Social Influence Cascade | mysimulator.uk

A behavior that spreads only after enough neighbors have it

Some behaviors spread like a simple contagion — one exposure is enough, as in a rumor. Many others, though, only spread once a person sees enough of their social contacts already doing it: joining a protest, adopting a new technology, switching political allegiance. Mark Granovetter's 1978 threshold model formalises exactly this: every individual i has a threshold θ_i, and adopts the behavior once the fraction of their neighbors who have already adopted meets or exceeds it.

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The update rule

On a social network, each node updates independently, checking the state of its neighbors and flipping to 'adopted' once the threshold condition is met. The process runs in discrete rounds until it reaches a stable state where no more nodes can be activated:

node i adopts at time t+1 if:

   ( number of adopted neighbors of i at time t )
   -----------------------------------------------  >=  theta_i
   ( total number of neighbors of i )

process halts when no inactive node meets its threshold  ->  final cascade size

Two nodes with identical neighborhoods but different thresholds can behave completely differently — a low-θ 'early adopter' flips almost immediately, while a high-θ conformist waits until nearly everyone around them has already switched. The distribution of thresholds across the whole population, not any single individual's threshold, is what determines whether a cascade takes off or fizzles.

Why the threshold distribution matters more than the average

Granovetter's classic illustration is a riot: identical crowds with slightly different threshold distributions can produce wildly different outcomes — one crowd disperses peacefully, another erupts, even though the average threshold in both crowds might be nearly the same. What matters is whether there exists a small, unbroken chain of increasing thresholds starting near zero: a few θ=0 instigators trigger the θ=0.1 group, who trigger the θ=0.2 group, and so on, each threshold met by the growing pool of already-adopted neighbors. A single gap in that chain — no one with a threshold between 0.3 and 0.5, say — can stop a cascade dead even if plenty of people beyond the gap would happily join if only they saw enough others already on board.

Network structure changes everything

The same threshold distribution produces very different cascades depending on network topology. In a network with strong local clustering — tight-knit friend groups with few bridges to other groups — a cascade can ignite easily within one cluster (neighbors reinforce each other quickly) but struggle to jump to the next cluster, because a node bridging two clusters may have very few adopted neighbors relative to its total degree, even if that node's own cluster is fully converted. Sparse, well-connected random or small-world networks propagate global cascades more easily precisely because they have fewer such bottlenecks, at the cost of being less able to build the concentrated local reinforcement a high-threshold cluster needs to tip in the first place.

Beyond riots: technology adoption and information cascades

The same threshold logic has been applied well beyond Granovetter's original riot example: adoption of a new product or platform (network effects mean your threshold to join often depends on how many friends already use it), migration decisions, strikes, and diffusion of new agricultural or medical practices. In marketing and viral-growth contexts, the practical lesson is that seeding a cascade effectively is less about reaching the most people and more about reaching a set of low-threshold nodes whose adoption is visible to, and sufficient to trigger, the next threshold tier — the 'chain of dominoes' framing that Granovetter's original model makes mathematically precise.

Frequently asked questions

How is the Linear Threshold Model different from simple contagion spread?

In simple contagion, one adopting neighbor can be enough to spread a behavior, similar to disease transmission. In the threshold model, a node only adopts once the fraction of its neighbors who have already adopted reaches its own individual threshold, so isolated or minority exposure is not enough — it requires a critical mass among that node's specific neighbors.

Why can two nearly identical crowds produce completely different outcomes?

Because cascade size depends on the fine structure of the threshold distribution, not just its average. If there is an unbroken chain of increasing thresholds from a few zero-threshold instigators up through the rest of the population, the cascade can spread step by step; a single missing threshold value in that chain can stall it, even with a similar average threshold overall.

Does a more connected network always spread a behavior further?

Not necessarily. Densely clustered local networks can ignite a cascade quickly within one tight group but struggle to cross bridges to other groups, since a bridging node may have too few adopted neighbors relative to its total degree. Sparser, more evenly connected networks often propagate a cascade further globally, even though they build up local reinforcement more slowly.

Try it live

Everything above runs in your browser — open Social Influence Cascade and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.

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Social Cascades: Granovetter's Threshold Model of Collective Behavior