Now that we've seen a number of different ways of finding central nodes in a network, today we're going to look at an example where we compare how the different centrality measures that we've looked at rank nodes differently. We're going to be looking at this particular network and we're going to run the different algorithms that we looked at on this particular network. Let's start with the most basic way of thinking about centrality in a network, and that is your in-degree, how many nodes point to you? If we use this measure on this network what we would be able to say is that nodes 1 and 6 have the highest in-degree, so they are the most central. They have in-degree of four. Then all the other nodes are second, because all the other nodes have in-degree 2. The in-degree centrality is only able to say that nodes 1 and 6 are the most central and everything else is the same. I'm going to be looking at all the other measures, and just like I did for in-degree I'm going to be putting the nodes ranked by highest to lowest and I'm going to be using red lines to indicate when the ties break. In this example nodes 1 and 6 are the most central nodes, and then everything else comes second. I'll indicate that using this red line here. Now let's look at closeness centrality. Just remember that closeness centrality says that nodes who are central are a short distance away from all the other nodes in the network. Using this measure we'll find that five is the most central node. You can see that this makes sense because node 5 is in the middle of the network. Most nodes can reach node 5 in just a small number of hops. For other nodes such as three or four, it is going to take a larger number of hops to reach them. For example, it takes multiple hops to go from node A to node 3, so it makes sense that five has a high closeness centrality. Nodes 1 and 6 come next, and you can see that they are also in the center of the network but not as central as five. Then come nodes 2 and 7, which are closer to the center of the network, then nodes 4, 3 and 8 and 9 but less central than 5, 1 and 6. Finally, the nodes with the lowest closeness centrality are nodes 4, 3, 8, and 9. Closest centrality can not distinguish between them. You can see why nodes 3 and 4 have the same closest centrality by noticing that to get to one of those nodes you must reach node 2 and at that point you are one hop away from reaching nodes 3 and 4. A similar argument holds for nodes 8 and 9. However, note that there is a structural difference between nodes 3 and 4. For example, well, we can go from node 3 to 2 directly. In order to get from node 4 to 2, we must pass through node 3. However, closeness centrality does not capture this difference. Next we'll look at betweenness. As a reminder, betweenness says that central nodes are those that show up in the shortest paths between different pairs of nodes in the network. The node with the highest betweenness is node 5. Again, this makes sense, it's pretty central in network. You can tell that five does show up in the shortest path between many pairs of nodes. Then next will come 1 and 6, just like with closeness, and again, this makes sense. Then we have nodes 2 and 7. Again, just like with betweenness centrality. Next will be nodes 3 and 8. Unlike closeness betweenness centrality is able to capture the structural differences between nodes 3 and 4. As we noted earlier, in order to get from node 2 to 4 we must pass through node 3, whereas node 3 can reach node 2 directly. Therefore, node 3 has higher betweenness centrality than node 4. Then finally, nodes 4 and 9 have the lowest betweenness centrality. Betweenness turns out very similar to closeness, but betweenness is able to capture those structural differences between nodes 3 and 4 that closeness does not capture. Next, let's look at PageRank. Again, PageRank has these useful interpretation which says that nodes who are central are the ones that if you were to take a random walk on this network then you would pass by them a lot or you would end up landing on them a lot. The nodes with the highest PageRank in this network are nodes 1 and 6 and then node 5. Unlike betweenness, which says that 5 is the most central node, PageRank has 1 and 6 and then 5. Now why this may be? Well, if you notice node 5 here gives all its PageRank to nodes 1 and 6, whereas nodes 1 and 6 give somewhat their PageRank to node 5 but they also give to other nodes. This's part of the reason why node 5 comes second to 1 and 6. Then you have the exact same thing, you have 2, 7, 3, 8, and 4, 9. In this case, PageRank comes out very similar to betweenness but it flips the nodes 1 and 6 and 5. Now let's look at the authority scores from the HITS algorithm that computes authority and hub scores for every node. This just like PageRank puts 1 and 6 at the top and then come nodes 4 and 9, which is surprising at first because you would imagine, well, what happened to node 5 and what happened to nodes 2 and 7 which are clearly centrally in this network? Why are they not coming before 4 and 9? We'll see that in a minute. But for the authority score next you have nodes 3 and 8, 2, 7 and then finally, you have node 5. The node with the lowest authority score here is five even though for many of the other centrality measures it had a very high centrality. Why might this be the case? Well, if you remember, the HITS algorithm gives every node an authority score and a hub score. In order to understand what the HITS algorithm is saying you have to look at those scores together. What happens is that when you look at the hub scores of this network 2, 5 and 7 which were the nodes that we're wondering why they wouldn't have high centrality, high authority, well, it's because they have high hub score. The way that the HITS algorithm analyzes network is that it says that the authorities are 1 and 6 and 2, 5 and 7 are the nodes with a very high hub score. To interpret the scores you really have to take them together. Then next will come 3 and 8, 4, 9 and 1 and 6. What we see here is that all of these measures give different rankings, although there are some commonalities. They all have nodes 1, 5 and 6 with high scores generally. But there are some differences as well. If we summarize we find out in this example no pair of centrality measures produces the exact same ranking, but there are some commonality so you are able to pick out some of the nodes that are very central. Of course, this centrality measures make different assumptions about what it means to be a central node and so that's why they produce different rankings. To figure out what the best centrality measure is really depends on the context of the network that you're analyzing. Usually the best thing to do to identify central nodes is to take multiple centrality measures and figure out which nodes come out central in many of them rather than relying on a single one to do this. I hope this gives you some context into how these different centrality measures compare and look at the differences between them as well. That's all for this video, and we'll see you next time.
