So now, let's see if we can come up with a type of graph that has the same properties as this apparent small world phenomenon. Right, so having some amount of philmophely, but also having small world in the fact that, in the sense that we can get from one node to another in a small amount of hops on average. So having such a model is obviously going to help us explain what the networks going to look like. There going to give rise to this marvel of phenomenon in the first place. Then we can say that the sixth degree of separation makes sense, if we can come up with a model to explain it. So we need this small world graph we come up with to be small. The question is how do we measure the network size to say that it's small in the first place. There's, there's a couple different options here. One would be to go with the diameter. What you have explained before is the longest of the shortest path lengths. Now, that might sound like a mouthful, but really, it's just taking each pair of nodes, we find the shortest path between the pair of nodes and we take the longest of those. So what's the worst case scenario basically. All right, so if we have a group of nodes, for instance, just do a very simple example of five nodes. Suppose they're all connected, these guys are all connected this way. And maybe this node is just connected to this node here. So, the longest of the shortest paths is going to be just getting from this node to this node. It would be 1, 2, 3. So, the diameter of this graph here would be 3. All right, but the diameter is really a very extreme measure, because suppose that this wasn't all the nodes in the graph. Suppose there were like billions of nodes here, right, and suppose that also the shortest path length was also going between this node and this node, or really between this node and this node, it wouldn't make a difference. So the diameter is the same whether we consider each of those two pairs in order to find the diameter. But now suppose that we added just one more node here, with a link like this. 'Kay, so now the diameter has gone up to four, right, because it's going to take 1, 2, 3, 4 to get from this node to this node, or from this node to this node, but. So, we've, we've just added one node. And we said before that there's still billions of other nodes, that doesn't even matter. Because this before was already the longest path length, still. So all the other nodes are clustered closely. So just by adding one node, we've increased it from 3 to 4, which is a high percent increase for this, just for this this graph. Especially considering that there's billions of nodes, which I haven't drawn here. But, so, we can consider another quantity that would not be such an extreme measures, this is a very extreme measure, the diameter. Which is the average shortest distance that we looked at before in the context of Facebook. It's not as extreme of a quantity, it's still. Is more extreme than other things we could measure, but basically what this is going to look at is, now we take all the shortest s, s, distances between all pair of nodes. Rather than finding the maximum of each of those, we just take the average of all of them, right. So still, we're still taking extreme in one sense. And I hope this isn't, doesn't get too confusing, but basically, there'd be two levels of extreme here. In this case, we're doing both levels because we're taking the maximum of the shortest path length. In this case, we're just taking the average of the shortest path length. So, in the once, the fact that we're just looking at the shortest path is extreme, because there are other ways to get between the pairs of nodes. But we only really care about the shortest path which is why we take that one. But then we're averaging them, so we're not doing that second level of the extreme which would have been to take the maximum of all the shortest path lengths. We're just averaging them. So this isn't as extreme of a quantity. So, just to give an example of how to compute the average shortest distance we'll just take a look at an example of a simple network here. So the diameter of this network if we look, is just going to be 3, because the longest of the shortest path lengths is going to be 3. Right, to get from A to C to D is 3, to get from E to B is 3. There's, there's many examples of there being three there. Well let's, let's find the, the average shortest distance here. And to do that, we have to consider all the different node pairs. So we'll do A to B first, so from A to B there's a direct link, right? Making the shortest path from A to B just one. From A to C there's also a direct link, which makes the shortest path one. Now, from A to D, we can't get there in just one. Oh and by the way, I said the diameter was 3 to 4. My apologies, it's actually two, because we just go A, C, D, which makes the diameter of the graph, just two not three. So the diam, the diameter of the graph in this case is 2. So we'll write diameter equals two. Sorry about that. So, now from A to D, right, the shortest path is going to be over A, C, D, right? So that's the shortest path which has a length of 2. From to A to E the same thing the same length A, C, E. So, it's the shortest path of two. Now, we can similarly do it for all the other pairs in the graph. For instance, we can say from B to C, that's just going to be 1. From B to D, that's going to be 2, right, going from B, C to D. From B to E, that's also going to be 2. From C to D, that's going to be 1. From C to E is also going to 1 and from D to E is also going to 1. Note that we don't have to consider the reverse paths, because going from A to E is the same, as going from E to A, so there's no point is considering everything twice. So, to find the average, all we have to do is just add them and divide them by the total number. So we do 1 plus 1 plus 2 plus 2 plus 1 plus 2 plus 2 plus 1 plus 1 plus 1. We divide that by 10, since there's 10 node pairs here, and if we do that out, you get 14 over 10, which is 1.4. So the answer here is 1.4. So the average shortest distance in this case, average shortest distance is equal to 1.4. So what does this tell us? What it's saying is that with the judicious, judicious choice of path. So, if the nodes know how to get from shortest destination correctly, and they can choose the path correctly through some, some way, it's going to take an average of 1.4 steps to get from one node to another node, on average. So, that's going to be, that's intuitive here right, because all of the shortest path line, so either one or two. So, obviously it's got to be somewhere between ore or two. But, the question is, what's going to happen as the number of nodes grows, and we get to a really large number. So, as the number of nodes grows, knowing what this length is can be important, and in certain circumstances it can also be pretty surprising. And so we'll take a look at that next