two blocks with 1 arc back: where a walk spends its time, against where the arcs are

At its defaults it draws two blocks with 1 arc back: where a walk spends its time, against where the arcs are. Two distributions over the 24 vertices of a two blocks with 1 arc back. The taller bar at each vertex is the stationary distribution of the random walk, computed by 824 power iterations with no subtraction anywhere and checked against Pᵀφ = φ at 9.69·10⁻¹⁵; the narrower one is the out-degree divided by the total, which is what an undirected intuition reaches for. They differ by up to 0.0616, which is 62 per cent of the largest entry. The difference is what makes a directed normalised Laplacian a global object: the degree of a vertex is local and how often a walk visits it is not.

stationary-vs-degree is one function in lib/figures/digraph.js — arrows — a laplacian that is not symmetric, and the walk that has to be computed before it can be. 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.

Two distributions over the 24 vertices of a two blocks with 1 arc back. The taller bar at each vertex is the stationary distribution of the random walk, computed by 824 power iterations with no subtraction anywhere and checked against Pᵀφ = φ at 9.69·10⁻¹⁵; the narrower one is the out-degree divided by the total, which is what an undirected intuition reaches for. They differ by up to 0.0616, which is 62 per cent of the largest entry. The difference is what makes a directed normalised Laplacian a global object: the degree of a vertex is local and how often a walk visits it is not.

Two distributions over the 24 vertices of a two blocks with 1 arc back. The taller bar at each vertex is the stationary distribution of the random walk, computed by 824 power iterations with no subtraction anywhere and checked against Pᵀφ = φ at 9.69·10⁻¹⁵; the narrower one is the out-degree divided by the total, which is what an undirected intuition reaches for. They differ by up to 0.0616, which is 62 per cent of the largest entry. The difference is what makes a directed normalised Laplacian a global object: the degree of a vertex is local and how often a walk visits it is not.

Two distributions over the 32 vertices of a balanced digraph on 32. The taller bar at each vertex is the stationary distribution of the random walk, computed by 388 power iterations with no subtraction anywhere and checked against Pᵀφ = φ at 1.01·10⁻¹⁴; the narrower one is the out-degree divided by the total, which is what an undirected intuition reaches for. On this family they agree to 1.64·10⁻¹⁴, because the digraph is balanced and the closed form says they must. That is the second route the whole construction is checked by.

Two distributions over the 24 vertices of a random strongly connected on 24. The taller bar at each vertex is the stationary distribution of the random walk, computed by 157 power iterations with no subtraction anywhere and checked against Pᵀφ = φ at 7.76·10⁻¹⁵; the narrower one is the out-degree divided by the total, which is what an undirected intuition reaches for. They differ by up to 0.0806, which is 70 per cent of the largest entry. The difference is what makes a directed normalised Laplacian a global object: the degree of a vertex is local and how often a walk visits it is not.

Two distributions over the 24 vertices of a directed cycle on 24. The taller bar at each vertex is the stationary distribution of the random walk, computed by 1 power iterations with no subtraction anywhere and checked against Pᵀφ = φ at 0; the narrower one is the out-degree divided by the total, which is what an undirected intuition reaches for. On this family they agree to 2.08·10⁻¹⁷, because the digraph is balanced and the closed form says they must. That is the second route the whole construction is checked by.

Two distributions over the 24 vertices of a symmetric digraph on 24. The taller bar at each vertex is the stationary distribution of the random walk, computed by 51 power iterations with no subtraction anywhere and checked against Pᵀφ = φ at 3.53·10⁻¹⁵; the narrower one is the out-degree divided by the total, which is what an undirected intuition reaches for. On this family they agree to 2.23·10⁻¹⁵, because the digraph is balanced and the closed form says they must. That is the second route the whole construction is checked by.

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.

13 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.

The rule does not apply to it. It factorises nothing, so there is no residual it could be withholding. That is worth stating rather than leaving blank: a site that reported the rule as satisfied by every generator would be counting mostly generators the rule never reached.

Across the library: the rule bites on 214 of 397 generators — 194 print a residual and 20 are exempt with a published reason; 183 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.