Mathematical Theory of Social Conformity I: Belief Dynamics, Propaganda Limits, and Learning Times in Networked Societies

This paper develops a novel probabilistic theory of belief formation in social networks, departing from classical opinion dynamics models in both interpretation and structure. Rather than treating agent states as abstract scalar opinions, we model them as belief-adoption probabilities with clear decision-theoretic meaning. Our approach replaces iterative update rules with a fixed-point formulation that reflects rapid local convergence within social neighborhoods, followed by slower global diffusion. We derive a matrix logistic equation describing uncorrelated belief propagation and analyze its solutions in terms of mean learning time (MLT), enabling us to distinguish between fast local consensus and structurally delayed global agreement. In contrast to memory-driven models, where convergence is slow and unbounded, uncorrelated influence produces finite, quantifiable belief shifts. Our results yield closed-form theorems on propaganda efficiency, saturation depth in hierarchical trees, and structural limits of ideological manipulation. By combining probabilistic semantics, nonlinear dynamics, and network topology, this framework provides a rigorous and expressive model for understanding belief diffusion, opinion cascades, and the temporal structure of social conformity under modern influence regimes.