Trust as a Stochastic Phase on Hierarchical Networks: Social Learning, Degenerate Diffusion, and Noise-Induced Bistability

Empirical debates about a "crisis of trust'' highlight long-lived pockets of high trust and deep distrust in institutions, as well as abrupt, shock-induced shifts between them. We propose a probabilistic model in which such phenomena emerge endogenously from social learning on hierarchical networks. Starting from a discrete model on a directed acyclic graph, where each agent makes a binary adoption decision about a single assertion, we derive an effective influence kernel that maps individual priors to stationary adoption probabilities. A continuum limit along hierarchical depth yields a degenerate, non-conservative logistic--diffusion equation for the adoption probability u(x,t), in which diffusion is modulated by (1-u) and increases the integral of u rather than preserving it. To account for micro-level uncertainty, we perturb this dynamics by multiplicative Stratonovich noise with amplitude proportional to u(1-u), strongest in internally polarised layers and vanishing at consensus. At the level of a single depth layer, Stratonovich–Itô conversion and Fokker–Planck analysis show that the noise induces an effective double-well potential with two robust stochastic phases, u ≈ 0 and u ≈1, corresponding to persistent distrust and trust. Coupled along depth, this local bistability and degenerate diffusion generate extended domains of trust and distrust separated by fronts, as well as rare, Kramers-type transitions between them. We also formulate the associated stochastic partial differential equation in Martin–Siggia–Rose–Janssen–De Dominicis form, providing a field-theoretic basis for future large-deviation and data-informed analyses of trust landscapes in hierarchical societies.