# Locating the Epidemic Source in Complex Networks with Sparse Observers
## 4 Conclusions  
Source localization problem is one of the most important topics in complex networks, including information source estimation, rumor source identification and epidemic source localization. In particular, locating the epidemic source in complex network is a classical application, helping us to deal with epidemic problems in a way. In this paper, the main purpose was to propose a method that can locate the epidemic source in complex networks with sparse observers. First, we did some derivation about the probability distribution of infected time of observers. In consideration of the randomness of the spread process, we modeled infected time using probability and then found that the infected time of two adjacent nodes in the network is independent and obeys the geometric distribution of the same parameter. Thus, the infected time from the source to any observer obeys the Pascal distribution. Next, we proposed an epidemic source localization algorithm based on Pascal distribution. We found that when there is no prior knowledge with epidemic source and observers; the posterior probability of epidemic source was equal to the product of infected time of each observer, so that it located the epidemic source through maximizing the posterior probability. After analysis of the initial time range, the algorithm only needed to optimize and solve in a very small search space, which greatly improved the computational efficiency.  
To verify the performance of the algorithm, the epidemic source localization experiments were carried out in the synthetic networks and empirical networks and we chose average error distance as the evaluation index. The results showed that the epidemic source localization accuracy of our algorithm in most cases was better than other several comparison algorithms; meanwhile, when the randomness increased, the performance of our algorithm behaved better. The randomness mainly reflected which node was infected by which node through which path. The results of sensitivity analysis showed that the source localization performance of this algorithm was closely related to the parameters in the SIR model. However, obtaining spread parameters from real data has always been a challenge, especially for a new epidemic, whose spread parameters could only be inferred after a large outbreak in a population.  
Although our proposed method provides a new reference for source localization problem in complex networks, more considerable works remain to be done. How to identify the multiple sources in complex networks now is a hot topic and useful in practical applications. Compared with the single-source localization problem, the case of multiple sources localization becomes more difficult because it needs to locate the sources at the same time in most cases. However, there are some works [6; 28; 29] dealing with multi-source localization via two-step strategies that first divide the network into several communities and then apply the single-source localization method to locate sources in each community. In addition, the network topology we consider in this paper is static and just single layer. In fact, the topology in the real world remains dynamic as time goes by or has multiple layers. As we know, few theoretical and practical works have proposed source localization in multi-layer networks [30] and temporal networks [31].  
Conceptualization, C.Z., X.L. and D.Y.; methodology, X.L. and X.W.; software, X.L. and X.W.; validation, X.W. and X.L.; formal analysis, X.W.; investigation, X.Z.; resources, X.W.; data curation, X.W.; writing--original draft preparation, X.L.; writing--review and editing, X.Z.; visualization, X.L. and X.W.; supervision, X.W. and X.L.; project administration, C.Z.; funding acquisition, C.Z. This paper was prepared the contributions of all authors. All authors have read and approved the final manuscript.  
This work is supported by the National Key R&D Program of China (Grant No. 2017 YCF1200301).  
We thank [PERSON] for valuable discussions.  
The authors declare no conflict of interest.