# Locating Multiple Sources of Contagion in Complex Networks under the SIR Model
## 3 Simulation and Analysis
### Sources Localization in Synthetic Networks  
To test the efficiency of the PCL method, we first carry out some experiments in synthetic networks, that is, the Radom (ER) network [26], the Watts-Strogtz small world(WS) network [27], and the scale-free (BA) network [28]. The ER network and WS network are homogeneous networks, and the BA network is a heterogeneous network. We focus our attention on the influence the network parameter has on sources localization accuracy. The main parameters are the scale of network \(N\), average degree \(<k>\) and the number of sources \(s\). This paper mainly considers the sources localization problem, there is no denying that the number of sources is the most important network parameter. At first, we examine the effects the number of sources has on the performance of sources localization. Figure 4 shows that when the number of sources increases, the sources localization accuracy has a decrease tendency for all the methods, that is,  
Figure 3: Sources localization accuracy with various frequencies of iteration in nine networks. (**a**–**i**) The relationship between _precision_ and \(\beta\) in synthetic networks and real networks. Without loss of generality, we suppose \(\alpha=1\). \(t1\) denotes the number of iterations. PCL, LPSI, DC, JC and UBC. When the number of sources becomes large, multiple sources may be too closed to identify them easily. To find the number of sources accurately is the first problem we need to solve urgently. In a short, in the above three synthetic networks, PCL behaves better than the other four methods in sources localization accuracy. With the increasing of the number of sources, the sources localization accuracy of PCL only declines slightly, reflecting its strong robustness. Figure 4: Sources localization accuracy of different numbers of sources with five methods in synthetic networks. (**a–c**) The relationship between _precision_ and \(\alpha(\beta=-1)\) for different number of sources in ER network. The number of sources is \(s=2,3,5\) respectively, which is similar to Figure 5. The scale of the network is \(N=100\) and the average degree is \(<k>=4\). (**d–f**) The difference is that the experiments are carried out in the WS network. (**g–i**) Similarly, tests are done in the BA network. Besides, all contrast experiments (including subsequent experiments) involve the above five methods, that is, PCL, LPSI, DC, JC and UBC, which is represented by five different colors. Figure 5: _Cont._  
From Figure 6, we find that the average degree has little influence on the sources localization accuracy with almost all methods. The results of the four methods in the ER network distinguish that PCL > DC > JC > UBC > LPSI; meanwhile those in the WS and BA networks distinguish that PCL > JC > DC > UBC > LPSI. All in all, PCL can always solve the sources localization problems, no matter whether the network is sparse or not. In other words, the accuracy of PCL keeps very robust when the number of edges in the network increases or decreases. For different scales of networks, Figure 7 indicates that the sources localization accuracy has a mild fluctuation with the increasing of network scale for all the methods except for DC. In the ER network, the DC method can get a higher accuracy when the scale of network increases. Of course, the  
Figure 5: The relationship between sources localization precision and \(\alpha(\beta=-1)\) with five methods in real networks. (**a**–**f**) represent the multiple sources localization performance in Karate, Lesmis, Adjnoun, Football, Jazz and USAir network respectively. The number of sources is \(s=3\) in all networks. Figure 6: Sources localization accuracy of different average degree with five methods in synthetic networks. (**a**–**c**) Results in ER network. The average degree is \(<k>=4,6,8\) respectively, the scale of network is \(N=100\). The number of sources is \(s=3\), the same as Figure 7. (**d**–**f**) Results in WS network. (**g**–**i**) Results in BA network. accuracy of PCL keeps robust when network size changes.