So, now let's summarize the importance score findings, and compare again back to the in degree measurement that we had in the beginning. So, and here we're showing in each node the importance scores that we got. For W 4 over 31, X is 6 over 31, Y 9 over 31, Z 12 over 31. You can see how they split out. So 4 over 31 goes to 2 and 2 and so on. And then 6 goes to 3 and 3, 12 goes to 4, 4, and 4, and things along those sorts. So, here are the equations again, that we have, and you can verify that each of these equations are satisfied, right? Because we were using those other equations that we got, the intermediate values in terms of z, to solve for these important scores, because it was just easier to plug back into them. But you can go back through and say for instance w has to be equal to z over 3, and you said z, z was 12 over 31. So, 12 over 31 divided by 3 is 4 over 31. Which we said is the important square of w. X is w over 2, which is 2 over 31. This is 2 over 31 plus z over 3. Which is giving you 4 over 31. So, that'll give you 6 over 31, which is x's value here. Now, this equation 2, which we didn't actually use, if you recall. We didn't use this equation for y here. But it's w. Divided by 2 which is 2 over 31 plus x over 2, which is 3 over 31 plus z over 3, which is just going to be 4 over 31. And then that should give us 9 over 31 for y which is what we thought. And z is x over 2, which is 6 over 31 divided by 2, which is 3 over 31. Plus y which is 9 over 31, which is going to be 12 over 31. So you can see that each of these equations are valid, exactly as we thought. So they all do hold. So here's a, a tabulated summary of the difference between the In-degree and the PageRank calculations. The PageRank we're showing on the, on the right side over here. And we see all these important scores and these sum up to be one, and both of them do, actually. Both of the important score sides we sum them up to one. Just makes it more convenient to look at it that way. So the rank is fourth, third, second, and first. Whereas, over here you saw before w's still 4th. That didn't change, but now the order is a little different for the rest of them, right? So, we before we had x and z tying for 2nd. Because x and z both had In-degrees of 2. And we had y with the highest, because it had In-degree of 3 before. So, you can see a difference here. [COUGH] The way that we're getting these normalized values is by dividing by the total number of in-degrees, so the total number of length space just to get them to sum up to one, right? W is going to have a value of 1 8ths because it has one of the eight lengths pointing to it. X is going to get 2 8ths. z is going to get 2 8ths and y is going to get 3 8ths. That will all add up to 1. So, so the main differences from the in-degree first that is, y is no longer the most important. 'Kay, so before we had y being the most important because we said, well It has three links incoming. But, it's no longer the most important. The reason is, is because now we're taking into account who's pointing to who. And how important the nodes are that are pointing to me, right. So, the reason that y really isn't the most important anymore is that, z, which is now the most important node, is spreading only a third of it's importance to y. Right, this 4 over 31 that it gets. And it's only taking half from w and x which are, which are like the middle important nodes, so it's not even getting their full importance. So it's aggregating importance from everyone but it's only getting a small piece of that importance, all right. And particularly with z being the most important node. And now, why is z the most important node? Well. Y, y, we said is the, third, is the second most important now, and z's going to get all that importance score, nine over 31, all of it, because y is only pointing to z. So when you land on y, the next step has to be go to z, and it's also getting some from x. Right. So it has to be greater than y. It has to be more important than y, because not only does it have all of y's importance, but it has something a little greater. So, we can see already that z, needs to be greater than y importance, and that's, that's why that's the case here. Really, we should emphasize that again. It's the importance of the links that are pointing to you that matters. The importance of the nodes that are you pointing to you that matters, because then those links are going to get used more often, and therefore you're going to be a higher importance as your own nodes. So it's not just the number of the links that you have in, it's the quality of those links, in terms of importance, that makes us to ultimately determine which nodes are important and which aren't. At least in the context of Google, and this is, again, this is how Google's page ranking algorithm works, in terms of solving these equations, in terms of their importance scores. There's many different ways of defining importance. So, it's not an easy clear cut problem, so that's why consensus is hard. Because there's not easy to determine what the right answer should be.