Granovetter’s Threshold Model

A riot can begin with one person who persuades nobody. In Mark Granovetter’s 1978 model, the crowd does not need a leader or a shared appetite for violence. It needs a sequence: one person willing to join alone, another who joins after seeing one participant, a third who waits for two, and so on.

How the cascade works

Each person has a threshold, θᵢ: the number of participants required before joining becomes worthwhile. If nₜ people are active at step t, the next count is:

nₜ₊₁ = F(nₜ)

Here, F(n) is the number of people whose thresholds are at most n. The process stops when F(n) = n.

Granovetter’s example uses 100 people with thresholds 0, 1, 2, ... 99. The threshold-zero person joins first. That activates the threshold-one person, who activates the threshold-two person, until all 100 participate.

Change only one rung and the result collapses. A crowd with thresholds 0, 2, 2, 3, ... 99 stalls at one participant. Its average threshold is 49.51 rather than 49.50, yet its collective outcome changes from 100 participants to one.

The distribution matters more than the average. Aggregate behaviour can therefore reveal very little about average private preference.

What the model explains

Granovetter applied the mechanism to riots, strikes, migration, voting, rumours, and innovation. The same shape appears in concept network effects: adoption becomes more attractive after other people adopt.

Duncan Watts moved the rule onto random networks in 2002. People now observe neighbours rather than an entire population. His model found a “cascade window”: networks can be too sparse to carry activation or so densely connected that each person requires too many active neighbours. Between those regimes, a small seed can reach much of the network.

What’s contested

A threshold is difficult to measure before the event. Estimating it from observed participation risks explaining the cascade by assigning each person the threshold needed to produce that cascade.

The original model also treats action as binary and thresholds as stable. Real crowds contain friendship ties, police responses, repeated decisions, false signals, and people who leave. Network structure, modelled in concept preferential attachment, may matter as much as the threshold distribution.

One qualification matters: the deterministic model still needs a threshold-zero participant or an external seed. It removes the need for an organiser, not the need for a first action.

Why this crosses realms

A spiking neuron also waits until accumulated input crosses a threshold. In concept neuromorphic computing, millions of local firing rules can produce system-level activity without a central clock. The parallel is structural, not biological: both systems convert small differences in threshold placement into different collective states.

This also complicates concept power law. A large cascade need not come from a proportionally large cause; it can come from a small cause arriving at a susceptible threshold distribution.

An open question

If thresholds change while people watch one another, can any model separate a pre-existing disposition from a preference created by the cascade itself?

Key Sources

Mark Granovetter, “Threshold Models of Collective Behavior,” American Journal of Sociology 83(6), 1978, pp. 1420–1443. DOI: 10.1086/226707
Thomas C. Schelling, Micromotives and Macrobehavior (1978). The adjacent account of how local rules generate aggregate patterns.
Mark Granovetter and Roland Soong, “Threshold Models of Diffusion and Collective Behavior,” Journal of Mathematical Sociology 9(3), 1983. DOI: 10.1080/0022250X.1983.9989941
Duncan J. Watts, “A Simple Model of Global Cascades on Random Networks,” PNAS 99(9), 2002, pp. 5766–5771. DOI: 10.1073/pnas.082090499

Abhishek's take

The missing rung interests me more than the tipping point. Crowds that look equally willing on average can produce opposite outcomes because one person waits for two signals instead of one. I read the model as a warning against explaining a surge after it happens: the visible crowd hides the threshold sequence that made it possible.

Abhishek