# Locating Multiple Sources of Contagion in Complex Networks under the SIR Model
## 5 Conclusions and Discussion  
In this paper, we study multiple sources of the contagion localization problem under the SIR model. Given the snapshot of a network, we propose a fast and more accurate multiple sources localization method, namely Potential Concentration Label. What matters in this method is to find the nodes with the maximal value of the potential concentration label as the sources. Firstly, we assign the initial concentration label to each node according to its infection state; next, it begins the label propagation process, where the label of one node is determined by its neighbors' and its initial own, through two iterations; finally, we choose the nodes with the maximal value of the potential concentration label as the contagion sources. The experiments demonstrate that when the number of sources increases, the sources localization accuracy of our method decreases gradually. However, it keeps very robust as the average degree and network scale make a change. Compared to other benchmark methods, this method has a low time complexity and higher sources localization accuracy in synthetic networks and real networks. What is more, the anti-noise ability of our method is strong enough, which shows its effectiveness. Although our method provide a new reference for the problem of multiple sources localization in complex networks, much work still needs to be done. The issue of sources localization we proposed is \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Network** & **Noise** & **PCL** & **LPSI** & **DC** & **JC** & **UBC** \\ \hline \multirow{3}{*}{ER} & 0.05 & 56.83 & 11.68 & 30.07 & 31.07 & 12.43 \\  & 0.10 & 54.73 & 12.34 & 26.05 & 31.09 & 12.70 \\  & 0.20 & 51.61 & 11.50 & 19.30 & 30.92 & 12.58 \\ \hline \multirow{3}{*}{WS} & 0.05 & 52.81 & 10.24 & 20.47 & 31.24 & 13.09 \\  & 0.10 & 50.65 & 10.33 & 18.52 & 31.01 & 13.12 \\  & 0.20 & 45.19 & 10.36 & 15.22 & 30.96 & 13.12 \\ \hline \multirow{3}{*}{BA} & 0.05 & 56.17 & 10.60 & 21.16 & 30.78 & 12.31 \\  & 0.10 & 54.13 & 10.54 & 18.64 & 30.83 & 12.91 \\  & 0.20 & 49.78 & 10.45 & 15.97 & 30.84 & 12.28 \\ \hline \multirow{3}{*}{Karate} & 0.05 & 49.45 & 11.50 & 28.12 & 33.34 & 16.69 \\  & 0.10 & 48.01 & 11.38 & 28.15 & 33.39 & 17.44 \\  & 0.20 & 43.87 & 11.74 & 25.66 & 32.82 & 15.73 \\ \hline \multirow{3}{*}{Lesmis} & 0.05 & 36.91 & 10.33 & 20.50 & 31.43 & 13.87 \\  & 0.10 & 36.61 & 10.51 & 19.69 & 31.41 & 13.99 \\  & 0.20 & 33.85 & 10.69 & 17.23 & 31.17 & 13.90 \\ \hline \multirow{3}{*}{Adjnoun} & 0.05 & 54.46 & 10.48 & 26.44 & 30.94 & 12.34 \\  & 0.10 & 52.12 & 10.45 & 21.46 & 30.83 & 12.01 \\  & 0.20 & 47.89 & 10.45 & 17.74 & 30.81 & 12.13 \\ \hline \multirow{3}{*}{Football} & 0.05 & 55.75 & 10.24 & 34.24 & 30.73 & 12.34 \\  & 0.10 & 52.60 & 10.24 & 31.12 & 30.86 & 12.46 \\  & 0.20 & 49.69 & 10.27 & 27.40 & 30.95 & 12.22 \\ \hline \multirow{3}{*}{Jazz} & 0.05 & 40.27 & 10.15 & 21.61 & 30.63 & 11.17 \\  & 0.10 & 40.84 & 10.18 & 18.43 & 30.36 & 11.08 \\  & 0.20 & 36.43 & 10.12 & 16.21 & 30.44 & 11.38 \\ \hline \multirow{3}{*}{**USAir**} & 0.05 & 40.09 & 10.06 & 14.83 & 30.23 & 11.05 \\  & 0.10 & 39.52 & 10.18 & 14.26 & 30.22 & 10.66 \\ \cline{1-1}  & 0.20 & 36.37 & 10.24 & 13.27 & 30.30 & 10.75 \\ \hline \end{tabular}
\end{table} 
Table 3: The performance of sources localization accuracy with PCL vs. LPSI vs. DC vs. JC vs. UBC in synthetic networks and real networks under noise of various intensities. The _precision_ is the mean value when \(\alpha=0.1\):0.1:1. The number of sources is \(s=3\) in all networks, \(<k>=4\), \(N=100\) in the ER, WS and BA networks, the connect probability is 0.1 in the WS network. Moreover, the strategy of dealing with unknown infection information chooses Rand-inf. based on an undirected network, while this method may extend to the directed network. Moreover, the network topology we consider in this paper is static and known to us. In fact, the topology in the real world remains dynamic as time goes by. This makes us improve our model so as to solve the sources localization problem in a temporal network [29; 30].