Contracting by 0.7668 where the folk rate says 0.85

At its defaults it draws contracting by 0.7668 where the folk rate says 0.85. The PageRank power iteration on a connected fifty-vertex graph, at α = 0.85, plotted as the change between successive iterates. The upper dashed line is αᵏ, the rate every account of the method quotes. The lower one is (α·λ₂(P))ᵏ, which is what the second eigenvalue of the Google matrix actually is: λ₂(P) = 0.90238 here. The measured contraction over the run is 0.76683. The two agree exactly when λ₂(P) = 1, which happens exactly when the link graph has more than one closed communicating class — which every real one does, and which is why a statement that is false of connected graphs has never been noticed to be. The answer is checked against one elimination of I − αP̃, which agrees to 1.04·10⁻¹⁶.

pagerank-converge 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.

The PageRank power iteration on a connected fifty-vertex graph, at α = 0.85, plotted as the change between successive iterates. The upper dashed line is αᵏ, the rate every account of the method quotes. The lower one is (α·λ₂(P))ᵏ, which is what the second eigenvalue of the Google matrix actually is: λ₂(P) = 0.90238 here. The measured contraction over the run is 0.76683. The two agree exactly when λ₂(P) = 1, which happens exactly when the link graph has more than one closed communicating class — which every real one does, and which is why a statement that is false of connected graphs has never been noticed to be. The answer is checked against one elimination of I − αP̃, which agrees to 1.04·10⁻¹⁶.

The PageRank power iteration on a link graph in two closed pieces, at α = 0.85, plotted as the change between successive iterates. The upper dashed line is αᵏ, the rate every account of the method quotes. The lower one is (α·λ₂(P))ᵏ, which is what the second eigenvalue of the Google matrix actually is: λ₂(P) = 1 here. The measured contraction over the run is 0.84123. The two agree exactly when λ₂(P) = 1, which happens exactly when the link graph has more than one closed communicating class — which every real one does, and which is why a statement that is false of connected graphs has never been noticed to be. The answer is checked against one elimination of I − αP̃, which agrees to 1.39·10⁻¹⁶.

The PageRank power iteration on a link graph in two closed pieces, at α = 0.5, plotted as the change between successive iterates. The upper dashed line is αᵏ, the rate every account of the method quotes. The lower one is (α·λ₂(P))ᵏ, which is what the second eigenvalue of the Google matrix actually is: λ₂(P) = 1 here. The measured contraction over the run is 0.48946. The two agree exactly when λ₂(P) = 1, which happens exactly when the link graph has more than one closed communicating class — which every real one does, and which is why a statement that is false of connected graphs has never been noticed to be. The answer is checked against one elimination of I − αP̃, which agrees to 5.2·10⁻¹⁷.

The PageRank power iteration on a link graph in two closed pieces, at α = 0.9, plotted as the change between successive iterates. The upper dashed line is αᵏ, the rate every account of the method quotes. The lower one is (α·λ₂(P))ᵏ, which is what the second eigenvalue of the Google matrix actually is: λ₂(P) = 1 here. The measured contraction over the run is 0.89177. The two agree exactly when λ₂(P) = 1, which happens exactly when the link graph has more than one closed communicating class — which every real one does, and which is why a statement that is false of connected graphs has never been noticed to be. The answer is checked against one elimination of I − αP̃, which agrees to 1.53·10⁻¹⁶.

The PageRank power iteration on a connected fifty-vertex graph, at α = 0.85, plotted as the change between successive iterates. The upper dashed line is αᵏ, the rate every account of the method quotes. The lower one is (α·λ₂(P))ᵏ, which is what the second eigenvalue of the Google matrix actually is: λ₂(P) = 0.90238 here. The measured contraction over the run is 0.76683. The two agree exactly when λ₂(P) = 1, which happens exactly when the link graph has more than one closed communicating class — which every real one does, and which is why a statement that is false of connected graphs has never been noticed to be. The answer is checked against one elimination of I − αP̃, which agrees to 1.04·10⁻¹⁶.

The PageRank power iteration on a connected fifty-vertex graph, at α = 0.99, plotted as the change between successive iterates. The upper dashed line is αᵏ, the rate every account of the method quotes. The lower one is (α·λ₂(P))ᵏ, which is what the second eigenvalue of the Google matrix actually is: λ₂(P) = 0.90238 here. The measured contraction over the run is 0.89332. The two agree exactly when λ₂(P) = 1, which happens exactly when the link graph has more than one closed communicating class — which every real one does, and which is why a statement that is false of connected graphs has never been noticed to be. The answer is checked against one elimination of I − αP̃, which agrees to 2.84·10⁻¹⁶.

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.

25 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 walkSpectrum, pagerankSolve, 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.