Diffusion Source Inference for Large-Scale Complex Networks Based on Network Percolation

This article studies the diffusion-source-inference (DSI) problem, whose solution plays an important role in real-world scenarios such as combating misinformation and controlling diffusions of information or disease. The main task of the DSI problem is to optimize an estimator, such that the real source can be more precisely targeted. In this article, we assume that the state of a number of nodes, called observer set, in a network could be investigated if necessary, and study what configuration of those nodes could facilitate a better solution for the DSI problem. In particular, we find that the conventional error distance metric cannot precisely evaluate the effectiveness of varied DSI approaches in heterogeneous networks, and thus propose a novel and more general measurement, the candidate set, that is formulated to contain the diffusion source for sure. We propose the percolation-based evolutionary framework (PrEF) to optimize the observer set such that the candidate set can be minimized. Hence, one could further conduct more intensive investigation or search on only a few nodes to target the source. To achieve that, we first theoretically show that the size of the candidate set is bounded by the size of the largest component cover, and demonstrate that there are some similarities between the DSI problem and the network immunization problem. We find that, given the associated direction information of the diffusion is known on observers, the minimization of the candidate set is equivalent to the minimization of the order parameter if we view the observer set as the removal node set. Hence, PrEF is developed based on the network percolation and evolutionary algorithm. The effectiveness of the proposed method is validated on both synthetic and empirical networks in regard to varied circumstances. Our results show that the developed approach could achieve much smaller candidate sets compared to the state of the art in almost all cases, e.g., it is better in 26 out of 27 empirical networks and 155 out of 162 cases regarding the critical threshold. Meanwhile, our approach is also more stable, i.e., it works well irrespective of varied infection probabilities, diffusion models, and underlying networks. More importantly, we provide a framework for the analysis of the DSI problem in large-scale networks.