Exact Scaling Theory of Social Tipping Phenomena in Finite Populations
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 $\rho^*$. Excluding fully mobilized trajectories, $\rho^* - \langle \rho_\infty \rangle$ 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.