Google PageRank Algorithm

Linear Algebra (Matrices) · Axiom Academy

The page you click first is decided by one eigenvector — the dominant eigenvector of a giant web matrix. Build it, run it, and watch importance fall out. Billions of pages, no editor ranking them — yet Google puts the right one on top. The trick from 1998: a page is important when important pages link to it . That circular rule is exactly an eigenvector equation, and a tiny 4-page web (A, B, C, D) shows the whole idea. Each page passes its importance equally along its out-links. Stack those rules into a column-stochastic link matrix S , then mix in a small chance the surfer just jumps to a random page. Drag the damping and watch the Google matrix G form — its columns always sum to 1. Power iteration finds the eigenvector The PageRank vector is the one importance distribution that survives a step of G : G = . Because G is column-stochastic its dominant eigenvalue is =1 , so just multiply by G over and over. Start everyone equal and step it. The ranking is where a random surfer lives Here's the payoff. Turn a crowd of surfers loose to wander by those same rules forever. The share of time they spend on each page settles onto the very same eigenvector — so PageRank literally measures how much of the web's attention a page captures . Crank the crowd up and watch it lock onto the ranking.

This is the written version of the interactive lesson above. See the full Linear Algebra (Matrices) course.