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

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 α and β, we map how these micro-level predispositions aggregate into macro-level collective outcomes. Here, a small α represents a high proportion of low-threshold, highly susceptible agents, while a small β marks a significant density of high-threshold, conservative individuals. In the infinite-population limit, a phase transition occurs at the critical parameter α=1, which separates an inactive phase from a regime of widespread mobilization. For a power threshold distribution (β=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 (β<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 (lnN)−1. These analytical findings establish a mathematical benchmark for finite-size effects in behavioral cascades.