Ranking pages.

Step 1 of 6

The random surfer

The web here has eight pages, and each arrow is a link from page to page. When you search for something, a search engine finds millions of matching pages, and it has to decide which comes first. The idea of PageRank, on which Google was built, is to see the links as a directed graph and imagine a random surfer: at each step it clicks one of the current page’s links, at random, or, now and then, jumps to any page at random. A page’s PageRank is the share of its time the surfer spends there in the long run. The height of each page’s plinth is its rank.

Right now Chance of clicking a link is 0%: the surfer only jumps. Predict what happens when it starts clicking.

Step 2 of 6

Quiet links

No page links to your page, so the surfer reaches it only by jumping. How much will the page gain from links?

Step 3 of 6

A link from the hub

Each page splits its rank among the links it gives. So what matters is not how many links reach a page, but where from: the formula is

$PR(p) = \frac{1 - d}{N} + d \sum_{q \to p} \frac{PR(q)}{L(q)}$$PR(p) =$$\frac{1 - d}{N} + d \sum_{q \to p} \frac{PR(q)}{L(q)}$⁠

where $d$⁠ is the chance of clicking a link, $N = 8$⁠ the number of pages, and $L(q)$⁠ the number of links page $q$⁠ gives. A page’s rank depends on the ranks of the pages linking to it, which depend on the ranks of the pages linking to them, and so on.

Step 4 of 6

Climbing

The links to your page come in a fixed order: three quiet pages, then the hub, then the news site. Climb the ranking.

Step 5 of 6

What a link passes on

Read the formula from right to left: page $q$⁠ passes the amount $\dfrac{d \cdot PR(q)}{L(q)}$⁠ through each of its links. The $\dfrac{1-d}{N}$⁠ at the front is the random jumps, which give every page a little, even a page nobody links to.

Step 6 of 6

The surfer settles

How do you work out the ranks, if each depends on the others? You can simply let the surfer walk, and count. Look inside brings in the surfer, and beside each plinth a bar of the share of visits that landed on that page. So far it has made only a few dozen clicks.