What Is It?

This model demonstrates Stuart Kauffman's "buttons and threads" thought experiment, one of the simplest illustrations of a phase transition in a random network.

Imagine N buttons scattered on a floor. You pick two at random and tie a thread between them. Repeat. At first, you get isolated pairs and small clusters. Then, around the moment when the number of threads reaches half the number of buttons, something dramatic happens: a single giant cluster emerges and rapidly absorbs most of the network.

This is the same mathematics behind Erdos-Renyi random graphs. The transition at threads/N = 0.5 is sharp, sudden, and universal. It does not depend on which buttons you pick. It depends only on the ratio.

How It Works

N buttons are arranged in a circle. At each step, two buttons are chosen uniformly at random and connected by a thread. A union-find data structure tracks which buttons belong to the same connected component.

The largest connected component is colored green. All other components are colored red. The chart below the network plots the fraction of buttons in the largest component as threads accumulate.

How To Use It

• Play / Pause runs the simulation continuously.
• Step adds a single thread so you can watch the network evolve one connection at a time.
• Reset clears the network and generates a fresh random sequence.
• Speed controls how many threads are added per animation frame.
• Buttons sets N, the number of buttons. Changing this resets the simulation.
• Keyboard: Space = play/pause, → = step, R = reset.

Things To Notice

• For the first ~N/2 threads, the largest component stays small. The chart line is nearly flat along the bottom.
• Right around threads/N = 0.5 (the dashed orange line), the curve kicks upward sharply. This is the phase transition.
• After the transition, a few more threads cause the giant component to swallow almost everything. The red buttons vanish quickly.
• The number of separate components drops precipitously at the same threshold.
• Run it several times. The transition always happens near the same ratio, even though every run uses different random connections.

Things To Try

• Run with a small number of buttons (20-30) and use Step mode. Watch how individual mergers accumulate into the giant component.
• Increase to 200-300 buttons. Notice how the transition becomes sharper and steeper with larger N.
• Pause right at threads/N = 0.5 and look at the network. You can often see the moment where two large clusters are about to merge.
• Reset several times at the same N and compare the chart curves. The shape is remarkably consistent.

Extending The Model

• What happens if you only allow connections between nearby buttons (local wiring) instead of random pairs?
• What if each button has a maximum number of threads it can hold?
• What if you remove threads at random while adding new ones?
• Kauffman extended this idea to autocatalytic sets in chemistry, where molecules catalyze each other's formation. Above a connectivity threshold, a self-sustaining network "crystallizes" from random interactions.

Why It Matters

Kauffman used buttons and threads to argue that the origin of life may not require an astronomically unlikely event. Instead, once molecular diversity crosses a threshold, a connected web of mutual catalysis becomes virtually inevitable. Order, in this view, is free.

The same phase transition appears in epidemiology (disease outbreaks), social networks (information cascades), ecology (food web stability), and percolation physics (fluid flow through porous rock). The math is identical. Only the substrate changes.

Related Models

Erdos-Renyi random graphs. Percolation on lattices. Kauffman's NK fitness landscapes and autocatalytic sets. Barabasi-Albert preferential attachment networks (a different growth rule that produces scale-free rather than random topology).