Google’s founding idea: a page is important if important pages link to it. Imagine a surfer clicking random links forever — the fraction of time spent on each page IS its rank. Importance flows along links and pools where the network points. It’s a self-referential definition solved by just iterating until it settles. Slide the iterations and watch the ranks converge.
PageRank models a ‘random surfer’: with probability d (~0.85) follow a random outlink, else jump to a random page. Each node’s rank is PR(v) = (1−d)/N + d·Σ(PR(u)/outdeg(u)) over inlinks u — the STATIONARY distribution of that Markov chain, found by power iteration until it converges. Rank flows along links and concentrates on well-linked nodes; the damping term guarantees a unique solution (and handles dangling nodes). It ranked the early web. A fail-loud self-check throws unless the ranks converge, sum to 1, and the most-linked node ranks highest. ◆ real network science, node-verified.
A tiny 4-node graph illustrates it; real PageRank runs on billions of nodes and adds dangling-node and teleport handling. The stationary-distribution definition and convergence are exact.