📢 Social Influence Cascade

Linear Threshold Model (Granovetter): each node adopts a behavior when fraction of neighbors exceeding its threshold θ_i. Cascade size depends on threshold distribution and network topology.

SocietyInteractive
Click node to toggle seed · Run Cascade to propagate · Step for round-by-round

How it Works

The Linear Threshold Model (LTM) is a fundamental model of social contagion introduced by Mark Granovetter (1978). Each agent (node) in a network has a private threshold θ_i drawn from a distribution F(θ). An agent adopts a behavior or opinion when the fraction of its neighbors who have already adopted exceeds its threshold.

The simulation proceeds in discrete rounds. In each round, all non-adopted nodes check whether the fraction of their adopted neighbors exceeds their threshold. If yes, they adopt. This continues until no further adoptions occur (fixed point) or all nodes have adopted.

Node i adopts at round t+1 if: |{j ∈ N(i) : adopted(j,t)}| / deg(i) ≥ θ_i Cascade condition (Watts 2002): A global cascade is possible when the vulnerable cluster percolates the network. Vulnerable: θ_i ≤ 1/deg(i)

The cascade size undergoes a phase transition: for low mean thresholds or high connectivity, nearly all nodes adopt; for high thresholds or sparse networks, the cascade dies out quickly. The "cascade window" in (k, θ) parameter space separates these regimes.

Frequently Asked Questions

What is the Linear Threshold Model?

The Linear Threshold Model (LTM), introduced by Granovetter, describes how behaviors spread through a network. Each node has a threshold θ_i and adopts a behavior when the fraction of its neighbors who have adopted exceeds θ_i.

What causes a social cascade?

A social cascade occurs when initial adopters trigger a chain reaction of adoption through the network. Even small initial seeds can cause global cascades if the threshold distribution and network topology are favorable.

How does network topology affect cascade size?

Highly connected hubs accelerate cascades. Random networks with low average thresholds see large cascades. Scale-free networks are particularly vulnerable because hubs can spread adoption rapidly to many neighbors.

What is a threshold distribution?

The threshold distribution F(θ) describes the probability that a random node has threshold below θ. A uniform distribution means thresholds are spread evenly between 0 and 1. Lower mean thresholds lead to larger cascades.

Who introduced the cascade model?

Mark Granovetter introduced the threshold cascade model in his 1978 paper "Threshold Models of Collective Behavior". It has since been extended by Watts and Dodds and many others.

What is the difference between innovators and imitators?

In cascade models, innovators (or seeds) adopt independently of neighbors. Imitators only adopt when enough neighbors have adopted. The seed set choice critically determines whether a cascade can ignite.

Can cascades be stopped once started?

Cascades can be stopped by removing high-degree hubs (network immunization) or by introducing nodes with very high thresholds that block propagation. This is the basis of epidemic containment strategies.

What real phenomena does this model capture?

The LTM captures viral marketing, political mobilization, bank runs, technology adoption, riot behavior, and the spread of social norms through communities.

What is a global cascade condition?

Watts (2002) showed a global cascade is possible when the "cascade window" is open: nodes with low thresholds and many low-threshold neighbors form a vulnerable cluster spanning the network.

How is cascade size measured?

Cascade size is the fraction of nodes that ultimately adopt. It ranges from 0 (no spread) to 1 (full adoption). The cascade size as a function of seed fraction shows a phase transition at a critical threshold.

About this simulation

This simulation builds a network of nodes — random, scale-free, or small-world — assigns each one a private adoption threshold θ_i, and seeds a handful of early adopters to watch whether a cascade of orange "adopted" nodes sweeps the graph. Round by round, a node flips to adopted only once the fraction of its already-adopted neighbours clears its personal threshold, exactly as in Granovetter's 1978 model.

🔬 What it shows

A force-laid-out graph where edges glow orange once one endpoint adopts, node thresholds are printed beneath each dot (for small networks), and the stats panel tracks adopted count, cascade percentage, round number, and active edges.

🎮 How to use

Set node count, average threshold θ, average degree k, and seed fraction with the sliders, pick a network type from the dropdown, then hit Run Cascade or Step; click any node directly on the canvas to toggle it as a seed adopter.

💡 Did you know?

Whether a cascade goes global or fizzles out barely depends on the exact seed set — it hinges on whether a "vulnerable cluster" of low-threshold nodes (θ_i ≤ 1/deg(i)) percolates the whole network, a result formalised by Duncan Watts in 2002.

Frequently asked questions

Why does changing the network type change the outcome so much?

Scale-free (Barabási-Albert) networks concentrate connections in a few hubs, so once a hub adopts it can flip many low-degree neighbours in a single round; random (Erdős-Rényi) and small-world (Watts-Strogatz) networks spread connectivity more evenly, usually slowing the cascade.

What does the "Active edges" stat actually count?

It counts edges where exactly one endpoint has adopted and the other hasn't — the live frontier of the cascade. As this number falls to zero, the simulation has reached a fixed point and stops changing.

Why does raising the average threshold θ shrink cascades?

Each node only adopts once the fraction of adopted neighbours meets its own θ_i; a higher average threshold means more neighbours must already be on board before any given node flips, which the simulation's per-node threshold labels make directly visible.

What happens when I click a node mid-simulation?

Clicking toggles that node's adopted state immediately, letting you manually inject or remove seeds and watch how the next Step or Run Cascade round responds — useful for testing whether a single well-placed hub can restart a stalled cascade.

Why does seed fraction matter less than it seems?

Below the cascade window, even large seed fractions die out because thresholds are too high or the network too sparse; above it, even a single well-connected seed can trigger near-total adoption — the transition is sharp rather than gradual.