Now we went through a lot of different definitions, a lot of different aspects of networks. I threw a lot of different definitions and things to give you an idea so how do the social network analysis is about and why it is important because it helps us to make predictions and to describe network structure. One last point and I hear we'll need your attention. You might have to watch the segment several times, has to do with network centrality and that's a very important question. Another question is, who is the center of the network? So one of the things, the first things that I made it last in this presentation in this lecture, one of the first things you often calculate, I also often calculate and analyze network, is the centrality of the different nodes and the distribution of the centrality and has to do because you want to know who is in the center of the network that later on you calculate your triads, and your clusters, and your partitions, and so forth. But the center, last but not least, is a very important question. It is not an easy question to answer and actually spoiler alert, it turns out that there are different kinds of definitions of who's in the center. Again, it depends on what you're actually after and what you try to model. So let's have a look at this question. So here I have a network and my question is, which node is in the center? One, two, or three? What would you say just as a guts feeling? But it might be that you said that node number two, that it is in the center it is, it is a broker actually that brokers two parts of this network. All right, so that's if you define a broker as being in the center but then there might be more but that's a reasonable definition. Let's see you some formal definitions. The most popular definition of center is what's called degree centrality and that's the most connected node actually. That makes a lot of sense. It's kind of like the most popular kid on the block, that kid is in the center. So if you go to your high school there's this one popular kid that's connected to most people, so that kid is in the center, it has the most connections. So my network analysis ends that is reasonable to assume or a celebrity. A celebrity mainly people know the celebrity. So it has a lot of connection that person is in the center or a politician, a decision-maker, an agent of change that has a lot of connections. So yes it makes sense. So let's look at our network. Here one two and three, how many degrees, how many connections does each one of them have? While node number two, how many degrees? Two, it goes to one and it goes to three. Okay so two [inaudible] , Node number three, how many degrees? Well three, we've got three links now in-degrees, out-degrees, you're going to distinguish, very good point. It has three and node number one. How many? Four, right? So actually the most popular kid on the block here, the one with the most connections, is node number one. So we would say formerly node number one has the highest degree centrality now and directed networks as I already alluded to there's in-degree, out-degree so you can have highest in-degree centrality, highest out-degree centrality. |A celebrity for example might have a high in-degree centrality but actually not know so many people because they keep the celebrity in a bubble, so the celebrity doesn't know a lot of people, so many people know this person. So there might be a high in-degree but a small out-degree. But that's not the only definition of degree, the most popular kid on the block. There also some another reasonable definition is to say which is the node that is closest to everybody else, and that is not necessarily the same. You don't have to have the most connections, but you can still be very distant to somebody because you've got to like in some corner of the network and you have a lot of connections there but to get to everybody in the network takes you a long time because you are in this particular corner, even so in this corner you are highly connected. So there's a difference between having a lot of connections and being close to everybody else. So how do we count the closeness? Well basically we count the degrees of separation. Well we're good at that already, degrees of separation we got that down. All right, so let's have a look at our network and let's count our degrees of separation. So I start with node number one. What are the degrees of separation to my left three nodes? How many steps do I need to get from node number one to my left three nodes? Node one, one-step. There's one degree of separation between node one and the left three nodes. So one times three, so I count them one step to three nodes. So there are three degrees of separation, one, one, and one in order to get to these three nodes. To get from one to one there's no degree of separation, I am already right there. To get from one to two, there's one degree of separation to get from one to three. There are two degrees of separation, I go from one to two and then to three. Then what about the upper right? Well the first one on the upper right hand, I need three steps, second one in four steps, and the last one in five steps. One, two, three, four, five. What about the lower three right? How many steps do I need to get the lower three right? Well good to know that you're still awake. Yes also three, four, five, of course that's the same. Now we can add them all up and see what the total degrees of separation that I need to get from node one to everybody else. So I have three plus one, four; plus two, six, and then I have three plus four, plus five, that's seven 12; two times, 24; plus six onto our right, we have 30. Okay good. So in order to get from one to every other node, we need 30 steps, okay. That's the number. Let's do it for node number two. So for node number two to get to the left three nodes, I need two steps. I have two degrees of separation between node number two and the left three nodes. I have this three times, so two steps times three nodes. To get to one, I need one step. To get from two to two, nothing; from two to three, I only also get one step. Now to the upper three nodes, I need two steps, three steps, four steps; and to the lower three nodes, I need two steps, three steps, four steps as well. So in total then, the sum is 26. Good, I don't have you as asleep that like something still working great, I'm happy it is, you just sum it up and you get 26. So in order to get from node two to everybody else, I need 26 steps or the total of 26 degrees of separation. So that's actually closer. Number two only needs 26 degrees of separation, whereas node number one has 30 degrees of separation between itself and everybody else. So node number two has a higher closeness centrality, that's how we say. What about node number three? Please try to do it by yourself. Do exactly what we did for node number one, node number two, do it for each one, write them down, and then sum them up. What do you get? So for the left three nodes, in order to get through them from node number three, I need three steps. Go to two, to one, and then get to the left three nodes. So I need three steps times three. So that's what I have there. To get to node number, I need two steps, node number two one step, node number three I'm already there. Then the upper right I need one, two, three steps; and the lower right as well, one two, three steps. So in total if I sum that up, I get 24. So it turns out that three is actually closest to everybody else. That's because three is close to this big group on the right-hand side where you have the upper and the lower, so its [inaudible] can reach them very quickly, and this group is bigger than on the other side. So it turns out that if I want to get quickly to all the nodes, If I have a message that quickly goes to all the nodes, node number three is my candidate. So node number three has the highest closeness centrality actually. So these numbers as they go down, so usually you take the reciprocal you divide it by this number and then you get a high closeness centrality for low number of degrees of separation, all right you put it underneath the fraction. So that's how you would calculate that. So node number three has the highest closeness centrality. Node number one has the highest degree centrality. So going back to our question, which one is the center of the network, well that depends on what you're interested in.
