Generator

perron-sign

One function in the graphwalk library, called 10 times across 5 essays. Below: what it draws at its defaults, what it draws at every value an essay asks for, the 9 claims it put to the test while drawing them, and where it stands against the rule this site is named for.

At its defaults it draws 4 entries of a strictly positive vector are negative. The Perron eigenvector of a twelve-vertex clique with a tail of 24, normalised to its largest entry, plotted as absolute values on a logarithmic axis. Perron–Frobenius says every entry of this vector is strictly positive, and it is a theorem about a connected nonnegative matrix rather than a statement about a good day. Each step along the tail divides the entry by about the Perron root 11.0077, so the entries fall by a decade a step and reach the rounding level after sixteen. Beyond it 4 of them are negative — the smallest at -1.507·10⁻²⁰ — and they are drawn as open circles. The bracket printed beside it is Collatz–Wielandt: for any strictly positive x, the Perron root lies between the smallest and largest of (Ax)ᵢ/xᵢ, so it is a two-sided bound that is proved, from one product, and after forty steps it is 1.2·10⁻⁴ wide.

perron-sign is one function in lib/figures/graphwalk.js — the walk and the ranking — a rate that is not the published one, a positivity the arithmetic loses, and places nothing settles. Everything below came out of it during this build, at arguments taken from the essays rather than invented for this page. A figure here is the figure a reader meets in an essay, and if the generator changes, this page changes with it.

At its defaults

Drawn even though every essay passes arguments — which on this site is every essay, at 100% of placements since the standard pass. A default nothing exercises is a trap for the next essay to call this with none, and this is the page where a default that has drifted from the figures around it becomes visible.

The Perron eigenvector of a twelve-vertex clique with a tail of 24, normalised to its largest entry, plotted as absolute values on a logarithmic axis. Perron–Frobenius says every entry of this vector is strictly positive, and it is a theorem about a connected nonnegative matrix rather than a statement about a good day. Each step along the tail divides the entry by about the Perron root 11.0077, so the entries fall by a decade a step and reach the rounding level after sixteen. Beyond it 4 of them are negative — the smallest at -1.507·10⁻²⁰ — and they are drawn as open circles. The bracket printed beside it is Collatz–Wielandt: for any strictly positive x, the Perron root lies between the smallest and largest of (Ax)ᵢ/xᵢ, so it is a two-sided bound that is proved, from one product, and after forty steps it is 1.2·10⁻⁴ wide.

`tail: 24`

The arguments are the ones A chain with no stationary vector passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.

`tail: 6`

The arguments are the ones An eigenvector that must not change sign passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.

The Perron eigenvector of a twelve-vertex clique with a tail of 6, normalised to its largest entry, plotted as absolute values on a logarithmic axis. Perron–Frobenius says every entry of this vector is strictly positive, and it is a theorem about a connected nonnegative matrix rather than a statement about a good day. Each step along the tail divides the entry by about the Perron root 11.0077, so the entries fall by a decade a step and reach the rounding level after sixteen. Every entry here is still positive. The bracket printed beside it is Collatz–Wielandt: for any strictly positive x, the Perron root lies between the smallest and largest of (Ax)ᵢ/xᵢ, so it is a two-sided bound that is proved, from one product, and after forty steps it is 1.78·10⁻¹⁵ wide.

`tail: 14`

The Perron eigenvector of a twelve-vertex clique with a tail of 14, normalised to its largest entry, plotted as absolute values on a logarithmic axis. Perron–Frobenius says every entry of this vector is strictly positive, and it is a theorem about a connected nonnegative matrix rather than a statement about a good day. Each step along the tail divides the entry by about the Perron root 11.0077, so the entries fall by a decade a step and reach the rounding level after sixteen. Every entry here is still positive. The bracket printed beside it is Collatz–Wielandt: for any strictly positive x, the Perron root lies between the smallest and largest of (Ax)ᵢ/xᵢ, so it is a two-sided bound that is proved, from one product, and after forty steps it is 8.88·10⁻¹⁵ wide.

