Synergistic coevolution of opinion formation and message diffusion: Critical phenomena induced by relative timescales

In the real world, public opinion is inevitably shaped by various forms of message diffusion, and vice versa. However, these two dynamics operate with different timescales and the effect of relative timescale in the coevolution dynamics has not been explored. Here, we propose a coupled noisy threshold voter–UAU model that incorporates an adjustable relative timescale between message diffusion and binary decision-making. We find that the system undergoes a first-order phase transition with respect to the message transmission rate, accompanied by hysteresis and bistability, once the relative timescale exceeds its critical value. Although the system exhibits marked non-monotonic and discontinuous changes with the relative timescale in different situations, there are common features: the prevalence of positive opinion exhibits a slight decline as the relative timescale increases within the low positive opinion phase, whereas it shows an upward trend in the high positive opinion phase. Consequently, the forward critical message transmission rate increases with the relative timescale, while the backward one decreases. Notably, the model illustrates how a minority opinion can abruptly overturn the majority once a critical tipping point is crossed. This study advances our understanding of the role of relative timescale in synergistically coupled dynamics and provides potential strategies for public opinion stabilization.