# Locating the Source of Diffusion in Complex Networks via Gaussian-Based Localization and Deduction
## 1 Introduction  
Diffusion dynamics taking places on complex networks has been a long-term hot topic with importance value to help us better understand many ubiquitous natural phenomena and social behaviors. In general, many different diffusion processes that occur in daily life are harmful and may result in huge losses to society. Prototypical examples include outbreaks of epidemics among humans [1], the spreading of rumor over social networks [2], cascading failures of power grids around country wide [3], etc. If decision-makers, such as managers and politicians, can identify the diffusion sources as early as possible, they are more likely to make the right decision in time and avoid economic losses and social panic due to the associated tragedies. In order to better resist the potential terrible consequences induced by those diffusion processes, there is a great need for us to develop efficient strategies to locate the source of diffusion and devise control methods as early as possible. Despite great effort that has been made in this field, the problem of identifying the zero patient is still challenging work. The main difficulty lies in two aspects, i.e., the incomplete information and the stochastic nature. In one aspect, the exact number of sources and the zero time at which diffusion first occurred are usually unknown to us. In addition, even if we can get full access to such information, theinevitable randomness of the diffusion process still weakens our ability to accurately locate the source. For example, in the epidemic spreading process, such as the susceptible-infected process, different initial conditions may lead to the same observations, making it extremely hard for us to identify the real source. Models and methods for source localization have been studied in a number of literature works. [PERSON] and [PERSON] [4] were two pioneers who first provided a systematic study of the problem about how to detect the origins of a computer virus in a network. They modeled the process of virus spreading within a network via a susceptible-infected (SI) model and presented the Rumor centrality as the maximum likelihood estimator for a class of graphs. Inspired by their work, [PERSON] and [PERSON] [5] developed a sample-path-based approach named Jordan centrality to detect the information source in a network under susceptible-infected-recovered (SIR) dynamics. Later, two Bayesian inference solutions, namely dynamic message-passing [6] and belief propagation [7], were successively designed to measure the probability distribution for the observations and identify the zero patient of the networks. [PERSON] and [PERSON] [8] studied global disease dynamics and proposed a new index named effective distance to locate the origin of complex spatiotemporal patterns. [PERSON] and [PERSON] [9] developed the Short-Fat Tree algorithm to locate the diffusion source with an independent cascade model. [PERSON] et al. [10] developed a framework for optimal source localization in arbitrarily weighted networks with arbitrary distributions of sources based on controllability theory and compressive sensing. Several other methods for source localization [11; 12; 13; 14; 15; 16] and observation selection [17; 18] have also been proposed based on different considerations. Those excellent works are mainly based on the knowledge of the diffusion status for a proportion of nodes at specific snapshots. However, the time information or the timestamp when diffusion first arrives at some nodes has not been fully investigated by many scholars. In [19], [PERSON] et al. presented the Gaussian heuristic as a maximum likelihood estimator of source localization problem on arbitrary trees, but its performance could not be guaranteed in general graphs. [PERSON] et al. [20] formulated the source localization problem as a ranking problem on graphs and proposed two ranking algorithms, namely cost-based ranking and tree-based ranking, to locate the diffusion source in networks. Recently, [PERSON] et al.