Bootstrap Percolation

Bootstrap percolation is a cellular automaton model of threshold activation on networks, where a node becomes active only when at least r of its neighbors are already active. Originally introduced as a model of ferromagnetism, it has become a central tool for understanding how local activation rules produce global cascades — from social contagion and viral marketing to the k-core decomposition of networks and the failure of financial institutions under correlated shocks. The dynamics are deceptively simple but analytically treacherous: the final active set depends sensitively on the initial seed configuration and the network topology, and the phase transition between complete inactivation and full activation can be first-order, continuous, or hybrid depending on the rule and the graph. Bootstrap percolation is the prototype for all threshold models, and its behavior exposes the lie that local rules with global effects are ever simple to predict.