So one factor we did not take into account with degree centrality was the distance between nodes in the graph. And here what we mean by distance is the number of links that are on the path between two nodes. And a path is simply a set of links that are connecting two nodes. So we'll refer to a path by the nodes that we visit in traversing it from start to end. And since we have names here for nodes, we're going to abbreviate them by the first letters for convenience. So the link from Ben to Anna, for instance, we're going to denote as B, A here. And that's, of course, a path too; it's just a path with only one link. On the other hand the, path from Ben to Frank that goes through Anna, Cara, Evan and then Dana. So we're going to go from Ben to Anna then to Cara to Evan to Dana and then to Frank. Would be B, A, C, E, D, F and alternatively, we could go to Dana before we go Evan because Dana and Evan are both connected, and they're in turn both connected to Cara, so we could of gone from Ben to Anna to Cara to Dana, to Even to Frank, which would be B, A, C, D, E, F. But we could have also gotten there using fewer lengths because there's no need to go through both Dana and Evan, right? Because both Dana and Evan are connected to Frank. So why would we bother going through both of them? And for instance, we could have taken B, A, C, D, F; so we can write that down. B, A, C, D, F which you'd be going from Ben to Anna to Cara to Dana to Frank, and therefore bypassing Evan or B, A, C, E, F, which is another one. This is an illustration of the shortest path between two nodes. And this is the path between two nodes that has the shortest distance. Again there's two shortest paths here between Ben and Frank. Both of these are shortest paths. And all four of these are paths. So all shortest paths or paths, not the other way around. Obviously, because not all paths are shortest paths that we showed here. Paths are not going to be unique, because in general we could traverse multiple different ways to get from one node to another. Path lengths are also not going to be unique because on different paths between given set of nodes could have different lengths, as shown here. Shortest paths are also not unique. So we should write this down. They're not unique either. because there could be multiple shortest paths between each nodes. But the shortest path length, meaning the length of the shortest path or the shortest paths, is always going to be the same. Therefore, the shortest path length is itself unique. Now if we find the lengths of the shortest paths between each pair of notes, so, we find the shortest path between Ben and Anna. Between Ben and Cara. Ben and Evan. Ben and Dana. Ben and Frank. Anna, Cara. Anna, Dana. And so forth between all the different pairs. And find the longest of all those shortest paths. Consider all the pairs. That's the diameter of the network. It's the longest of the shortest paths, considering all the different node pairs. In this case, the diameter in this network is actually four. And that should make sense because Ben and Frank are the farthest away, no matter how you cut it. There's no other pair of nodes you can find that's going to be of distance four away from one another on the shortest paths between the pairs. So the diameter is four in this case. Now this is going to bring us to our second interpretation of importance, which is to look at how far a node is from its neighbors. And this is the closest centrality for a node. So specifically, what we do is, we first find the shortest path lengths to all other nodes. So we take a given node, say Ben, and we want to find Ben's closeness centrality. So we find the shortest path length from Ben to Anna, from Ben to Cara, from Ben to Dana, from Ben to Evan, and from Ben to Frank. We get all the shortest path lengths, and we write them down. And we find the average of those. So rather than finding the longest of those as we would do if we were looking for diameter, in one step to get to the diameter, we're going to take the average of these in the case of closeness centrality. And then the closeness centrality is simply one over that average shortest path length. So when we take this average, it becomes the averages shortest path length. And the reason that we take the reciprocal is because they're, they're inverse ideas. Right? We want to he closest to go up as the average shortest path length goes down. So a, as a notice closer to all other nodes, the average shortest path lengths is going to go down. Therefore, we wanted to have more of a higher closeness centrality and vice versa. So as the average shortest path length goes up, that means the node is farther away, and therefore should be less central and have less of a closeness centrality.
