The random surfer
Six web pages. Click a page, then click another, to add or remove a link between them; the link points from the first page you click to the second. Press One step to watch the ranks move, or Run until it settles. A surfer starts anywhere and keeps clicking random links. Where does the surfer spend their time? That fraction is the page's rank, and it is how Google first ordered the web.
What's going on
Each page shares its rank equally among the pages it links to. If page A has rank 0.3 and three outgoing links, each of those pages receives 0.1 from A. Add up what every page receives, and you have the ranks after one more click. That is a Markov chain: the next state depends only on the current one, through a fixed table of probabilities called the transition matrix.
Repeat the step and, usually, the numbers stop changing. A vector that no longer moves is a steady state. The step button lets you watch it happen; the readout shows the largest change each time, shrinking toward zero. The Run button stops when that change drops below one millionth, and says so honestly if it gives up first.
Two problems and one fix. A page with no links out is a dead end, and a group of pages linking only to each other is a trap that would swallow all the rank. So with probability 1 − d the surfer ignores the links and jumps to a page chosen uniformly at random; d is the damping factor, and Google used 0.85. A dead-end page is treated as if it linked to everyone. For any d strictly less than 1 every entry of the matrix is positive, and then the chain has exactly one steady state and reaches it from any starting ranks. That is the guarantee; it needs 0 ≤ d < 1.
Try it: make every page link to page 1 and watch it dominate. Then add a single link from page 1 to page 2 and see how much rank flows along it. A link from an important page is worth more than a link from an obscure one, which is exactly the idea.
Experiment at d = 1. Press Star web: page 1 links to every other page and each of them links back. Slide d to 1, so the surfer never jumps, and press Run. The ranks swing between (1/6, 1/6, …) and (5/6, 1/30, …) forever, because every path from page 1 back to page 1 has even length, so odd and even steps never agree. The chain still has a steady state, (1/2, 1/10, 1/10, 1/10, 1/10, 1/10), but repeated stepping never finds it, and the last vector you see is not the ranking. Now nudge d down to 0.95 and watch the swing shrink step by step. Positive entries break the rhythm.