Learning from a Teacher: Andrew the First-Called and Thomas the Doubter

We present a nonlinear reaction–diffusion model of belief propagation in hierarchical networks, incorporating logistic self-amplification and degenerating diffusion. Inspired by contrasting apostolic archetypes, Andrew the First-Called and Thomas the Doubter, the model captures both immediate social adoption and delayed, self-sustained belief. We show that when influence diffusivity is low, FKPP-type pulled waves fail to reach the network periphery, whereas degenerating diffusion (DD), despite being slower, enables spatially correlated and temporally synchronized activation in distal layers. Analytical traveling-wave solutions reveal that belief fronts in the DD regime retain memory of initial conditions and saturate without direct influence from the source. Numerical simulations confirm that belief saturation propagates through indirect coordination rather than sequential diffusion, illuminating the non-intuitive dynamics of large-scale influence. Applications to digital misinformation, political persuasion, and collective decision-making are discussed. Mathematics Subject Classification (2020) 35K57 · 91D30 · 60J28