sparsifier-quality

At its defaults it draws 344 of 1225 edges keep every eigenvalue and no degree. A fifty-vertex complete graph sparsified by keeping 400 samples drawn in proportion to each edge's effective resistance and reweighted by the reciprocal of its probability, which leaves 344 distinct edges of 1225. The filled curve is the ratio of each eigenvalue of the sample's Laplacian to the original's, in ascending order: it stays between 0.432 and 1.67, so every quadratic form, every cut and every solve is preserved to that factor. The open curve is the ratio of each vertex's degree, which runs from 0.562 to 1.5 — half the graph's degrees are wrong by more than a third. And the graph's diameter goes from 1 to 3: no shortest path survives. A sparsifier preserves what the Laplacian says and nothing else about the graph, and the two are easy to confuse because they are the same object.

sparsifier-quality is one function in lib/figures/graphres.js — resistance, sparsification and elimination — three routes to one number, and a count that outgrows the format. 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.

A fifty-vertex complete graph sparsified by keeping 400 samples drawn in proportion to each edge's effective resistance and reweighted by the reciprocal of its probability, which leaves 344 distinct edges of 1225. The filled curve is the ratio of each eigenvalue of the sample's Laplacian to the original's, in ascending order: it stays between 0.432 and 1.67, so every quadratic form, every cut and every solve is preserved to that factor. The open curve is the ratio of each vertex's degree, which runs from 0.562 to 1.5 — half the graph's degrees are wrong by more than a third. And the graph's diameter goes from 1 to 3: no shortest path survives. A sparsifier preserves what the Laplacian says and nothing else about the graph, and the two are easy to confuse because they are the same object.

A fifty-vertex complete graph sparsified by keeping 400 samples drawn in proportion to each edge's effective resistance and reweighted by the reciprocal of its probability, which leaves 344 distinct edges of 1225. The filled curve is the ratio of each eigenvalue of the sample's Laplacian to the original's, in ascending order: it stays between 0.432 and 1.67, so every quadratic form, every cut and every solve is preserved to that factor. The open curve is the ratio of each vertex's degree, which runs from 0.562 to 1.5 — half the graph's degrees are wrong by more than a third. And the graph's diameter goes from 1 to 3: no shortest path survives. A sparsifier preserves what the Laplacian says and nothing else about the graph, and the two are easy to confuse because they are the same object.

A fifty-vertex complete graph sparsified by keeping 200 samples drawn in proportion to each edge's effective resistance and reweighted by the reciprocal of its probability, which leaves 187 distinct edges of 1225. The filled curve is the ratio of each eigenvalue of the sample's Laplacian to the original's, in ascending order: it stays between 0.289 and 2.02, so every quadratic form, every cut and every solve is preserved to that factor. The open curve is the ratio of each vertex's degree, which runs from 0.375 to 1.75 — half the graph's degrees are wrong by more than a third. And the graph's diameter goes from 1 to 3: no shortest path survives. A sparsifier preserves what the Laplacian says and nothing else about the graph, and the two are easy to confuse because they are the same object.

A fifty-vertex complete graph sparsified by keeping 100 samples drawn in proportion to each edge's effective resistance and reweighted by the reciprocal of its probability, which leaves 96 distinct edges of 1225. The filled curve is the ratio of each eigenvalue of the sample's Laplacian to the original's, in ascending order: it stays between 0.132 and 2.57, so every quadratic form, every cut and every solve is preserved to that factor. The open curve is the ratio of each vertex's degree, which runs from 0.25 to 2 — half the graph's degrees are wrong by more than a third. And the graph's diameter goes from 1 to 4: no shortest path survives. A sparsifier preserves what the Laplacian says and nothing else about the graph, and the two are easy to confuse because they are the same object.

A fifty-vertex complete graph sparsified by keeping 800 samples drawn in proportion to each edge's effective resistance and reweighted by the reciprocal of its probability, which leaves 599 distinct edges of 1225. The filled curve is the ratio of each eigenvalue of the sample's Laplacian to the original's, in ascending order: it stays between 0.611 and 1.59, so every quadratic form, every cut and every solve is preserved to that factor. The open curve is the ratio of each vertex's degree, which runs from 0.719 to 1.53 — half the graph's degrees are wrong by more than a third. And the graph's diameter goes from 1 to 2: no shortest path survives. A sparsifier preserves what the Laplacian says and nothing else about the graph, and the two are easy to confuse because they are the same object.

A fifty-vertex complete graph sparsified by keeping 1600 samples drawn in proportion to each edge's effective resistance and reweighted by the reciprocal of its probability, which leaves 887 distinct edges of 1225. The filled curve is the ratio of each eigenvalue of the sample's Laplacian to the original's, in ascending order: it stays between 0.654 and 1.38, so every quadratic form, every cut and every solve is preserved to that factor. The open curve is the ratio of each vertex's degree, which runs from 0.719 to 1.31 — half the graph's degrees are wrong by more than a third. And the graph's diameter goes from 1 to 2: no shortest path survives. A sparsifier preserves what the Laplacian says and nothing else about the graph, and the two are easy to confuse because they are the same object.

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.

It draws a decomposition and prints its residual. It calls laplacianEig, sparsify, whatSurvives, 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.