So the general formula for the clustering coefficient of a regular graph, without going into the details of how this is derived, is going to be 3 times the number of lengths per node, minus 2 over 4 times number of lengths per node, minus 1. And so, this is going to be the clustering coefficient for a regular graph, and we, we can plot this. So for the, as the number of lengths per node increases, what the clustering coefficient does, and remember the number of lengths per node has to be even because it's symmetric. So we go 2 then this jumps to 4, 6, 8, 10, 12 and so forth. Now from this equation that we can verify very quickly the two cases that we just did. And the case with links per node is 2 and then we get a ring. We would have 3 times 2 minus 2 over 4 times 2 minus 1. The top is going to be 0 because we're getting 3, 0 which is 0. That makes sense with what we had computed before. And then, on the other hand in the other case where we had number of links per node being 4, which is the second case that we did, we had 3 times 4 minus 2 over 4 times 4 minus 1. And that one's going to come out to be, this is 3 times 2 over 4 times 3, which the 3's are going to cancel out and it's going to leave us one-half. So, that makes sense also with what we computed before. And it's shown here in this graph. So, as we used to have two links per node. That's going to give us clustering coefficient of 0. Shutting up the 4 is going to gives us the clustering coefficient of one-half. And we get up to 0.6 with 6 and so forth. And then it's going to peak out at roughly 0.75. That's going to be there limit here as we increase the number of links per node. And the largest possible is 0.75. So if, if we take the limit as the number of links per node approach is just the number of nodes we're going to get 0.75. And this is going to be, this is realistic for social networks. This is much higher than what we saw in the case of the random graph pattern way. Potentially 0 for the clustering coefficient. The downside is that the regular graph is also going to have a large average shortest distance. So the average shortest path length is also going to be large. And the reason for that is simply because, well, there's only short-range connections here, because as we've said, each node is only going to be establishing to their nearest neighbor nodes, right? So in this case of having four links per node, we just have a connecting to his nearest two followed by his second nearest two and, and so forth. So, in order for A to get to B, it's to get across half the graph onto this side. It's going to have to jump, it'll hop one, two, three, four. Already four hops to get to B. And considering the total number of nodes it's a pretty large number of hops that's required. And again, as we said, this is because that there's only short range connections in this regular graph. So the question you're probably asking is, why not just increase the number of links per node. This should be, sorry, this should be links per node. Because as the links per node goes up clearly the average shortest path distance is going to go down, right, because we're going to have more connections. And if A for instance had 8 of his neighbors connected, maybe that would be a much shorted path then to get to B because you could jump either right to B or to someone right next to B. And additionally the clustering coefficient is just going to go up, so we're going to get the best of both worlds in that case. But you have to think of is what that would imply, and really what the number of links per node is going to be in a realistic network setting. All right. Because on Facebook what that would be implying is that, if the links per node is approaching the number of nodes, that you're connected with the majority of the population, which is obviously not going to be the case. You're only have maybe 300 friends as opposed to the billions that are out there. So that's also not going to be realistic. So we can't just increase the number of lengths per node, because that's going to violate the other constraint that we have here, realistically, of us needing to keep the number of length per node small relative to the number of nodes. [SOUND] So now if we compare the random and the regular graph, we see that each of them has one property that we like and one property that we don't like. Right? The random graph has a small average shortest path distance, but it also has a small clustering coefficient which we don't want. The regular graph has a, has a large average shortest distance which we don't want, but it also has a large clustering coefficient. What we want is we want to be able to have a model where we have a small average shortest path distance and a large clustering coefficient without making there be too many lengths per node as we just said. So the next question we have to ask is it possible to have a hybrid graph? So hybrid graph. That combines properties of the random and the regular graphs to get the best of the both worlds. Having the large clustering coefficient, but a small average shortest distance, and we're going to look at such a model next.
