Exact Scaling Theory of Social Tipping Phenomena in Finite Populations

Abstract

Granovetter's threshold model provides a classical framework for social mobilization, where collective action spreads through cascades as individuals join once movement size reaches their personal threshold. Here, we characterize social tipping points-the minimum seed required for global mobilization-using the initial fraction of instigators, ${\rho }_0$, as a control parameter. For a finite population of size $N$ with Beta-distributed thresholds $(\alpha ,\beta )$, we present an exact analytical study of the cascade dynamics. By evaluating the asymptotic active fraction ${\rho }_{\infty }$, we map the thermodynamic phase diagram separating partial cascades (${\rho }_0\le {\rho }_{\infty }<1$) from complete mobilization (${\rho }_{\infty }=1$), revealing continuous and discontinuous transition lines that meet seamlessly at a critical endpoint. For interior-peaked distributions ($\alpha >1,\beta >1$), the regimes are separated by a hybrid phase transition combining a first-order discontinuity with second-order bottleneck singularities. Combining exact finite-$N$ combinatorial formulations with large-deviation theory, we establish how finite-size fluctuations smooth these singularities. For power-law thresholds ($\alpha >1,\beta =1$), the critical scaling window shrinks as $N^{-1/3}$, while the expected inactive fraction vanishes as $N^{-1/3}$ at criticality. For interior-peaked distributions ($\beta >1$), the order parameter is bimodally distributed: realizations either achieve full mobilization or stall near a bottleneck ${\rho }^{\ast }$. Excluding fully mobilized trajectories, ${\rho }^{\ast }-⟨{\rho }_{\infty }⟩$ vanishes as $N^{-1/4}$ and the scaling window compresses to $N^{-1/2}$. Together, these results establish an exact finite-size scaling theory for threshold-driven tipping phenomena.