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Graph diffusion problems such as the propagation of rumors, computer viruses,or smart grid failures are ubiquitous and societal. Hence it is usually crucialto identify diffusion sources according to the current graph diffusionobservations. Despite its tremendous necessity and significance in practice,source localization, as the inverse problem of graph diffusion, is extremelychallenging as it is ill-posed: different sources may lead to the same graphdiffusion patterns. Different from most traditional source localizationmethods, this paper focuses on a probabilistic manner to account for theuncertainty of different candidate sources. Such endeavors require overcomingchallenges including 1) the uncertainty in graph diffusion source localizationis hard to be quantified; 2) the complex patterns of the graph diffusionsources are difficult to be probabilistically characterized; 3) thegeneralization under any underlying diffusion patterns is hard to be imposed.To solve the above challenges, this paper presents a generic framework: SourceLocalization Variational AutoEncoder (SL-VAE) for locating the diffusionsources under arbitrary diffusion patterns. Particularly, we propose aprobabilistic model that leverages the forward diffusion estimation model alongwith deep generative models to approximate the diffusion source distributionfor quantifying the uncertainty. SL-VAE further utilizes prior knowledge of thesource-observation pairs to characterize the complex patterns of diffusionsources by a learned generative prior. Lastly, a unified objective thatintegrates the forward diffusion estimation model is derived to enforce themodel to generalize under arbitrary diffusion patterns. Extensive experimentsare conducted on 7 real-world datasets to demonstrate the superiority of SL-VAEin reconstructing the diffusion sources by excelling other methods on average20% in AUC score.

Source Localization of Graph Diffusion via Variational Autoencoders for Graph Inverse Problems