Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

mixing-rate

One function in the graphwalk library, called 13 times across 6 essays. Below: what it draws at its defaults, what it draws at every value an essay asks for, the 11 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 122 steps to forget where it started, and λ₂ said 131. A random walk on a sixty-vertex graph with two blocks joined at a density of 0.03, started at one vertex, measured as the total variation distance from its stationary distribution. It reaches 10⁻⁶ after 122 steps. The dashed line is |λ₂|ᵏ with λ₂ = 0.900071 from the walk matrix, and the number of steps that predicts is 131. The distance is not the eigenvalue: for the first several steps the walk is leaving one block and the decay is faster than the asymptote, and the two curves become parallel only once the slowest mode is all that is left. The eigenvalue is a rate, not a schedule, and the difference between them is 9.22 steps.

mixing-rate 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.

122 steps to forget where it started, and λ₂ said 131A random walk on a sixty-vertex graph with two blocks joined at a density of 0.03, started at one vertex, measured as the total variation distance from its stationary distribution. It reaches 10⁻⁶ after 122 steps. The dashed line is |λ₂|ᵏ with λ₂ = 0.900071 from the walk matrix, and the number of steps that predicts is 131. The distance is not the eigenvalue: for the first several steps the walk is leaving one block and the decay is faster than the asymptote, and the two curves become parallel only once the slowest mode is all that is left. The eigenvalue is a rate, not a schedule, and the difference between them is 9.22 steps.0173451688510211910⁻⁶10⁻⁵10⁻⁴10⁻³10⁻²10⁻¹1steps of the walkdistance from stationarya rate read as a timeλ₂ of the walk0.9steps measured122steps predicted131ratio0.93the dashed line is the eigenvaluethe curve is the walk

A random walk on a sixty-vertex graph with two blocks joined at a density of 0.03, started at one vertex, measured as the total variation distance from its stationary distribution. It reaches 10⁻⁶ after 122 steps. The dashed line is |λ₂|ᵏ with λ₂ = 0.900071 from the walk matrix, and the number of steps that predicts is 131. The distance is not the eigenvalue: for the first several steps the walk is leaving one block and the decay is faster than the asymptote, and the two curves become parallel only once the slowest mode is all that is left. The eigenvalue is a rate, not a schedule, and the difference between them is 9.22 steps.

pOut: 0.005

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.

1002 steps to forget where it started, and λ₂ said 1055A random walk on a sixty-vertex graph with two blocks joined at a density of 0.005, started at one vertex, measured as the total variation distance from its stationary distribution. It reaches 10⁻⁶ after 1002 steps. The dashed line is |λ₂|ᵏ with λ₂ = 0.986994 from the walk matrix, and the number of steps that predicts is 1055. The distance is not the eigenvalue: for the first several steps the walk is leaving one block and the decay is faster than the asymptote, and the two curves become parallel only once the slowest mode is all that is left. The eigenvalue is a rate, not a schedule, and the difference between them is 53.3 steps.0143286429572715858100110⁻⁶10⁻⁵10⁻⁴10⁻³10⁻²10⁻¹1steps of the walkdistance from stationarya rate read as a timeλ₂ of the walk0.99steps measured1002steps predicted1055ratio0.95the dashed line is the eigenvaluethe curve is the walk

A random walk on a sixty-vertex graph with two blocks joined at a density of 0.005, started at one vertex, measured as the total variation distance from its stationary distribution. It reaches 10⁻⁶ after 1002 steps. The dashed line is |λ₂|ᵏ with λ₂ = 0.986994 from the walk matrix, and the number of steps that predicts is 1055. The distance is not the eigenvalue: for the first several steps the walk is leaving one block and the decay is faster than the asymptote, and the two curves become parallel only once the slowest mode is all that is left. The eigenvalue is a rate, not a schedule, and the difference between them is 53.3 steps.

pOut: 0.2

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.

16 steps to forget where it started, and λ₂ said 17A random walk on a sixty-vertex graph with two blocks joined at a density of 0.2, started at one vertex, measured as the total variation distance from its stationary distribution. It reaches 10⁻⁶ after 16 steps. The dashed line is |λ₂|ᵏ with λ₂ = 0.450568 from the walk matrix, and the number of steps that predicts is 17. The distance is not the eigenvalue: for the first several steps the walk is leaving one block and the decay is faster than the asymptote, and the two curves become parallel only once the slowest mode is all that is left. The eigenvalue is a rate, not a schedule, and the difference between them is 1.33 steps.0246810121416182010⁻⁶10⁻⁵10⁻⁴10⁻³10⁻²10⁻¹1steps of the walkdistance from stationarya rate read as a timeλ₂ of the walk0.45steps measured16steps predicted17ratio0.92the dashed line is the eigenvaluethe curve is the walk

A random walk on a sixty-vertex graph with two blocks joined at a density of 0.2, started at one vertex, measured as the total variation distance from its stationary distribution. It reaches 10⁻⁶ after 16 steps. The dashed line is |λ₂|ᵏ with λ₂ = 0.450568 from the walk matrix, and the number of steps that predicts is 17. The distance is not the eigenvalue: for the first several steps the walk is leaving one block and the decay is faster than the asymptote, and the two curves become parallel only once the slowest mode is all that is left. The eigenvalue is a rate, not a schedule, and the difference between them is 1.33 steps.

pOut: 0.03

The arguments are the ones A ranking that is an eigenvector 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.

