A quantum-like rumor propagation model captures bimodal spreading dynamics in social networks.

With the proliferation of online social networks, rumors can spread rapidly, amplifying public-opinion risks and threatening social stability. To address this issue, we develop a six-state quantum-like rumor propagation model on a static social network by coupling ternary beliefs with binary behaviors. Unlike traditional discrete-state models, the proposed model uses a density operator to characterize individual states and their temporal evolution, allowing multiple belief-behavior tendencies to coexist. Decoherence models coherence loss induced by environmental noise, while a time-dependent multi-channel relaxation mechanism captures late-stage suppression and drives the dynamics from diffusion dominance to inhibition dominance. We further show that the resulting single-node evolution map is completely positive and trace-preserving, ensuring a valid probabilistic interpretation. Simulations show that forwarding-related states rise rapidly in the early stage and then approach a stable regime dominated by disbelief and non-forwarding as decoherence and dissipation increase. At the belief level, neutral groups differentiate quickly and gradually evolve toward disbelief dominance, while belief and neutral states persist as low-level long tails. At the behavioral level, a bimodal forwarding pattern emerges, and later forwarders decay into a long tail. Parameter sensitivity analysis shows that rumor coupling, coherence, and relaxation jointly shape spreading intensity, peak structure, and long-tail behavior. An illustrative comparison with a real rumor sequence and two classical baseline models, namely the susceptible-infected-recovered (SIR) model and the susceptible-exposed-infected-recovered (SEIR) model, further shows that the proposed model can capture early-stage bimodal spreading and middle-to-late-stage long-tail decay in a representative case while offering an interpretable framework for analyzing rumor propagation and suppression mechanisms.