12 points at 3 values of p
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.
With nothing chosen
A triangle appears when the count says it should
The chance of being connected, against the chance of an edge
The moment a giant piece appears
One threshold narrowing, one staying wide
One piece, and connected, are different thresholds
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 4 points, counted twice by different means ×4
- above the threshold the measured share matches the equation's root ×1
- and below it the largest piece is a vanishing share ×1
- and below the threshold the equation has only the zero root ×1
- and it is nonetheless not connected ×1
- and it stays put to within half ×1
- and the chance of at least one never exceeds the expected count, as Markov requires ×1
- and the triangle's stays wider than it ×1
- and well above it a triangle is nearly certain ×1
- being connected is at most as likely as having no isolated point ×1
- between four and forty samples at each size ×1
- between one and four sizes, each at most six points ×1
- between three and five sizes, each between fifty and twelve hundred ×1
- between twenty and two hundred samples per point ×1
- between two and four edge chances, each strictly between zero and one ×1
- between two and four sizes, each between twenty and a hundred and forty ×1
- connectivity's transition narrows relative to its threshold as the graph grows ×1
- dividing by the two-thirds power is the division that stays put ×1
- each sweep runs from mostly-absent to mostly-present ×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 component grows with the size, as it must ×1
- the count of isolated points follows n·e⁻ᶜ ×1
- the drawn graphs have between six and twenty points ×1
- the enumeration runs on between two and six points ×1
- the measured triangle count matches the expectation it is supposed to ×1
- the number of points is between three and twenty ×1
- the predicted share solves its own equation ×1
- the range drawn reaches the connectivity threshold ×1
- the sampled graphs have between sixty and three thousand points ×1
- the sampled graphs have between two hundred and three thousand points ×1
- the seed is a whole number the graphs can be drawn from ×1
- the view is one the family draws ×1
- well below the threshold a triangle is rare ×1
- well past average degree one, most of the graph is in one piece ×1
Where it is called
Every figure on this list is drawn by the same rule, so a change to the rule changes all of them at once. That is why the list is published.
Finding a threshold with two moments
Every monotone property of a random graph has a threshold, and locating one is nearly always the same two calculations — count what the property needs, and check the count does not concentrate on rare cases. The triangle is where the method is cleanest.
ProbabilitySharp, or merely a threshold
Every monotone property of a random graph has a threshold. Some of them turn on over a range that shrinks relative to the threshold as the graph grows, and some do not — and which kind a property is turns out to be decided by whether it is about a local structure or about the whole graph.
ProbabilityThe moment a giant appears
Raise the chance of an edge slowly and a random graph does nothing for a long time, then in a narrow window acquires a component holding a definite fraction of everything. The fraction is the root of an equation, and the equation says why the transition is where it is.
ProbabilityThe 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.
ProbabilityThe window where the giant is born
Below the threshold the largest piece is a few dozen points, above it a definite fraction of everything. At the threshold it is neither, and the size it does take — the two-thirds power — is an exponent with no elementary derivation that a measurement finds immediately.
ProbabilityTwo thresholds, not one
A random graph acquires a piece holding most of its points at average degree one, and is still not connected. Connectivity waits until the average degree reaches the logarithm of the size, and what holds it up is the very last isolated point.