PageRank

PageRank scores web pages by importance using one idea: a page is important if important pages link to it. Picture a surfer clicking links at random forever; PageRank is the fraction of time spent on each page in the long run. Popular pages, and pages cited by popular pages, accumulate visits.

Concretely, build a column-stochastic link matrix where column j spreads page j's vote equally among the pages it links to. To guarantee a unique answer, blend in random teleportation: M = d * P + (1 - d) * (1/n) * J, where d is around 0.85, J is all-ones, and 1/n is a uniform jump. M is now a positive stochastic matrix, so Perron-Frobenius gives a unique positive dominant eigenvector.

PageRank is exactly the stationary distribution of M: the eigenvector for eigenvalue 1, equivalently the limit of M^k applied to any start. In practice you compute it with the power method, iterating r -> M r until it settles, which is feasible even for billions of pages because M is sparse plus a rank-one correction.

Why it matters: PageRank turned eigentheory into a search empire and the same recipe ranks anything modeled as a graph (citations, social influence, road importance). The caveat is that it is gameable (link farms) and topic-agnostic, which is why modern ranking layers many other signals on top.