Enhancing Maximum Likelihood Estimation of Infection Source Localization

The studies on the spreading processes over complex networks have increasingly wide applications. Sometimes this is very important to find quickly the origin of spread, e.g. the patient zero in epidemics or the origin of fake news forwarded in social network. The method of maximum likelihood estimation proposed by Pinto et al. (PTV) solves the important case of this problem in which a limited set of nodes act as observers and report times at which the spread has reached them. While the PTV algorithm has been shown to be optimal on trees there are several challenges remaining on general graphs. One important issue is the complexity O(N) where N is the size of the network and α ∈ (3, 4) depending on the network topology and the number of observers. We address this issue with a new approach in which observers with low quality information (i.e., with large spread encounter times) are ignored and potential sources are selected based on the likelihood gradient from high quality observers. Our Gradient Maximum Likelihood Algorithm (GMLA) reduces this complexity to O(Nlog(N)). The other issue we address here is of precision on general graphs. The original PTV approach assumes the information travels via a single, shortest path, which by this assumption is supposed to be the fastest way. We show that such assumption leads to the overestimation of propagation time in networks where multiple potential traversal paths exist. We propose a new method of source estimation based on maximum likelihood principle that takes into account the existence of multiple shortest paths. We test our modifications extensively on both synthetic and real networks and show that they successfully address the aforementioned issues and thus enhancing the PTV method of locating the source of a spread on complex networks.

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