 So graph theory began with the Königsberg-Bridge problem, which required finding an Euler's circuit for something that isn't actually a graph. And that's because in a true graph, there is at most one edge between any two vertices. But in the Königsberg-Bridge problem, most of the vertices were multi-connected. And so we introduced the idea if some vertices are joined by two or more edges, you have a multi-graph. Now in practice, there isn't a lot of difference between how we treat graphs and how we treat multi-graphs. So we often don't worry about that distinction. But a more important distinction is the following. If some of our edges are one way and you could only proceed in a certain direction along them, you have a directed graph, also called a digraph. Now digraphs are fantastically important in the modern world because they are a good way to represent web pages on the Internet. A web page, or a website, is a node, and outgoing links are edges leading from the web page to another web page. And this leads to a question whose answer is worth billions of dollars. How do you rank web pages? So we can model Internet traffic using a discreet time model. The number of visitors to any given website at time t equals k depends on the number of visitors on pages that link to the website at t equals k minus 1. It also depends on the number of visitors who stay on the page. If we make the additional assumption that surfers aren't lost, this leads to a useful measure of the importance of a web page. How many visitors are on the web page? For example, let's consider the three web pages shown, and let's assume this is the entire Internet. And we'll make a few assumptions. First, we'll assume that nobody stays on a web page. That outgoing traffic is split evenly among the outgoing links. So if we begin with 1,000 visitors on each page at t equals 0, how many visitors will be on each page at t equals 10? So let's set up a table to track the number of surfers on each page, and at t equals 0, each page has 1,000 surfers. So the important thing to remember is that it's easier to figure out where you've come from than where you're going. So let's consider the people who end up at the three different web pages. So at t equals 1, website A is going to get all 1,000 surfers from website C. And that's because the only outgoing link from C goes to website A. Now to keep us from miscounting, it helps to think about these surfers as leaving C and en route to A. So they're in the wilds of the information superhighway. And the data packet is asking, are we there yet? Also at t equals 1, website B is going to get its visitors from A. But since there are two outgoing links from A, the 1,000 visitors on A will split into two equal parts so that B only gets half of 1,500 visitors. And at t equals 1, website C gets visitors from both A and from B. It gets the other half of the traffic from A, that's these 500 visitors. And since the only outgoing link from B goes to C, then C also gets all 1,000 visitors currently on B. And so now if we let these surfers actually get to their destination, we find A has 1,000 visitors, B has 500, and C has 1,500. So next, everyone on C will move towards A. Again, that's the only outgoing link. Since there's two outgoing links from A, half the people on A will move towards B. And the other half, and all of those on B, will move towards C. And when everyone gets to where they're going, A has 1,500, B has 500, and C has 1,000. And we'll continue the process. Everyone on C will move towards A. Half the people on A will move towards B. And the other half, and all of those on B, will move towards C. And when they arrive, A has 1,000, B has 750, and C has 1,250. And we can continue to find the numbers on the different websites. Now, notice that while the numbers seem to be random at first, they do settle down into a pattern. And we see that most of the traffic ends up at A or C, and much less ends up at B. So if we wanted to rank the importance of the webpages, we might rank them A is the most important, because it ends up with the most traffic, followed by C, which has almost as much, with B at the end. And here's how to make a billion dollars. With several billion webpages and millions more created every day, it's important for a web search to return relevant results. The discrete-time model allows you to rank webpages based on where web traffic ends up. So you can use it to rank webpages. A patent on this process could be worth billions. Too late, it's already been patented. This is, in fact, how Google ranks webpages. And here's how to lose a billion dollars. With the mathematics you can learn in half an hour, you could have invented Google, which should be very concerning if you're Google. Because the question that keeps Google shareholders up at night, is there a half-hour invention that will make Google obsolete? Now, this is something that's very concerning to them, and so their best solution is to make sure they employ the people most likely to make that half-hour invention.