About Granovetter's Threshold Model
In 1978, Mark Granovetter proposed the threshold model of collective behavior: each individual i has a personal threshold phi_i, and takes action only when the proportion of already-active neighbors exceeds phi_i. Unlike epidemic models where infection probability is uniform, the threshold model emphasizes heterogeneity -- even small differences in threshold distribution can lead to dramatically different outcomes, from complete stagnation to full cascade.
The threshold distribution critically shapes outcomes. With a uniform distribution, propagation is gradual and predictable. A bimodal distribution creates a clear tipping point -- once enough early adopters activate, a sudden cascade sweeps through the low-threshold population. Power-law distributed thresholds mean most people are easily influenced while a few resist, leading to rapid adoption through the majority.
Network topology determines how influence flows. Random (ER) networks produce predictable, averaged propagation. Small-world (WS) networks feature shortcuts that allow contagion to jump between distant clusters. Scale-free (BA) networks have hub nodes -- super-spreaders that can activate large portions of the network in a single step, but also create vulnerability: removing hubs can stop cascades entirely.
Select a network type, adjust node count and average degree, choose a threshold distribution, and set the initial activation ratio. Click Reset to create a new network, then use Play/Step to watch the contagion unfold. The adoption curve tracks what fraction of the network is active at each time step. Try the 5 presets to see how different combinations lead to all-or-nothing outcomes.