So the key point that we've yet to take away here, is that with additional random links. We get to preserve the large clustering coefficient, while still achieving that small world effect. But the question we're going to ask is, when we're establishing these long range links. Doesn't the, don't the additional links reduce the clustering coefficient? Because they're going to create more connected triples, but not triad closures. Right? Because if we're connecting people that are far away from each other, right? They're not likely going to have the same people connected, especially in this regular graph structure. Right. So this person is not likely going to have that connection out here unless, another random like is established in head created that triangle already. And the answer to that question is yes, it's going to reduce the clustering coefficient, but not as substantially and not as quickly as it's going to decrease the average shortest path distance. And to see that we can plot right here the case where we have 600 nodes, and six links per node. So it's for that type of a regular graph structure. As the probability varies, we're not going to mathematically derive anything about how the probability is going to change the clustering coefficient of the average for this path. But this is how it would in this case, for six nodes and six links per nodes. And it's very similar in the other cases as well for other types of nodes for other number of nodes and links per node. So, in this case here, we see the clustering coefficient in between this range of the clustering coefficient and the average shortest path distance are both going to decrease as the probability increases always. And that obviously, should make sense. Because as we establish for our long range links, we're going to able to get from one side of the graph to the other. Hence, reducing the average sort of assistance. And additionally, we're going to create more connected triples without creating triad closures, which is hence going to reduce the clustering coefficient. But, you have to look at how much quicker the clustering coefficient, or the average shortest distance, is decreasing relative to the clustering coefficient. So, the average shortest distance is going down very, very fast and whereas the clustering coefficient's going down much less quickly. And in this range right here of about 0.01 so there's about a 1% chance of establishing a link to 0.1 which is about a 10% chance of establishing a link in this range. And here we're going to have a large clustering coefficient and a short and a small average short assistance in so in any of this region here. So, when the probability is about 0.1 or right around there, we're going to get what we want, because the probability hasn't increased too much yet, where we're going to really see a decrease in the clustering coefficient. So a small amount of randomization is going to reduce the average short assistance dramatically. But it's not going to impact the clustering coefficient that much. So you're probably wondering why is this that the probabilities can reduce the one so much, but not going to impact the clustering coefficient that much. And fundamentally, it has to do with the very definition of our metrics that we talked about earlier. The shortest path that we have, the average shortest path, is extremal quantity. Right? And what that means is that, we only are looking at the shortest path. Right? So if we establish one long range link, there's going to be a lot of nodes that are going to have the shortest path reduced to other pairs in the other side of the network. And it has to do with the fact, that of taking that one extreme measure of using shortest rather than just an average distance. But note, that we're still averaging overall nodes, but it's just the quantity we are averaging is the shortest distance to begin with. On the other hand, the clustering coefficient is an average quantity. Because it's really the number of triangles divided by the number of connected triples, and as we saw for a regular graph, we can do that as an average of each node. So adding a small proportion of non-triangular connected triples is not going to hurt. The clustering coefficient that much, because for the majority of the other nodes it's still going to be the same situation.