# Locating the Source of Diffusion in Complex Networks via Gaussian-Based Localization and Deduction
## 4 Discussion
### Relationship with the TRBS Algorithm  
In previous experiments, the results demonstrate that the performances of GLAD-modified and TRBS are closely correlated. Generally, the performance of GLAD-modified is slightly better than that of TRBS under all metrics in most synthetic networks and real networks. In particular, we notice that the performance of the two methods for Football and Celegans is indistinguishable, which inspires us to expose whether certain potential relationships occur between GLAD-modified and TRBS.  
Figure 10: \(\gamma\)%-accuracy of GLAD-modified on three types of synthetic networks under different observer placement strategies with signal-to-noise ratio \(\mu/\sigma=4\). The number of nodes is 100 in all networks, and the rewiring probability in WS is 0.1.  
In [21], [PERSON] et al. developed the TRBS algorithm to locate the source of a diffusion-like process. TRBS starts the reversed diffusion process from each observer \(o_{k}\) to all nodes in the networks along the reversed direction of links. At each node \(s\), the reversed arrival time \(t_{o_{k}}-\hat{t}(s,o_{k})\) from each observer \(o_{k}\) should be recorded, and a vector \(T_{s}=[t_{o_{1}}-\hat{t}(s,o_{1}),t_{o_{2}}-\hat{t}(s,o_{2}),...,t_{o_{K}}-\hat {t}(s,o_{K})]^{T}\) is then obtained. Afterward, the node \(s^{*}\) with the minimum variance of \(Var(T_{s^{*}})\) is detected as the source. Along with the notations in our paper, the reversed arrival time vector \(T_{s}\) in TRBS is identical to \(w_{s}\) in Equation (5), thus the optimization problem of TRBS can be given by  
\[s^{*}=\underset{s\in V\backslash O}{argmin}\,Var(w_{s}). \tag{19}\]  
Recall the parameter estimation process in GLAD-modified in Equations (15)-(17). For each node \(s\), if extra prior knowledge that all observers \(\{o_{k}\}\) are uniformly distributed around \(s\) is provided, the distance \(d_{sk}\) between \(s\) and each observer \(o_{k}\) can be approximately considered as equal to \(d_{sk}\approx d_{s}\).  
Then, the corresponding MLE for \(t_{0}\) in Equation (15) should be  
\[\hat{t_{0}} =\sum_{k}d_{sk}^{-1}w_{sk}/\sum_{k}d_{sk}^{-1}\approx\sum_{k}d_{s} ^{-1}w_{sk}/\sum_{k}d_{s}^{-1}\approx\overline{w_{s}}. \tag{20}\]  
The auxiliary variable \(\tilde{z}_{s}\) in Equation (16) becomes  
\[\tilde{z}_{s} =\sum_{k}(w_{sk}-\hat{t_{0}})^{2}d_{sk}^{-1}\approx d_{s}^{-1}\sum _{k}(w_{sk}-\overline{w_{s}})^{2}. \tag{21}\]  
Finally, the optimal estimator for the source node in Equation (18) can be given by  
\[\tilde{s} =argmin_{s\in V\backslash O}\{K\log\tilde{z}_{s}+\sum_{k}\log d_ {sk}\}\] \[\approx argmin_{s\in V\backslash O}\{K\log d_{s}^{-1}+K\log\sum_{k }(w_{sk}-\overline{w_{s}})^{2}+\sum_{k}\log d_{s}\}\] \[\approx argmin_{s\in V\backslash O}\{K\log\sum_{k}(w_{sk}- \overline{w_{s}})^{2}\}\] \[\approx argmin_{s\in V\backslash O}\{\frac{1}{K}\sum_{k}(w_{sk}- \overline{w_{s}})^{2}\}\] \[\approx argmin_{s\in V\backslash O}Var(w_{s}). \tag{22}\]  
By adding this prior knowledge, the optimization problem in GLAD-modified is similar to that of TRBS, which explains from the side of why GLAD-modified often performs better than TRBS. Compared with TRBS, the performance of GLAD-modified is more sensitive to the relative locations between the source and observers, which also provides insights on the strange phenomenon in Figure 4 that the average error distance of GLAD-modified is larger than TRBS in BA trees. In BA trees, the distance between two nodes changes considerably, which severely limits the effectiveness of GLAD-modified.  
However, it is worth noting that this prior knowledge is not rigorous. Even if all observers \(\{o_{k}\}\) can be uniformly distributed around one node \(s\), they are almost impossible to be uniformly distributed around another node at the same time. For example, the observers can never be uniformly distributed around the leftmost or rightmost node in a line graph. The prior knowledge we attached here only acts as one possible bridge between GLAD-modified and TRBS, and more work on this issue should be done in the future.