Abstracts: This paper develops a dynamic framework for behavior-based belief diffusion in networked environments, where individuals choose whether to support, reject, or ignore content according to a softmax utility based on evolving private beliefs, plat- form incentives, and alignment with institutional goals. Crucially, belief updates occur through localized social learning, but are filtered by a potentially distorted inference process—mapping observed behavior to inferred beliefs—which evolves recursively over time. We show that such endogenous filtering distortion can accumulate, leading to structural irreversibility in belief dynamics and a collapse of effective incentive reach (EIR). Leveraging tools from dynamic programming and control theory, we derive optimal incentive trajectories that internalize both strategic constraints and interpretability feedback. Our theoretical results characterize the structure of viable incentive paths and identify necessary conditions for long-run belief convergence. Simulations demonstrate that only mechanisms which maintain interpretive coherence—via structural compression, belief anchoring, and recursive repair—can sustain robust control. In contrast, exposure-only or delayed strategies result in irreversible collapse due to epistemic misalignment. Our framework establishes interpretability not as a refinement, but as a structural prerequisite for influence and robust mechanism design in decentralized systems.

Abstracts: The long-term behaviour of complex systems is typically captured by the analysis of non-linear models and their statistical properties. Recent works by ecologists and climate scientists have highlighted that these asymptotic equilibria may, from time to time, be subject to extreme disruptions due to loss of stability.

In the past decade mathematicians have sought to identify and quantify this emergent phenomena using the language of dynamical systems and bifurcation theory. In the first of this two parts talk we will discover how catastrophic events that occur in nature and society can be described in the developing framework of tipping points under restrictive assumptions. We will follow this in Part 2 by introducing a novel area of research that deals with the detection of signs of early-warning that precede such catastrophes.