Recently, as arising from online social network services such as Facebook and Twitter, people are more actively using social networks to exchange their new information. In this consideration, finding the information source becomes one of the indispensable and useful tasks in detecting a malicious agent and an influential person in the networks. A seminal work by Shah and Zaman in 2010 showed that the detection probability cannot be beyond 31% even for regular trees if time goes to infinity. From the study, extensive researches have been done for this problem, whose major interests lie in constructing an efficient estimator and providing theoretical analysis on its detection performance. However, most of the works assumed the homogeneous diffusion rate of the information, where the diffusion rate does not change at all times over the network. In practice, it is reported that information has a lifetime and it becomes less attractive over time. In this paper, we study the problem of detecting the information source when the diffusion rate decreases by the distance from the source in the network. As a result, we obtain analytical detection performance of Maximum Likelihood Estimator (MLE) and validate our theoretical findings over the regular tree, random and real-world networks.
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