Chung's Laplacian against the undirected one, on four families

At its defaults it draws chung's laplacian against the undirected one, on four families. The worst entrywise difference between the directed normalised Laplacian and the undirected normalised Laplacian of the same graph with its arrows removed. On the balanced and symmetric families the difference is at the rounding level at every size — 10⁻¹⁴ and below — because a balanced digraph's stationary distribution is its degree distribution and the construction then collapses to the undirected one. On the two-block family with a single arc back the difference is 0.163 at n = 30, which is the size of what the direction adds. The two curves are the same computation and the flat one is a closed form, so this figure is the field's second route rather than a comparison.

chung-reduction 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.

The worst entrywise difference between the directed normalised Laplacian and the undirected normalised Laplacian of the same graph with its arrows removed. On the balanced and symmetric families the difference is at the rounding level at every size — 10⁻¹⁴ and below — because a balanced digraph's stationary distribution is its degree distribution and the construction then collapses to the undirected one. On the two-block family with a single arc back the difference is 0.163 at n = 30, which is the size of what the direction adds. The two curves are the same computation and the flat one is a closed form, so this figure is the field's second route rather than a comparison.

The worst entrywise difference between the directed normalised Laplacian and the undirected normalised Laplacian of the same graph with its arrows removed. On the balanced and symmetric families the difference is at the rounding level at every size — 10⁻¹⁴ and below — because a balanced digraph's stationary distribution is its degree distribution and the construction then collapses to the undirected one. On the two-block family with a single arc back the difference is 0.163 at n = 30, which is the size of what the direction adds. The two curves are the same computation and the flat one is a closed form, so this figure is the field's second route rather than a comparison.

The worst entrywise difference between the directed normalised Laplacian and the undirected normalised Laplacian of the same graph with its arrows removed. On the balanced and symmetric families the difference is at the rounding level at every size — 10⁻¹⁴ and below — because a balanced digraph's stationary distribution is its degree distribution and the construction then collapses to the undirected one. On the two-block family with a single arc back the difference is 0.1902 at n = 48, which is the size of what the direction adds. The two curves are the same computation and the flat one is a closed form, so this figure is the field's second route rather than a comparison.

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.

13 distinct claims across 3 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 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.