-Attribute Nodes in Complex Networks. *Symmetry*. 2021; 13(4):731.
https://doi.org/10.3390/sym13040731

**Chicago/Turabian Style**

Karczmarczyk, Artur, Jarosław Jankowski, and Jarosław Wątrobski.
2021. "Multi-Criteria Seed Selection for Targeting Multi-Attribute Nodes in Complex Networks" *Symmetry* 13, no. 4: 731.
https://doi.org/10.3390/sym13040731