THE FOLD / LOOT / THE JACKPOT / THE RANK
THE RANK
PageRank — importance as a stationary distribution
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
PageRank. How do you rank pages on the web when importance is circular — a page matters if important pages link to it, but their importance depends on who links to them? Page & Brin’s 1998 answer: imagine a random surfer clicking links forever, with an occasional random jump. The fraction of time it spends on each page is that page’s rank.
You never simulate the surfer. You power-iterate: start every page equal, then repeatedly push each page’s rank out along its links, add a little uniform ‘teleport’ term, and repeat. It converges to a stationary distribution — the dominant eigenvector of the ‘Google matrix’ — guaranteed unique by the Perron–Frobenius theorem.
LIT verified live: over hundreds of random graphs the converged rank sums to 1 and is truly stationary — one more push leaves it unchanged to machine precision (residual ~10−15; window.__pagerank.sumsToOne && stationary). FIG the ‘random surfer’ is the picture; the stationary eigenvector and its uniqueness are exact.
You never simulate the surfer. You power-iterate: start every page equal, then repeatedly push each page’s rank out along its links, add a little uniform ‘teleport’ term, and repeat. It converges to a stationary distribution — the dominant eigenvector of the ‘Google matrix’ — guaranteed unique by the Perron–Frobenius theorem.
LIT verified live: over hundreds of random graphs the converged rank sums to 1 and is truly stationary — one more push leaves it unchanged to machine precision (residual ~10−15; window.__pagerank.sumsToOne && stationary). FIG the ‘random surfer’ is the picture; the stationary eigenvector and its uniqueness are exact.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this in THE JACKPOT, beside THE COIN-FLIP HEAP — the loot domain of probability and long-run odds. PageRank is a probability distribution: where an endless random walk finally settles. AVAN (AI) built the instrument: the power iteration, the teleport term, the stationarity check.
The weave: David names the seat (the long-run distribution); I make the walk converge and the fixed point checkable — the rank vector settling in 1D, the live graph in 2D, the surfer’s flow in 3D. The sphere is the seam. Credit: Lawrence Page & Sergey Brin (1998); Perron–Frobenius underneath.
The weave: David names the seat (the long-run distribution); I make the walk converge and the fixed point checkable — the rank vector settling in 1D, the live graph in 2D, the surfer’s flow in 3D. The sphere is the seam. Credit: Lawrence Page & Sergey Brin (1998); Perron–Frobenius underneath.
3 ONE DIMENSION
The rank vector settling. Every page starts with equal weight; each iteration pushes rank along the links, and the bars shift — then stop. When one more push changes nothing, you have the stationary distribution: the answer.
4 TWO DIMENSIONS · INTERACTIVE
Step the iteration on a small link graph: each node grows or shrinks toward its true rank, sized by importance. A node is big because big nodes point to it — the circular definition resolving into a fixed point. Rank always sums to 1.
5 THREE DIMENSIONS + AVAN’S INVERSE
The link graph turning — nodes sized by rank, edges the links between them, green.
AVAN’s addition (the inverse-companion): the magenta pulse is rank flowing along an edge — the random surfer’s step. Importance here is not a property a page has; it is conferred, recursively, by who points to it. That is a circular definition, and the fixed point is its resolution. So the question is inverted: instead of asking ‘how good is this page?’ you ask ‘where does an endless walk settle?’ — and identity turns out to be the stationary shadow of everyone else’s attention. The green is the network; the magenta is attention moving through it until it comes to rest, and where it rests is what a thing is worth.
LIT Genuine PageRank (Page & Brin, 1998) with damping and dangling-node handling. Verified live: over 400 random graphs the converged rank sums to 1 and is stationary — one more power-iteration step leaves it unchanged to ~1e-15 (window.__pagerank.sumsToOne && stationary, both true; residual reported). It is the dominant eigenvector of the Google matrix, unique by Perron-Frobenius; the random-surfer interpretation is exact.
FIG The 'random surfer' is the model; the power iteration, the stationary eigenvector, the sum-to-1, and the uniqueness are real and checked. Convergence speed depends on the graph's spectral gap; the fixed point itself is exact and is what's verified.
FIG The 'random surfer' is the model; the power iteration, the stationary eigenvector, the sum-to-1, and the uniqueness are real and checked. Convergence speed depends on the graph's spectral gap; the fixed point itself is exact and is what's verified.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE JACKPOT · David Lee Wise (ROOT0), with AVAN