# Protection Strategy for Edge-Weighted Graphs in Disease Spread

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## Abstract

**:**

## 1. Introduction

## 2. Basic Definitions

#### 2.1. Graphs

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

#### 2.2. DIL-W${}^{\alpha}$ Ranking

**Definition**

**6**

**.**The degree centrality of ${v}_{i}\in V$ of an edge-weighted graph $(G,w)$, denoted by ${C}_{D}^{w\alpha}\left({v}_{i}\right)$, is defined as

**Definition**

**7**

**.**The importance of an edge ${e}_{ij}\in E$, denoted by ${I}^{\alpha}\left({e}_{ij}\right)$, is defined as

**Definition**

**8**

**.**The contribution that ${v}_{i}\in V$ makes to the importance of the edge ${e}_{ij}$, denoted by ${W}^{\alpha}\left({e}_{ij}\right)$, is defined as

**Definition**

**9**

**.**The importance of a vertex ${v}_{i}\in V$, denoted by ${L}^{\alpha}\left({v}_{i}\right)$, is defined as

## 3. Strategy Protection

**Definition**

**11.**

**Definition**

**12.**

## 4. Simulation of Disease Spread with Protection and Analysis Results

#### 4.1. Data and Methodology

#### 4.2. Simulation of Disease

- The probability (${P}_{I}\left({v}_{i}\right)$) that a susceptible vertex ${v}_{i}$ is infected by one of its neighbors is given by$${P}_{I}\left({v}_{i}\right)=\sum _{{v}_{j}\in {N}_{I}\left({v}_{i}\right)}\rho \Delta t\xb7{w}_{ij},$$
- The probability (${P}_{R}\left({v}_{i}\right)$) that an infected vertex ${v}_{i}$ at time t will recover is given by$${P}_{R}\left({v}_{i}\right)=\delta \Delta t,$$

#### 4.3. Survival Rate

#### 4.4. Scale-Free Network

**Remark**

**1.**

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

Zachary Karate Club Network | |||||||
---|---|---|---|---|---|---|---|

DIL-W${}^{0}$ | DIL-W${}^{0.5}$ | DIL-W${}^{1}$ | Strength | Closeness | Betweenness | Laplacian | |

5% | 0.8391 | 0.8434 | 0.8354 | 0.8406 | 0.8028 | 0.8016 | 0.8353 |

10% | 0.8945 | 0.8912 | 0.8941 | 0.9050 | 0.8210 | 0.8889 | 0.9015 |

15% | 0.9392 | 0.9367 | 0.9359 | 0.9437 | 0.8989 | 0.9333 | 0.9437 |

20% | 0.9510 | 0.9525 | 0.9558 | 0.9568 | 0.9276 | 0.9497 | 0.9551 |

25% | 0.9543 | 0.9579 | 0.9577 | 0.9593 | 0.9548 | 0.9518 | 0.9567 |

30% | 0.9578 | 0.9585 | 0.9572 | 0.9589 | 0.9580 | 0.9540 | 0.9597 |

40% | 0.9591 | 0.9603 | 0.9659 | 0.9677 | 0.9593 | 0.9681 | 0.9606 |

50% | 0.9657 | 0.9655 | 0.9667 | 0.9695 | 0.9603 | 0.9705 | 0.9662 |

Wild Birds Network | |||||||
---|---|---|---|---|---|---|---|

DIL-W${}^{0}$ | DIL-W${}^{0.5}$ | DIL-W${}^{1}$ | Strength | Closeness | Betweenness | Laplacian | |

