In the early stages of rumor dissemination, accurately locating the source of transmission is crucial for the management and control of information flow. However, the inherent uncertainty in information dissemination complicates precise source localization. Although incorporating transmission direction can alleviate some of this uncertainty, thereby facilitating source localization, traditional methods still depend on the often inaccurate informed timestamps of observed nodes. To address this limitation, this paper develops a method to infer all rumor sources using a single snapshot of observed nodes at a specific time and the direction of transmission toward these nodes, without requiring prior knowledge of the informed timestamps. First, we conduct a theoretical analysis demonstrating how the network structure can be pruned based on the status and transmission direction of the observers. Subsequently, we propose the Reduce Candidate Source algorithm, which operates within the pruned subgraph to identify the node with the minimum sum of shortest paths to the informed observers as the potential source of propagation. Additionally, we propose the Reduce Candidate Source by Deleting All algorithm for propagation models characterized by high certainty, retaining only the informed observers and those nodes indicating the transmission direction to further narrow the candidate source range. Finally, extensive experiments confirm that our two proposed methods align with the theoretical analysis, effectively identifying the sources of rumor dissemination in the early stages, even under conditions where the propagation model and informed are unknown.
Ma et al. (Thu,) studied this question.