A Mathematical Framework for Trust Dynamics in Small-Scale Risk-Sharing Communities

This paper develops a rigorous mathematical framework for analyzing trust dynamics and statistical properties in small-scale risk-sharing communities. We establish that small pools with interdependent risks exhibit fundamentally different mathematical properties than large insurance systems, with volatility exceeding stability thresholds by a factor of √(N/Nc) and correlation structures reducing effective pool size by up to 89%. We formalize trust as a mathematically tractable variable with threshold stability properties, proving the existence of critical values TRcritical ∈ [0.65, 0.75] that create bifurcation points in system behavior. Our mathematical analysis demonstrates that network density directly determines trust propagation speed according to precise mathematical relationships. We prove that trust response exhibits asymmetric properties, with negative experiences having 1.5-2.5 times stronger impact than positive experiences of equal magnitude, creating hysteresis effects in system stability. By developing differential equations governing trust evolution and applying network diffusion models, we establish exact conditions for system stability and characterize phase transitions under parameter variation. The mathematical framework enables precise quantification of correlation penalties, network effects, and trust thresholds with applications to community-based risk-sharing systems where conventional statistical approaches fail. Our results transform qualitative concepts of trust and social capital into quantifiable mathematical variables with specific dynamics and stability properties.