5% | 0.7355 | 0.7394 | 0.6447 | 0.6188 | 0.6696 | 0.6411 | 0.6212 |

10% | 0.7871 | 0.7942 | 0.7375 | 0.7040 | 0.7326 | 0.6952 | 0.7022 |

15% | 0.8721 | 0.8863 | 0.8760 | 0.8625 | 0.8149 | 0.7966 | 0.8654 |

20% | 0.9011 | 0.9059 | 0.9076 | 0.8943 | 0.8486 | 0.8399 | 0.8980 |

25% | 0.9135 | 0.9155 | 0.9208 | 0.9130 | 0.8826 | 0.8676 | 0.9078 |

30% | 0.9150 | 0.9226 | 0.9261 | 0.9303 | 0.8980 | 0.8941 | 0.9114 |

40% | 0.9361 | 0.9327 | 0.9481 | 0.9540 | 0.9018 | 0.9457 | 0.9272 |

50% | 0.9500 | 0.9626 | 0.9710 | 0.9762 | 0.9046 | 0.9766 | 0.9462 |

Sandy Authors Network | |||||||
---|---|---|---|---|---|---|---|

DIL-W${}^{0}$ | DIL-W${}^{0.5}$ | DIL-W${}^{1}$ | Strength | Closeness | Betweenness | Laplacian | |

5% | 0.9301 | 0.9292 | 0.9296 | 0.8395 | 0.8817 | 0.9312 | 0.8329 |

10% | 0.9671 | 0.9637 | 0.9613 | 0.9535 | 0.9434 | 0.9615 | 0.9382 |

15% | 0.9738 | 0.9727 | 0.9692 | 0.9706 | 0.9513 | 0.9686 | 0.9616 |

20% | 0.9749 | 0.9799 | 0.9760 | 0.9779 | 0.9558 | 0.9764 | 0.9645 |

25% | 0.9771 | 0.9819 | 0.9822 | 0.9798 | 0.9568 | 0.9808 | 0.9773 |

30% | 0.9795 | 0.9837 | 0.9832 | 0.9827 | 0.9700 | 0.9829 | 0.9791 |

40% | 0.9824 | 0.9852 | 0.9848 | 0.9839 | 0.9680 | 0.9853 | 0.9824 |

50% | 0.9825 | 0.9870 | 0.9860 | 0.9860 | 0.9670 | 0.9860 | 0.9837 |

CAG-mat72 Network | |||||||
---|---|---|---|---|---|---|---|

DIL-W${}^{0}$ | DIL-W${}^{0.5}$ | DIL-W${}^{1}$ | Strength | Closeness | Betweenness | Laplacian | |

5% | 0.7214 | 0.7641 | 0.7632 | 0.7734 | 0.7714 | 0.7256 | 0.7779 |

10% | 0.8947 | 0.8886 | 0.8792 | 0.8685 | 0.8451 | 0.8077 | 0.8706 |

15% | 0.9185 | 0.9308 | 0.9330 | 0.9230 | 0.8993 | 0.9085 | 0.9090 |

20% | 0.9323 | 0.9452 | 0.9458 | 0.9446 | 0.9285 | 0.9094 | 0.9302 |

25% | 0.9464 | 0.9562 | 0.9581 | 0.9588 | 0.9444 | 0.9131 | 0.9351 |

30% | 0.9578 | 0.9657 | 0.9672 | 0.9631 | 0.9646 | 0.9312 | 0.9426 |

40% | 0.9710 | 0.9745 | 0.9750 | 0.9719 | 0.9720 | 0.9456 | 0.9679 |

50% | 0.9770 | 0.9809 | 0.9807 | 0.9779 | 0.9781 | 0.9458 | 0.9758 |

Scale-Free Network | |||||||
---|---|---|---|---|---|---|---|

DIL-W${}^{0}$ | DIL-W${}^{0.5}$ | DIL-W${}^{1}$ | Strength | Closeness | Betweenness | Laplacian | |

