pagerank-converge
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.
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.
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.
kind: "reducible"
The arguments are the ones A chain with no stationary vector passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.
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.
alpha: 0.5
The arguments are the ones A chain with no stationary vector passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.
The PageRank power iteration on a connected fifty-vertex graph, 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) = 0.90238 here. The measured contraction over the run is 0.45086. 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 3.82·10⁻¹⁷.
alpha: 0.99
The arguments are the ones A chain with no stationary vector passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.
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.
alpha: 0.85
The arguments are the ones A chain with no stationary vector passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.
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.
alpha: 0.7
The arguments are the ones A ranking that is an eigenvector passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.
The PageRank power iteration on a connected fifty-vertex graph, at α = 0.7, 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 0.63139. 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 6.25·10⁻¹⁷.
What it checked while drawing
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.
22 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 observed contraction is α·λ₂(P) = 0.767 — asserted 6 times
a graph with at least two vertices
a planted partition with no isolated vertex
a positive weight
a teleportation parameter a power iteration can be run at
a teleportation parameter strictly inside (0, 1)
an arc inside the vertex set
every endpoint inside the vertex set
Jacobi needs a symmetric matrix
LU is for square matrices
no arc given twice
no edge given twice
no isolated vertex, which the normalisation divides by
no self-link
no self-loop
one of the two link graphs
the iteration and the elimination agree
Against the rule
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 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.
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.
A chain with no stationary vector
A page with no outgoing links loses forty per cent of the walker's probability in six hundred steps. A directed cycle never converges at all. And on a graph whose links only run one way, the entire rank of half the vertices is exactly one minus the teleportation parameter.
The matrix that is a graphA ranking that is an eigenvector
PageRank is the stationary vector of a walk that follows links with probability α and jumps at random otherwise. The iteration and the elimination agree to 4·10⁻¹⁷. What α is set to changes which pages come third, fourth and fifth.
The matrix that is a graphA ranking whose order is not determined
Of the fourteen adjacent comparisons in a top-fifteen, fourteen survive perturbing the arithmetic at the rounding level, eleven survive moving the teleportation parameter across its usual range, and three survive removing one link. The computation is the strongest part of the answer.
The matrix that is a graphAn eigenvector that must not change sign
Perron's theorem says the leading eigenvector of a connected nonnegative matrix is strictly positive. On a clique with a long tail, four of its thirty-six entries come back negative — and beside them is the one two-sided bound on this site that is proved rather than estimated.
The matrix that is a graphThe rate is the second eigenvalue
A walk forgets where it started at a rate the graph's second eigenvalue names exactly. Across three orders of magnitude in the step count the prediction is five per cent high — and the published rate for PageRank is right for a reason nobody states, which is that a link graph is in pieces.