`tail: 18`

The Perron eigenvector of a twelve-vertex clique with a tail of 18, normalised to its largest entry, plotted as absolute values on a logarithmic axis. Perron–Frobenius says every entry of this vector is strictly positive, and it is a theorem about a connected nonnegative matrix rather than a statement about a good day. Each step along the tail divides the entry by about the Perron root 11.0077, so the entries fall by a decade a step and reach the rounding level after sixteen. Every entry here is still positive. The bracket printed beside it is Collatz–Wielandt: for any strictly positive x, the Perron root lies between the smallest and largest of (Ax)ᵢ/xᵢ, so it is a two-sided bound that is proved, from one product, and after forty steps it is 7.87·10⁻¹¹ wide.

`tail: 34`

The Perron eigenvector of a twelve-vertex clique with a tail of 34, normalised to its largest entry, plotted as absolute values on a logarithmic axis. Perron–Frobenius says every entry of this vector is strictly positive, and it is a theorem about a connected nonnegative matrix rather than a statement about a good day. Each step along the tail divides the entry by about the Perron root 11.0077, so the entries fall by a decade a step and reach the rounding level after sixteen. Beyond it 10 of them are negative — the smallest at -1.032·10⁻²⁰ — and they are drawn as open circles. The bracket printed beside it is Collatz–Wielandt: for any strictly positive x, the Perron root lies between the smallest and largest of (Ax)ᵢ/xᵢ, so it is a two-sided bound that is proved, from one product, and after forty steps it is 9.15 wide.

What it checked while drawing

Every figure above asserted its own claims on the way to being drawn, and a claim that failed would have failed the build rather than drawn a wrong picture. Those assertions used to leave no trace at all: a passing one returned true and the only evidence the figure had checked anything was that nothing crashed. The list below is what they actually said, collected by running this generator with an observer installed — not a description of what it is believed to check.

9 distinct claims across 6 sets of arguments, grouped below by shape — because most of them are one sentence with a different number in it, and how many separate times that sentence was put to the test is the informative part.

a graph with at least two vertices

a positive weight

a strictly positive vector, which is what the theorem asks for

a tail long enough to reach the rounding level and short enough to draw

every endpoint inside the vertex set

Jacobi needs a symmetric matrix

no edge given twice

no self-loop

the Collatz–Wielandt bracket contains the Perron root

Against the rule

It draws a decomposition and prints its residual. It calls perronVector, and every figure above carries the badge — which residualcheck verifies by looking for it in the emitted SVG rather than by finding the call that builds one. A badge that is constructed and then left out of the body is the failure that check exists for.

Across the library: the rule bites on 214 of 382 generators — 194 print a residual and 20 are exempt with a published reason; 168 factorise nothing. Read from lib/residual-rule.js, which is the same body the gate enforces from, and the gate's last check fails the build if this page and it disagree about any generator.

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.

The matrix that is a graph

A chain with no stationary vector

A page with no outgoing links loses forty per cent of the walker's probability in six hundred steps. A directed cycle never converges at all. And on a graph whose links only run one way, the entire rank of half the vertices is exactly one minus the teleportation parameter.

The matrix that is a graph

A ranking that is an eigenvector

PageRank is the stationary vector of a walk that follows links with probability α and jumps at random otherwise. The iteration and the elimination agree to 4·10⁻¹⁷. What α is set to changes which pages come third, fourth and fifth.

The matrix that is a graph

A ranking whose order is not determined

Of the fourteen adjacent comparisons in a top-fifteen, fourteen survive perturbing the arithmetic at the rounding level, eleven survive moving the teleportation parameter across its usual range, and three survive removing one link. The computation is the strongest part of the answer.

The matrix that is a graph

An eigenvector that must not change sign

Perron's theorem says the leading eigenvector of a connected nonnegative matrix is strictly positive. On a clique with a long tail, four of its thirty-six entries come back negative — and beside them is the one two-sided bound on this site that is proved rather than estimated.

The matrix that is a graph

The rate is the second eigenvalue

A walk forgets where it started at a rate the graph's second eigenvalue names exactly. Across three orders of magnitude in the step count the prediction is five per cent high — and the published rate for PageRank is right for a reason nobody states, which is that a link graph is in pieces.

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