Bi-objective vital node identification for outbreak detection and source inference in multi-layer networks
Timely detection of spreading events and accurate inference of their sources are central challenges in multi-layer networks, where heterogeneous topologies and interactions across layers shape diffusion. We formulate these coupled tasks as the Multi-layer outbreak Detection and source Inference (MDI) problem. MDI is a bi-objective vital-node identification problem that selects observer sets of fixed cardinality to minimize detection time and inference cost under limited sensing resources. We propose MOEA/D-TCM, a decomposition-based multi-objective evolutionary framework tailored to node-set optimization in multi-layer networks. It represents each solution as an observer set and combines a set-structure-adaptive evolutionary strategy with stable-state replacement and neighborhood adjustment to coordinate exploration and exploitation. Whereas the baseline methods return a ranked list or a fixed observer set, MOEA/D-TCM returns multiple non-dominated observer sets that represent different trade-offs between detection timeliness and inference cost. Experiments on 128 synthetic and empirical multilayer networks show that MOEA/D-TCM outperforms nine representative baselines on average, with mean improvements of 21.84% on synthetic networks and 28.90% on real-world networks. These results support bi-objective vital-node identification as a useful framework for monitoring and source localization in multi-layer diffusion systems.