# Locating the Source of Diffusion in Complex Networks via Gaussian-Based Localization and Deduction
## 3 Experiments and Analysis
### Results on Arbitrary Trees  
We first perform simulations on two types of arbitrary trees, i.e., ER trees and BA trees. These two tree networks are generated by random network (ER) and scale-free network (BA), respectively, which are all connected networks. To generate a BA tree, we firstly make the initial network contain only one isolated node. At time \(t\), add one new node to the network and meanwhile add an edge linking the new node with one existing node according to the preference attachment rule in the BA model. Relatively speaking, it is very difficult to directly use an ER model to generate a connected 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. Thus, a compromise is adopted here: Firstly, we generate a ER network with \(1.5N\) nodes and its average degree is \(1\). If the giant component in the network has \(N\) nodes (through many simulations, we do find the giant component), take the maximum spanning tree as the approximation of ER tree; if not, repeat the first step, generating another ER network, until we find the ER tree. Figure 2 shows the average ranking under different methods with different fraction of observers. In the experiments, the number of nodes in the tree is fixed to be \(100\), and the signal-to-noise ratio of the diffusion delay along edges \(\mu/\sigma\) are chosen from \(\{4,3,2\}\). A lower signal-to-noise ratio implies larger uncertainty or noise on edges. The average ranking of the source for GLAD-naive and GAU is slightly lower than that of the other two methods, which is consistent with the previous discussion, indicating that they are all optimal solutions in arbitrary trees. Since GLAD-modified does not fully exploit the path intersections between the source and each observer, it performs strictly worse than GLAD-naive. The same reason leads to the poor results of TRBS in one aspect. On the other hand, as TRBS does not model the diffusion delay along edges, its performance is not robust to large noise. As can be seen from left to right in the figure, with the increase of the noise, TRBS performs worse and worse compared to other methods. Figure 3 demonstrates the \(\gamma\)%-accuracy of different source localization algorithms on arbitrary trees. Compared with the previous simulations, the results of the four methods under this metric are more distinguishable that GLAD-naive \(>\) GLAD-modified \(>\) GAU \(>\) TRBS. Although the average ranking of GLAD-naive and GAU are nearly equivalent, the \(\gamma\)%-accuracy of the former algorithm is obviously better than that of the latter. This finding reveals one disadvantage of GAU that it cannot precisely locate the diffusion source in trees. In contrast, the proposed GLAD-naive and GLAD-modified estimate the diffusion parameters \(t_{0}\) and \(\sigma\) for each node, which will improve the ability to distinguish between the real source and its neighbors. As the signal-to-noise ratio decreases, the elements of the covariance matrix \(\Lambda_{s}\) in Equation (3) become increasingly significant. As GLAD-modified ignores the non-diagonal elements of \(\Lambda_{s}\), the performance gaps between GLAD-naive and GLAD-modified will also increase. The average distance between the estimated source and the real source is shown in Figure 4. Under this metric, GLAD-naive and GAU show the best performance among all methods, which is similar to the findings shown in Figure 2. More precisely, the performance of GLAD-naive is slightly better than GAU, especially in the trees with larger noise. A strange phenomenon is observed in which  
Figure 3: \(\gamma\)%-accuracy of different source localization algorithms on synthetic trees. Because the number of nodes is 100, this figure corresponds to a case in which the estimated source is also the real source. Figure 2: Average ranking of the real source in different source localization algorithms on synthetic trees.