# Locating the Epidemic Source in Complex Networks with Sparse Observers
## 3 Experiments and Analysis
### Source Localization Accuracy in Synthetic Networks  
In this section, we mainly consider the source localization performance of the algorithm in two synthetic tree networks, namely ER tree (Erdos-Renyi tree, ERT) and BA tree (Barabasi-Albert tree, BAT). These two networks are generated by random network (ER) and scale-free network (BA) respectively, which are all connected networks. To generate BAT, we first make the initial network contain only one isolated node, then add one node at a time and add an edge according to the preference attachment rule in the BA model. Relatively speaking, it is very difficult to directly use ER model to generate a connected tree network with \(N\) nodes. Although \(n-1\) edges can be randomly generated among the \(N\) nodes, there is no guarantee that the generated network is a connected network. Therefore, a compromise is adopted here: First, generate a ER network with \(1.5N\) node and \(1\) average degree; then, if the giant component in the network has N nodes (through the vast simulations, we do find the giant component), take the maximum spanning tree as the approximation of ERT. The basic topological properties of these two tree networks are shown in Table 2.  
Figure 2 shows the source localization performance under various infection probabilities with five various algorithms in ERT and BAT. In ERT, the epidemic source localization accuracy of PESL algorithm is consistently better than other comparison algorithms. In BAT, PESL algorithm can still achieve good performance of source localization, which reflects the advantages of PESL algorithm on the source localization in ERT and BAT. Intuitively speaking, the larger the infection probability \(\beta\) under the SIR model is, the earlier the infected time of the nodes near the epidemic source will be, so the infected time of the observers will be closer to the initial time and a better source localization algorithm should have higher accuracy. It can be found from the results in the figure that the localization accuracy \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Type** & **Name** & \(N\) & \(M\) & \(\langle k\rangle\) & \(k_{max}\) & \(H\) & \(\langle d\rangle\) \\ \hline Erdos-Renyi Tree & ERT & 100 & 99 & 1.980 & 5 & 1.240 & 9.855 \\ Barabasi-Albert Tree & BAT & 100 & 99 & 1.980 & 22 & 2.372 & 4.945 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Basic topological properties of two tree networks. Where \(N,M\) represent the number of nodes and edges respectively, \(\langle k\rangle\) and \(k_{max}\) denote the average degree and maximum degree respectively, \(H\) means index of degree heterogeneity in network and \(\langle d\rangle\) is the mean shortest path in network. of GAU, TRBS, and PESL increases with the increasing of infection probability, while DC and JC, which only use the location information of nodes, do not show such a trend. Figure 3: The relationship between epidemic source localization accuracy and reporting probability in tree networks. (**a–c**) represent the relationship between average error distance and reporting probability in ERT under the different infection probability of nodes \(\beta=0.2,0.5,0.8\) respectively. (**d–f**) represent the same results in BAT. The abscissa represents the reporting probability \(\delta\) of nodes. Figure 2: The relationship between epidemic source localization accuracy and infection probability in tree networks. (**a–c**) represent the relationship between average error distance and infection probability in ERT under the different reporting probability of nodes \(\delta=0.3,0.6,0.9\) respectively. (**d–f**) represent the same results in BAT. In the figure, the abscissa represents the infection probability \(\beta\) of SIR model, the ordinate denotes the distance between the real source and the source found by different source localization algorithms.