# Protection Strategy for Edge-Weighted Graphs in Disease Spread
## 5 Conclusions  
In this paper, we evaluate the effectiveness of the DIL-W\({}^{\alpha}\) ranking in the immunization of nodes that are attacked by an infectious disease that spreads on an edge-weighted graph using a graph-based SIR model. This protection strategy shows better survival rates in \(3\) of the four networks that we have considered, conditioned to the protection budget equal to \(10\%\) of the network nodes. In the case where DIL-W\({}^{\alpha}\) does not perform better than the other strategies, it is still a good alternative due to its lower computational complexity.  
When we modify the protection budget, the DIL-W\({}^{\alpha}\) ranking continues to show itself as one of the \(3\) main immunization strategies with the best survival rate. Even for more than a value of \(\alpha\). Again, when it is not in the top places, with respect to performance, its computational complexity makes DIL-W\({}^{\alpha}\) a good alternative to choose over the others.  
In the case of scale-free networks, the DIL-W\({}^{\alpha}\) ranking maintains the trend of obtaining better survival rates than the other compared rankings when the protection budget is modified. The size of the network does not affect this result, because DIL-W\({}^{\alpha}\) reaches the first place with \(\alpha=0\) or \(\alpha=1\). Something similar happens when modifying the average number of contacts, DIL-W\({}^{1}\) has the best performance up to \(30\%\) of total nodes.  
An interesting and complex task to solve is to determine which value of \(\alpha\) will be chosen in certain networks so that the ranking generated would be the optimal one. Not always the same value makes the best performance. However, when considering this method, there are as many rankings as there are numbers between \(0\) and \(1\).  
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.  
**Author Contributions:** The authors [PERSON], [PERSON] and [PERSON] contributed equally to this work. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This work was supported and funded by Agencia Nacional de Investigacion y Desarrollo (ANID), COVID0739 project.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The source code of our analysis and network files can be found at [[https://github.com/[PERSON]/Protection-DIL-W.git](https://github.com/[PERSON]/Protection-DIL-W.git)]([https://github.com/[PERSON]/Protection-DIL-W.git](https://github.com/[PERSON]/Protection-DIL-W.git)) from 5 May 2021.  
**Conflicts of Interest:** The authors declare that they have no conflict of interest. The funders had no role in the design of the study; in the collection, analysis, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.  
**Appendix A**  
Survival rates (\(\sigma\)), according to different protection budgets in each test network.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \multicolumn{6}{c}{**Scale-Free Network**} \\ \cline{2-9}  & **DIL-W\({}^{\mathbf{0}}\)** & **DIL-W\({}^{\mathbf{0},\mathbf{5}}\)** & **DIL-W\({}^{\mathbf{1}}\)** & **Strength** & **Closeness** & **Betweenness** & **Laplacian** \\ \hline
**5\%** & 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 \\ \hline \hline \end{tabular}
\end{table}
Table 6: \(\sigma\) according to different values of \(d\) in a Scale-free Network.