Exact Solution of Granovetter's Threshold Model for a Finite Population

Abstract

The Granovetter threshold model formalizes collective behavior by assuming that individual agents face a binary decision to join a movement, doing so only when the number of already active participants reaches or exceeds an intrinsic, personal threshold. In this work, we derive an exact analytical expression for the probability that a cascade halts with precisely $k$ active agents in a finite population of size $N$ triggered by a single initial instigator, and use this result to obtain the scaling corrections that govern the system near its critical boundaries. By parameterizing individual threshold heterogeneity via a Beta distribution with shape parameters $\alpha$ and $\beta$, we map how these micro-level predispositions aggregate into macro-level collective outcomes. Here, a small $\alpha$ represents a high proportion of low-threshold, highly susceptible agents, while a small $\beta$ marks a significant density of high-threshold, conservative individuals. In the infinite-population limit, a phase transition occurs at the critical parameter $\alpha = 1$, which separates an inactive phase from a regime of widespread mobilization. For a power threshold distribution ($\beta = 1$), the system exhibits a discontinuous, first-order phase transition where the active fraction jumps abruptly from 0 to 1, and the finite-size critical scaling window contracts as $N^{-1/2}$. In stark contrast, when the population features a persistent density of high-threshold agents ($\beta < 1$), the system undergoes an infinite-order phase transition characterized by an exceptionally smooth, continuous onset of collective activity, causing the finite-size critical region to contract at a drastically slower rate proportional to $(\ln N)^{-1}$. These analytical findings establish a mathematical benchmark for finite-size effects in behavioral cascades.