# Locating the Source of Diffusion in Complex Networks via Gaussian-Based Localization and Deduction
## 5 Conclusions  
A fundamental question in modern systems that undergo a diffusion-like process such as information propagation and epidemic spreading is where the origin is located. In this paper, our main purpose was to devise a method that can locate the diffusion source in complex networks with limited observations. To do so, we modeled the diffusion delay along each edge as a random variable that follows a Gaussian distribution \(\mathcal{N}(\mu,\sigma)\), and derived the corresponding likelihood function for the informed timestamps of observers. Thus, the source localization problem could be transformed to a parameter estimation and likelihood maximization problem. Since our optimizations were based on Gaussian assumption, and the diffusion parameters were deduced from observers, we named our method Gaussian-based localization and deduction (GLAD). We obtained the GLAD-naive as the optimal solution on arbitrary trees, and further derived the GLAD-modified as an approximate solution on general graphs. We compared the algorithm performances with two benchmark methods in terms of three types of abilities: average localization accuracy, high-precision localization, and approximate area localization. Extensive experiments on synthetic trees showed that GLAD-naive was superior to other methods, which was consistent with our conclusions because GLAD-naive was indeed an optimal solution in this case. The results on general graphs demonstrated that GLAD-modified performed the best among all methods. Furthermore, we studied the effects of different observer placement strategies for GLAD-modified, and the underlying relationship between GLAD-modified and TRBS. The main contribution of this work is that we employ the parameter estimation process into the optimization problem of source localization, which enables us to build a complex model with more parameters and achieve better results. Compared with one well-known source localization algorithm named GAU, our framework applies the same Gaussian assumption on diffusion delays and utilizes the same known information, while it outperforms GAU significantly on both arbitrary trees and general graphs. In addition, the computational complexity of GLAD-modified is just linear with the scale of networks. The extremely low cost will enable the wide application of our method in large-scale networks for source localization within a reasonable time. Although we only present the optimization process for undirected networks, the corresponding formula for directed networks can be easily achieved via slight modifications. Upon we finish our work, we notice a recent paper proposed by [PERSON] et al. [35] that also combines parameter estimation with source localization. Their approach is more complicated than ours since they treat more diffusion parameters as unknown and perform the optimization over a parameterized family of Gromov matrices. However, high computational cost restricts its further usage on large-scale networks. Although our proposed algorithm provides a new perspective for the problem of source localization in complex networks, considerable work remains to be done. Our proposed method relied strongly on diffusion delay that all diffusion delays \(\{\theta_{uv}\}\) along edges are i.i.d Gaussian variables \(\theta_{uv}\stackrel{{ i.i.d. }}{{\sim}}\mathcal{N}(\mu,\sigma^{2})\), which indicates to us to consider other delay distributions. The most intuitive task is how to identify multiple diffusion sources in networks. Compared with the single source localization problem, the case of multiple source localization is far more complicated because it is a combinatorial optimization problem. Several studies [36; 37; 38] have addressed the problem via a two-step strategy that first obtains a set of source candidate clusters and then applies single source localization algorithms to identify the source in each cluster. However, a general framework for simultaneously determining the number of sources and their locations in a large complex network is still lacking.