Generator

rgraph

A generator in the probability library, called 6 times across 1 essays. Below: what it draws at its defaults and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

rgraph is one function. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and if the generator changes, this page changes with it.

At its defaults

12 points at 3 values of p. Three random graphs on the same points drawn at increasing edge probability, each point coloured by the piece it belongs to and the pieces counted.

`show: "exact"`

The chance of being connected, against the chance of an edge. Curves of the exact probability that a random graph on three to six labelled points is connected, plotted against the probability of each individual edge.

`show: "isolated"`

Connected, against merely having no point left out — 6 points. Two exact probability curves for a random graph on a few labelled points: the chance it is connected and the chance no point is isolated, with the gap between them shaded.

What it checks while it draws

Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.

the connected graphs on 3 points, counted twice by different means ×4
being connected is at most as likely as having no isolated point ×1
between one and four sizes, each at most six points ×1
between two and four edge chances, each strictly between zero and one ×1
every edge present is connected with probability one ×1
every point is in exactly one piece ×1
no edges is connected with probability zero ×1
the chance of being connected rises with p ×1
the comparison runs on between three and six points ×1
the drawn graphs have between six and twenty points ×1
the enumeration runs on between two and six points ×1
the number of points is between three and twenty ×1
the seed is a whole number the graphs can be drawn from ×1
the view is one the family draws ×1

Where it is called

Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.

Probability

The moment everything joins up

Add edges to a set of points one chance at a time and the graph goes from dust to a single piece — not gradually, but over a window that narrows as the point count grows. The last obstacle is almost always a single point with no edge at all, and that is what fixes where the change happens.

The whole library · What the figures prove