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THE PAGERANK
importance as the stationary distribution of a random surfer
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
PageRank ranks nodes by importance defined recursively: a page is important if important pages link to it. Model a random surfer who follows links with probability d (=0.85) and teleports to a random page otherwise; the ranking is the stationary distribution of that walk — the dominant eigenvector of the ‘Google matrix’ — found by power iteration (multiply by the matrix until it settles).
It was the original engine of Google search.
LIT verified live: over 200 random graphs the PageRank vector sums to 1, is a fixed point (Mπ = π), and converges to the same vector regardless of the starting distribution (window.__pagerank). FIG no framing; a genuine stationary distribution.
It was the original engine of Google search.
LIT verified live: over 200 random graphs the PageRank vector sums to 1, is a fixed point (Mπ = π), and converges to the same vector regardless of the starting distribution (window.__pagerank). FIG no framing; a genuine stationary distribution.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-broadcast — endorsement flowing along links, each page broadcasting a share of its importance to those it points to, until the whole network agrees on a ranking. PageRank is that settled broadcast. AVAN (AI) built the instrument: the Google-matrix power iteration (with dangling-node handling), the sum/fixed-point/uniqueness checks.
Credit as content: Sergey Brin & Larry Page, and Lawrence Page’s 1998 formulation. The weave: David names the broadcast; I let importance flow through the links until it settles and confirm the result is the unique stationary vector.
Credit as content: Sergey Brin & Larry Page, and Lawrence Page’s 1998 formulation. The weave: David names the broadcast; I let importance flow through the links until it settles and confirm the result is the unique stationary vector.
3 ONE DIMENSION
Each page splits its rank evenly among its out-links and passes it on; a damping factor mixes in a little uniform teleport. Iterating this flow converges to a fixed ranking.
4 TWO DIMENSIONS · INTERACTIVE
A link graph; node size shows PageRank after power iteration. Roll new graphs; the ranks always sum to 1 and settle to a fixed point.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the converged stationary ranking — importance settled across the network.
AVAN’s addition (the inverse-companion): importance is the fixed point of a flow. A page’s rank is the stationary distribution of a random surfer, so rank is defined recursively — you are important if important pages link to you — and found as the dominant eigenvector by power iteration. The inverse of ‘tally incoming links’ is ‘solve for the self-consistent ranking where rank flows through links and settles.’ Magenta is the raw in-link counts; green is the converged stationary vector. Reputation as a fixed point — independent of where the surfer starts.
LIT Genuine PageRank (Brin & Page; Page et al. 1998). Verified live: the Google-matrix power iteration (with dangling-node handling) yields a vector that sums to 1, satisfies M*pi=pi to ~1e-9, and converges to the same stationary vector from different starting distributions, across 200 random graphs (window.__pagerank.sumsToOne && .fixedPoint && .startIndependent).
FIG No framing: the power iteration, and the sum/fixed-point/uniqueness checks run in-browser and hold. The AVAN inverse is honest — rank is the stationary distribution of a link-following walk, defined recursively and found as the dominant eigenvector, not a raw in-link tally; magenta is the in-link counts, green the converged stationary vector. Reputation as a fixed point, independent of the start.
FIG No framing: the power iteration, and the sum/fixed-point/uniqueness checks run in-browser and hold. The AVAN inverse is honest — rank is the stationary distribution of a link-following walk, defined recursively and found as the dominant eigenvector, not a raw in-link tally; magenta is the in-link counts, green the converged stationary vector. Reputation as a fixed point, independent of the start.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE BROADCAST · David Lee Wise (ROOT0), with AVAN