Graph theory and probability

Random connections can create surprisingly organised networks.

This prototype explores random graphs in the style of G(n, p), where each possible edge appears with probability p. Learners can change n and p, then watch connected components form and merge.

The aim is to connect graph theory with probability, networks and the idea of phase transitions in a visual, experimental way.

Random graph lab

Change edge probability and watch structure appear

This prototype generates a random graph in the style of G(n, p): start with n vertices, then include each possible edge with probability p.

Edges

Average degree

2.71

Components

Largest component

13 / 14

Controls

Number of vertices: 14

Edge probability p: 0.18

Rough connectivity threshold

For this number of vertices, log(n) / n is about 0.19. Around this region, connected graphs become much more likely, though randomness still matters.

The edge probability is below the rough connectivity threshold, so disconnected pieces are expected.

Component sizes

131

What this shows

Random graphs let us study structure that emerges from simple probability rules.

As p increases, isolated vertices and small components often join into one large connected component.

This connects graph theory, probability, networks, epidemics and computer science.

Guided tasks

Find a value of p where the graph is usually disconnected.

Increase p slowly. When does one giant component appear?

Keep p fixed and regenerate. What changes, and what seems predictable?

Joy in the process

The point is not just to finish. It is to notice, test and return.

These tools are invitations to explore. A good mistake, a surprising pattern or a question you cannot yet answer is part of the work, not a failure of it.

The challenge is deliberate: the site should support thinking, not remove the need for it.

Before changing a setting, pause and predict what you think will happen.

Change one thing at a time. What stayed the same, and what changed?

Try to create a surprising case, a broken case, or a beautiful pattern.

Ask what this connects to outside the page: maps, movement, nature, systems or decisions.

Reset, then try again with a new question in mind.

Future extensions

This can grow into a broader network science lab.

Add a repeated-trials view showing the probability of connectedness.

Add degree distribution histograms.

Add comparisons with real-world networks and social graphs.

Add epidemic spread simulations on the generated network.