◄ WORLD V · SONNY 5DART 201 · a helldive at the net

THE PAGERANK the web's dominant eigenvector, walked

PageRank is a Markov chain on the web: a random surfer follows a link at random, and with probability 1−d instead teleports to a page chosen uniformly. Rank = the stationary distribution of that walk — the long-run fraction of time the surfer spends on each page. The teleport term guarantees the chain is connected and aperiodic, so the power method converges to one answer whatever the link graph, dead ends and all.

THE TECHNIQUE rank = stationary dist of ( d·follow-link + (1−d)·teleport )

Power-iterate the Google matrix on a small link graph; the ranks settle to the surfer's residence times and sum to one. The most-linked-to page from strong pages wins — not just the most links: live demo







  
HISTORY & CREDIT Brin & Page, 1998

  
“Google invented ranking pages by their eigenvector.” — eigenvector centrality is decades older: Seeley 1949 (sociometry), Katz 1953, Pinski–Narin 1976 (journal-influence weights, the closest ancestor). What Brin–Page genuinely added was the teleport/damping term that turns a broken, dead-end-ridden link graph into one connected chain with a guaranteed unique answer. (That lineage is a retrospective tracing — Franceschet, CACM 2011 — not a citation Brin–Page made; their own papers framed PageRank as citation-counting plus Kleinberg's HITS.) credit both ways

  
1949–76 · Seeley (1949), Katz (1953), Hubbell (1965), Pinski & Narin (1976) — rank nodes by a recursive “important things point to you” eigenvector, long before the web.
1998 · Sergey Brin & Larry Page — add the random-teleport damping factor so the walk is ergodic on the real web, and compute it at scale; their own papers frame it as citation-counting plus Kleinberg's HITS, not as eigenvector centrality.
2011 · Franceschet (“Standing on the Shoulders of Giants”, CACM) and Vigna (“Spectral Ranking”) — place PageRank in the older eigenvector-centrality lineage after the fact; the ancestry above is their retrospective identification.

  
PageRank is a stationary distribution in disguise, and it is computed, not stored: the rank vector is wherever the surfer's walk lands after enough steps. Teleport is the one line that makes “enough steps” exist. Brin–Page 1998






  
RECOMMEND FOR I-13 Google matrix power-iterates to rank, computed

  
On the canonical compiler, a 3-page graph (0→1, 1→2, 2→{0,1}) with damping d=0.85 ranks the pages by residence time — page 1, fed by both others, wins:

  
$ i13 run pagerank.i13   # 3 pages, d=0.85, power iteration
rank[0] = 0.21481062747314866
rank[1] = 0.39739966082532496   -- highest: linked from both 0 and 2
rank[2] = 0.3877897117015262
total   = 0.9999999999999999   -- a probability distribution

  
Recommend: PageRank is LIT for I-13 — verified the Google matrix on a 3-page graph converges to (0.215, 0.397, 0.388) summing to 1.0, ranking page 1 top because it is pointed to from strong pages, not merely often. It is the stationary distribution plus one teleport term — the same power loop.