# Locating the Epidemic Source in Complex Networks with Sparse Observers
## 1 Introduction  
Epidemic spread [1; 2] in complex networks is an important topic with significant value to help us better understand the spread law and influence scope of contagion. In fact, epidemic is a serious threat to human safety, and it is a new one, drawing research attention. If leaders can identify the epidemic source in time, they prefer to make the right decision and avoid greater harm. To protect citizens from epidemic harm, there is an urgent need for us to develop efficient methods to locate the epidemic source with higher accuracy. In the past ten years, the source localization problem has attracted many scholars' attention and some good works have been done, but remains still a challenge. [PERSON] et al. [3] first proposed a rumor centrality, as maximum likelihood estimation of single-source, in tree network under SI model. Inspired by [PERSON], [PERSON] et al. [4] studied the source localization problem under the SIR model and presented a Jordan centrality. The node with the smallest Jordan centrality is a source, i.e., the largest distance between source to all infectious nodes remains the least. A similar concept, namely distance centrality, means the sum distance between one node and other all infectious nodes. The source always has the smallest distance centrality. [PERSON] et al. [5] calculated the dynamical age of each node based on the importance of node dynamical. They considered the eigenvalue drop rate of the adjacency matrix as dynamical age when a node was eliminated from this network and the sources are the nodes with highest dynamical age. Based on the SIR model and incomplete node information, [PERSON] et al. [6] proposed an advanced unbiased betweenness algorithm. They used a reverse propagation algorithmto build an extended infection graph and marked off several infection subgraphs where we can identify the source with the highest unbiased betweenness. The above methods consider the location information of all nodes or most nodes in complex networks and the location information and time information of sparse observers can also achieve a satisfactory source localization accuracy. The information source localization method based on observer placement was presented by [PERSON] [7], called the Gaussian-based method. They calculated the likelihood of each node when it is considered to be the source using the data from observers. The larger the likelihood, the higher probability the node became the source with. [PERSON] et al. [8] investigated a universal method, constructing a virtual backward diffusion tree to locate the source, called Time-Reversal Backward Spreading. By calculating the time of each node when observers reach it reversely, they chose the node with smallest variance as source. Furthermore, [PERSON] et al. [9] extended and applied it in temporal network. For multiple source localization, [PERSON] et al. [10] proposed a maximum-minimum strategy based on backward diffusion, which has been further extended by [PERSON] et al. [11] via integer programming. Moreover, [PERSON] et al. [12] presented the equivalent distance concept, changing the disease-spread problem to a plane wave diffusion process. The source is in the middle of plane wave. [PERSON] et al. [13] proposed a new source localization method-based Monte-Carlo simulation under the SIR model, through a lot of simulations. Based on source identification algorithm, [PERSON] et al. [14] presented label propagation, where they set up the initial label, propagated the label, and chose the nodes with the tallest labels as the sources. In this paper, we present the Pascal-based Epidemic Source Localization (PESL) algorithm as a simple and efficient framework for locating the epidemic source and finding the initial time in complex networks with sparse observers. We mainly consider the SIR spread model with time delay and provide a probabilistic method to locate the epidemic source via Bayesian Estimation and Maximum Likelihood Estimation.