有限种群中社会引爆现象的精确标度理论
Exact Scaling Theory of Social Tipping Phenomena in Finite Populations
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中文总结 AI 辅助
本文针对有限种群的社会引爆现象，基于Granovetter阈值模型，结合精确组合公式与大偏差理论，建立了阈值驱动引爆的精确有限大小标度理论，揭示了不同阈值分布下的相变特性。
中文摘要 AI 辅助
Granovetter的阈值模型是社会动员的经典框架，其中集体行动通过级联传播，当个体加入时，一旦运动规模达到其个人阈值。本文以煽动者的初始比例ρ₀作为控制参数，刻画社会引爆点——实现全球动员所需的最小种子。对于大小为N、阈值服从Beta分布(α, β)的有限种群，我们对级联动力学进行了精确分析研究。通过评估渐近活跃比例ρ_∞，我们绘制了热力学相图，将部分级联（ρ₀ ≤ ρ_∞ < 1）与完全动员（ρ_∞ = 1）区分开，揭示了连续和不连续的转变线在临界终点处无缝交汇。对于内部峰值分布（α > 1, β > 1），各区域由混合相变分隔，该相变结合了一阶不连续性与二阶瓶颈奇点。结合精确的有限N组合公式与大偏差理论，我们阐明了有限大小涨落如何平滑这些奇点。对于幂律阈值（α > 1, β = 1），临界标度窗口以N^(-1/3)收缩，而在临界点处，预期的非活跃比例以N^(-1/3)消失。对于内部峰值分布（β > 1），序参量呈双峰分布：实现完全动员的轨迹或停滞在瓶颈ρ*附近。排除完全动员的轨迹后，ρ* - ⟨ρ_∞⟩以N^(-1/4)消失，且标度窗口压缩至N^(-1/2)。这些结果共同建立了阈值驱动引爆现象的精确有限大小标度理论。
英文摘要
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, $ρ_0$, as a control parameter. For a finite population of size $N$ with Beta-distributed thresholds $(α, β)$, we present an exact analytical study of the cascade dynamics. By evaluating the asymptotic active fraction $ρ_\infty$, we map the thermodynamic phase diagram separating partial cascades ($ρ_0 \le ρ_\infty < 1$) from complete mobilization ($ρ_\infty = 1$), revealing continuous and discontinuous transition lines that meet seamlessly at a critical endpoint. For interior-peaked distributions ($α> 1, β> 1$), 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 ($α> 1, β= 1$), 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 ($β> 1$), the order parameter is bimodally distributed: realizations either achieve full mobilization or stall near a bottleneck $ρ^*$. Excluding fully mobilized trajectories, $ρ^* - \langle ρ_\infty \rangle$ 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.
