There have been several models for social search in the past decade beyond the original Watts-Strogatz model. And social search is the process again of discovering those short path lengths. So we're going to examine one here in particular called the Watts-Dodds-Newman model. So, Duncan Watts, who is very famous in the space by the way, part of the Watts-Strogatz model and also part of the Watts-Dodds-Newman model. And so here different people live in different roots of a binary tree. And so we're going to, let's explain what that means a little more. But basically a tree or a tree graph is basically one where we can go from a root node at the top and we can come down to some leaf nodes at the bottom. So basically every we can trace from top to bottom of the of the tree in other words. Of the graph. There's just a graph with a special type of structure. And, so for instance we wouldn't have this node being connected to another another node above it. So each of these nodes is going to have a certain number of nodes below it, and the nodes below it aren't going to be connected to any other nodes. They just have one parent node basically. So, this is a parent node, and these are the children of that parent node. So we have different levels of the tree. And a binary tree is one in which we've drawn here where there's only two children for each parent. So we could draw one where there's three children for each parent also. Which would be something looking like this. [SOUND] But here it is it's just a binary tree. This is two children for each parent. So here we have people differing, living in different leaf nodes of a binary tree and the leaf nodes are the once that are all the way at the bottom. So the leaf nodes have no children, they only have parent nodes. And so here these are the leaf nodes. We have, we've drawn eight of them in this case. But gen, in general the number of leaf nodes is going to be a variable. And also the number of the levels in a tree is going to be a variable also. So, within a leaf, we're going to say that everyone is connected. So, in Leaf 1, for instance, we have A, B, C, D. They're all connected, they all have connections with one another. They all know each other, they're close geographical proximity, however you want to call it. In two, E, F, G, H, they're going to know each other as well. And as we move up, so as we go, we're going to hit different levels of ancestry. Okay? So, this is going to be level one. Basically A and E are going to connect at level one ancestry. So basically their first common level of ancestry is one. On the other hand, A and I, their first common ancestry is going to be two. So, that's how this tree is built with these different levels of ancestry. So, for instance, there's, there's four ancestry nodes at level one. 1, 2, 3, 4, and there, each of the ancestry nodes is going to have two children or two leaf nodes. And at level two there's going to be two ancestry nodes, but each of them is going to have four leaf nodes. As we go up we're going to have lest ancestry nodes, but each of the ancestry nodes is going to encapsulate a larger fraction of the total numbers of leaf nodes at the bottom. And finally, at the top of this tree we just have a singe node that has all of the leaf nodes underneath it, all right, so level three, and the third level encapsulates all the leaves. And as we add more leaves, it has to be done in multiples of two to preserve symmetry. So, basically it's just adding ancestry level. So, here we have eight leaf nodes and we have, we need three levels. If we had 16 if we had 16 leaf nodes, we would need four levels, right, so we'd have to go to another level. Basically it'd be like taking this entire structure and putting it over here as well, on the other side. And then connecting it up to a, a fourth. And then if we had 32, we would take that entire thing and we would connect. We would basically multiply that by two and then connect two of those, and so forth. So, as the ancestry between the nodes increases, and this is the important part of the model, the further the nodes are in terms of social distance. So the higher the level becomes, the higher the first common level of ancestry becomes. As we said again, like A and M, their first common level of ancestry is three. So the higher that first level of common ancestry becomes, the, the further the people are in terms of ge, geography and occupation. And that's especially considering the original experiment that we looked at. So we would say that A and M would be very far apart. But there's still a chance that they would know each other, and that encapsulates the random link adding in the Watz-Strogatz model. Right, so this, the fact that A and M could still know each other, even though they're not close in terms of geography or occupational proximity. And the way to formalize that mathematically is we say if the probability of the nodes know each other gets smaller as we go up. So, as the level of ancestry increases, the probability of the nodes know each other gets smaller. So, A, B, C, D, as we said, they're always going to know each other. But A and E are going to have a smaller chance of knowing each other since their common is at one. A and I are going to have an even smaller chance of knowing each other, so it's more likely that A and E know each other than A and I. And it's if, and it's even less likely that A and M know each other than A and I or A and E. [SOUND]