Percolation, SIR Disease Model, and Bass Diffusion
Click any bank on the grid to trigger its failure and watch the cascade.
Sweeps connection density from 0 to 1 and plots largest cluster size, revealing the critical threshold.
Each cell is a "bank" connected to its neighbors with a given probability (connection density). When a bank fails, connected neighbors may also fail if the connection strength exceeds the failure threshold. Below a critical density (~0.5 for a square grid), failures remain local. Above it, a single failure can cascade through the entire system -- a phase transition.
The phase transition is sudden and dramatic: a small change in connectivity can mean the difference between a contained local failure and a system-wide collapse.
The SIR model divides a population into Susceptible (S), Infected (I), and Recovered (R). Each day, infected individuals spread the disease to susceptible neighbors with probability beta, and recover with probability gamma.
R0 = beta/gamma is the basic reproduction number. When R0 > 1, the disease spreads exponentially. When R0 < 1, it dies out. The herd immunity threshold is 1 - 1/R0: vaccinating above this percentage prevents epidemics.
The Bass Diffusion Model describes how new products are adopted. Two forces drive adoption:
p (innovation): External influence -- people adopt from advertising/media independent of others. Drives early adoption by "innovators."
q (imitation): Internal influence -- people adopt because of word-of-mouth from existing adopters. Creates explosive growth in the "early majority" phase.
The formula: f(t) = [p + q*F(t)] * [1 - F(t)], where F(t) is cumulative adoption fraction. This produces the classic S-curve for cumulative adoption and a bell-shaped curve for adoption rate.