122 steps to forget where it started, and λ₂ said 131
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.
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: 0.03
The arguments are the ones The rate is the second eigenvalue passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
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 nobody placed would be a picture no essay asked for and no claim was ever checked against.
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 nobody placed would be a picture no essay asked for and no claim was ever checked against.
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.
pOut: 0.08
The arguments are the ones The rate is the second eigenvalue passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
A random walk on a sixty-vertex graph with two blocks joined at a density of 0.08, started at one vertex, measured as the total variation distance from its stationary distribution. It reaches 10⁻⁶ after 36 steps. The dashed line is |λ₂|ᵏ with λ₂ = 0.698104 from the walk matrix, and the number of steps that predicts is 38. 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 2.44 steps.
pOut: 0.2
The arguments are the ones The rate is the second eigenvalue passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
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.
What it checked while drawing
Every figure above checked its own claims on the way to being drawn, and a claim that failed
would have stopped the picture rather than shipped a wrong one. Those checks 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 217
of 397 generators —
199 print a residual and
18 are exempt with a published reason;
180 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.