122 steps to forget where it started, and λ₂ said 131A random walk on a sixty-vertex graph with two blocks joined at a density of 0.03, started at one vertex, measured as the total variation distance from its stationary distribution. It reaches 10⁻⁶ after 122 steps. The dashed line is |λ₂|ᵏ with λ₂ = 0.900071 from the walk matrix, and the number of steps that predicts is 131. The distance is not the eigenvalue: for the first several steps the walk is leaving one block and the decay is faster than the asymptote, and the two curves become parallel only once the slowest mode is all that is left. The eigenvalue is a rate, not a schedule, and the difference between them is 9.22 steps.0173451688510211910⁻⁶10⁻⁵10⁻⁴10⁻³10⁻²10⁻¹1steps of the walkdistance from stationarya rate read as a timeλ₂ of the walk0.9steps measured122steps predicted131ratio0.93the dashed line is the eigenvaluethe curve is the walk

A random walk on a sixty-vertex graph with two blocks joined at a density of 0.03, started at one vertex, measured as the total variation distance from its stationary distribution. It reaches 10⁻⁶ after 122 steps. The dashed line is |λ₂|ᵏ with λ₂ = 0.900071 from the walk matrix, and the number of steps that predicts is 131. The distance is not the eigenvalue: for the first several steps the walk is leaving one block and the decay is faster than the asymptote, and the two curves become parallel only once the slowest mode is all that is left. The eigenvalue is a rate, not a schedule, and the difference between them is 9.22 steps.

pOut: 0.01

The arguments are the ones The rate is the second eigenvalue 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.

314 steps to forget where it started, and λ₂ said 329A random walk on a sixty-vertex graph with two blocks joined at a density of 0.01, started at one vertex, measured as the total variation distance from its stationary distribution. It reaches 10⁻⁶ after 314 steps. The dashed line is |λ₂|ᵏ with λ₂ = 0.958917 from the walk matrix, and the number of steps that predicts is 329. The distance is not the eigenvalue: for the first several steps the walk is leaving one block and the decay is faster than the asymptote, and the two curves become parallel only once the slowest mode is all that is left. The eigenvalue is a rate, not a schedule, and the difference between them is 15.3 steps.0459013518022527031510⁻⁶10⁻⁵10⁻⁴10⁻³10⁻²10⁻¹1steps of the walkdistance from stationarya rate read as a timeλ₂ of the walk0.96steps measured314steps predicted329ratio0.95the dashed line is the eigenvaluethe curve is the walk

A random walk on a sixty-vertex graph with two blocks joined at a density of 0.01, started at one vertex, measured as the total variation distance from its stationary distribution. It reaches 10⁻⁶ after 314 steps. The dashed line is |λ₂|ᵏ with λ₂ = 0.958917 from the walk matrix, and the number of steps that predicts is 329. The distance is not the eigenvalue: for the first several steps the walk is leaving one block and the decay is faster than the asymptote, and the two curves become parallel only once the slowest mode is all that is left. The eigenvalue is a rate, not a schedule, and the difference between them is 15.3 steps.

pOut: 0.02

The arguments are the ones The rate is the second eigenvalue 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.

169 steps to forget where it started, and λ₂ said 178A random walk on a sixty-vertex graph with two blocks joined at a density of 0.02, started at one vertex, measured as the total variation distance from its stationary distribution. It reaches 10⁻⁶ after 169 steps. The dashed line is |λ₂|ᵏ with λ₂ = 0.92517 from the walk matrix, and the number of steps that predicts is 178. The distance is not the eigenvalue: for the first several steps the walk is leaving one block and the decay is faster than the asymptote, and the two curves become parallel only once the slowest mode is all that is left. The eigenvalue is a rate, not a schedule, and the difference between them is 8.63 steps.02448729612014416810⁻⁶10⁻⁵10⁻⁴10⁻³10⁻²10⁻¹1steps of the walkdistance from stationarya rate read as a timeλ₂ of the walk0.93steps measured169steps predicted178ratio0.95the dashed line is the eigenvaluethe curve is the walk

A random walk on a sixty-vertex graph with two blocks joined at a density of 0.02, started at one vertex, measured as the total variation distance from its stationary distribution. It reaches 10⁻⁶ after 169 steps. The dashed line is |λ₂|ᵏ with λ₂ = 0.92517 from the walk matrix, and the number of steps that predicts is 178. The distance is not the eigenvalue: for the first several steps the walk is leaving one block and the decay is faster than the asymptote, and the two curves become parallel only once the slowest mode is all that is left. The eigenvalue is a rate, not a schedule, and the difference between them is 8.63 steps.

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.

11 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 between-block density that keeps the graph connected

a graph with at least two vertices

a planted partition with no isolated vertex

a positive weight

a walk that reached stationarity

an aperiodic walk, whose second eigenvalue is inside the circle

every endpoint inside the vertex set

Jacobi needs a symmetric matrix

no edge given twice

no isolated vertex, which the normalisation divides by

no self-loop

Against the rule

It draws a decomposition and prints its residual. It calls walkSpectrum, mixingRun, 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.

 The matrix that is a graph

The spectrum is not the graph

Two graphs on six vertices with the same Laplacian characteristic polynomial — as integer polynomials, not to fourteen digits. One contains a triangle; the other is bipartite. Every method in this field that reads only the spectrum is answering about the class.

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