5% | 0.2235 | 0.2359 | 0.2773 | 0.2845 | 0.1748 | 0.2320 | 0.2312 |

10% | 0.8610 | 0.5965 | 0.6893 | 0.8593 | 0.3901 | 0.6857 | 0.6264 |

15% | 0.9437 | 0.7585 | 0.9528 | 0.9168 | 0.6622 | 0.8734 | 0.9155 |

20% | 0.9763 | 0.9298 | 0.9636 | 0.9609 | 0.7431 | 0.9092 | 0.9174 |

25% | 0.9804 | 0.9442 | 0.9803 | 0.9684 | 0.7635 | 0.9637 | 0.9497 |

30% | 0.9853 | 0.9710 | 0.9852 | 0.9837 | 0.8509 | 0.9778 | 0.9519 |

40% | 0.9882 | 0.9755 | 0.9875 | 0.9857 | 0.9268 | 0.9863 | 0.9671 |

50% | 0.9897 | 0.9845 | 0.9887 | 0.9875 | 0.9359 | 0.9884 | 0.9830 |

Scale-Free Network | |||||||
---|---|---|---|---|---|---|---|

DIL-W${}^{0}$ | DIL-W${}^{0.5}$ | DIL-W${}^{1}$ | Strength | Closeness | Betweenness | Laplacian | |

2 | 0.9841 | 0.9667 | 0.9842 | 0.9838 | 0.9284 | 0.9847 | 0.9800 |

5 | 0.8411 | 0.6032 | 0.8410 | 0.8373 | 0.4783 | 0.8046 | 0.7692 |

10 | 0.1143 | 0.1170 | 0.1311 | 0.1118 | 0.1104 | 0.1193 | 0.1107 |

30 | 0.1032 | 0.1043 | 0.1059 | 0.1037 | 0.1046 | 0.1041 | 0.1061 |

50 | 0.1041 | 0.1045 | 0.1046 | 0.1050 | 0.1053 | 0.1048 | 0.1043 |

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**Figure 1.**The left side shows a graph without protection or infection. On the right side, we see an infected node (red) and a protected node (green) in the same graph.

**Figure 2.**On the

**right**, curve of infected with and without protection on each network. On the

**left**, only curves of infected with protection according to the different rankings.

**Figure 7.**Scale-free network with $N=15$ and $d=3$. In cyan, the first 3 places generated by DIL-W${}^{1}$ ranking. In purple, the first 3 places generated by Strength ranking.

Zachary Karate Club | Wild Bird | Sandy Authors | CAG-mat72 | |
---|---|---|---|---|

$\rho $ | $0.15$ | $0.23$ | $0.12$ | $0.0052$ |

**Table 2.**Survival rate in each network according to the protection ranking with a protection budget of 10% of the network nodes.

Zachary | Wild Birds | Sandy Authors | CAG-mat72 | |
---|---|---|---|---|

DIL-W${}^{0}$ | 0.8946 | 0.7871 | 0.9671 | 0.8947 |

DIL-W${}^{0.5}$ | 0.8912 | 0.7943 | 0.9637 | 0.8887 |

DIL-W${}^{1}$ | 0.8941 | 0.7375 | 0.9613 | 0.8792 |

Strength | 0.9051 | 0.7041 | 0.9536 | 0.8685 |

Closeness | 0.8211 | 0.7327 | 0.9435 | 0.8451 |

Betweenness | 0.8890 | 0.6952 | 0.9616 | 0.8078 |

Laplacian | 0.9016 | 0.7023 | 0.9382 | 0.8707 |

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**MDPI and ACS Style**

Manríquez, R.; Guerrero-Nancuante, C.; Taramasco, C.
Protection Strategy for Edge-Weighted Graphs in Disease Spread. *Appl. Sci.* **2021**, *11*, 5115.
https://doi.org/10.3390/app11115115

**AMA Style**

Manríquez R, Guerrero-Nancuante C, Taramasco C.
Protection Strategy for Edge-Weighted Graphs in Disease Spread. *Applied Sciences*. 2021; 11(11):5115.
https://doi.org/10.3390/app11115115

**Chicago/Turabian Style**

Manríquez, Ronald, Camilo Guerrero-Nancuante, and Carla Taramasco.
2021. "Protection Strategy for Edge-Weighted Graphs in Disease Spread" *Applied Sciences* 11, no. 11: 5115.
https://doi.org/10.3390/app11115115