0:00

Okay, hello. So, now we're looking at Centrality

Â Measures. We're going to talk about positioning

Â nodes inside a network, and understanding how they're positioned.

Â And in terms of the way we've been going through our definitions and trying to

Â understand structure of networks and so forth.

Â what we've done so far as we talked about, you know, basic patterns of

Â networks,degree disrtibutions, path lengths things like that get to raise

Â works. we talked a little bit about homophily

Â and effect that there can be segregation among nodes.

Â we talked a little bit about Local patterns, things like Clustering and

Â related concepts of transitivity. We'll talk a little bit about, you know,

Â how, how many of links are actually supported so they have friends in common

Â and so forth as we go forward in the, in the course.

Â so these are things which characterize the network itself and we'll also be very

Â interested in understanding how different nodes are positioned in a network.

Â And so how we can talk about whether a node is important or not or influential

Â or central, powerful, etcetra. And so the idea of how to describe a

Â position in a network you know, there's different aspects of individual

Â characteristics, some of which we have already talked about.

Â you know? Just, how, how connected it is.

Â How clustered its friends are. distance to other nodes.

Â but more generally, trying to capture centrality, influence, and power are

Â going to, you know, build on, on specific definitions which, keep track of, of a

Â node's position. So, in terms of looking at nodes

Â centrality. the most basic measure in just trying to

Â figure out how important a node is, or how influential it is and so forth.

Â It's just how connected it is and that's captured directly by degree.

Â So degree captures some notion of connectedness of a, of a node.

Â And you know, in order to make it a measure between zero and one, we can just

Â keep track of dividing through by n minus 1 the most possible links I could have

Â and then what fraction of people am I connected to compared to how many I could

Â be connected to. so if we look, you know, for instance at

Â the Medici, the, the Florentine data we had before.

Â here, what do we see from, from the different families we see that the Medici

Â here have a degree of six the Strozzi, degree of four, Guadagni, degree of four,

Â Albizzi's, degree of threes, so some of the most important families, the Medici

Â were better connected in terms of having higher degree.

Â It's not an enormous difference, but there's some difference there.

Â so degrees capturing some of what goes on.

Â [BLANK_AUDIO] but degree is actually going to miss a lot of what's going on

Â in, in terms of a network. And you know for instance here, degree

Â you know node three has the same degree as node one or node two, and in some

Â sense we might, just looking at the network think if three as being less

Â central than some of the other nodes. And how do we capture that fact that, you

Â know, degree isn't really gathering all of position.

Â It's just saying, you know, how big is your local neighborhood.

Â It's not saying where you are positioned in the network, or how central you are in

Â a, in a deeper sense. So in order to get at things like

Â centrality, we'll have different types of things that we can think about capturing.

Â So what I've done here is, sort of, break things down into four different

Â categories. And so degree is really just capturing

Â basic connectedness. Another thing we might worry about, is

Â how close you are to other nodes. So, closeness centrality measures, and

Â what we'll look at is in terms of decay, is sort of an ease of reaching of other

Â nodes. So, how far am I on average from other

Â nodes? between this, we talked about very

Â briefly, we'll look at that in a little more detail.

Â Role as an intermediary or connector. So are, do other pairs of nodes have to

Â go through me in order to reach themselves?

Â That's a very different concept then, then thinking how close I am to somebody

Â else. This is saying is, am I as important as a

Â connector of, of other individuals. then there will be a whole series of

Â influence or prestige or eigenvectors kinds of, of notions, which we'll try and

Â capture the idea that your are important if your friends are important.

Â So being well connected is something which depends on the connectedness of

Â one's friends. And so you know this is the old it's not

Â what you know, but who you know it's not necessarily important to have more

Â friends but to have well positioned friends, we'll take a look at the

Â definitions which capture that. So we have sort of four different

Â concepts of, of centrality or power and we'll try and incorporate these into

Â different definitions and see, I don't know, the differences between these

Â things. And one thing to emphasize here; there's

Â lots of different measures, and not one it, it, it, there's not one which is

Â best, in a sense that it dominates. These things are capturing different

Â ideas, different aspects of a position, and some of them are going to be more

Â important in making predictions in one setting than another.

Â And so, what we really do in terms of, of using one of these things, it's going to

Â depend very much on the context as to which one was important, most important.

Â Okay. So let's have a look at Closeness

Â centrality. So, Closeness centrality one basic

Â definition of it here is just to look here, this is the length of the shortest

Â path between two nodes i and j. And we can sum across all j.

Â So how far am I away from all the other nodes?

Â And then look at n minus 1 over this and it keeps track of sort of relative

Â distances to other nodes. And so the idea here is that if, if these

Â are very large numbers, then my closeness centrality is going to be a very small

Â number. so I'm dividing by larger numbers, it

Â makes this small. So how close I am to other people the

Â closer I am if, if I'm a distance 1 from everybody, then this thing normalizes to

Â 1, and otherwise it, it's going to become further and further.

Â this scales directly with distance so twice, twice as far from everybody makes

Â me half as central. right, so if I double all these things,

Â I'm going to get half. if I quadruple them, then I'll get a

Â quarter. And so forth.

Â So, it's scaling, with the distance. When we look at the closeness centrality

Â back in our, Medici data again. ignoring the Pucci now because if we add

Â them to everybody and we think of everybody has being infinitely distant

Â from them, then everybody would have closeness centrality of zero.

Â So, if we ignore them and just look at the remaining network, then the, the

Â Medici are 14 out of 25, Strozzi 14 out of 32, Guadagni 14 out of 26, and so

Â forth. So here you know, closeness centrality

Â gives us some differentiation between different families.

Â It's not it, it, it doesn't sort things enormously.

Â it gives us some feeling for who's further and, and who's closer.

Â another measure that we could use instead, is what's known as decay

Â centrality. And this is, designed to capture the idea

Â that, what I, I might get is, is value from being connected or indirectly

Â connected to other nodes. So I might have some value from a friend

Â a different value from a friend of a friend, and so forth.

Â And so the idea is that there's going to be some delta factor which is, generally

Â less than 1 and bigger than 0. And the centrality then of a node i under

Â this decay notion, is going to be, look at the distances to other nodes, and

Â raise delta to that power. So if I'm a direct friend I get a delta.

Â If I'm an indirect friend, distance 2 from somebody, I get delta squared.

Â distance 3 delta cubed. So if this were 0.5 then we're going to

Â get 0.25 here. and, and, and so forth.

Â 0.125. if this were 0.9 then these numbers would

Â be much closer to each other. If this was, you know, 0.05 then this

Â would be 0.0025 and so forth and so, so it would scale more dramatically.

Â So as delta becomes near 1, then this just sort of counts all the people that I

Â can reach indirectly. As delta goes close to 0 then this is

Â just going to become degree centrallity. All it's going to do is, is really

Â emphasise the direct connections and all the other ones are going to be much

Â smaller. And then somewhere in between it's, its

Â going to weight indirect connections compared to, to direct connections.

Â So you can think of varying this delta, as sort of how much do I think of, of it

Â being important to be close to many people, or, of how much do I get from

Â indirect connections, of different varying lengths.

Â [BLANK_AUDIO] . Okay.

Â So, you know? Here's a network, with, you know, sort

Â of, like, bow tie kind of network here. we've got, you know, node 4 in the

Â middle. Node 3 over here.

Â Node 1 over here. basically there is three different types

Â of nodes, so nodes two, six and seven all look node 1, node five looks like node 3.

Â So we're going to just keep track of these three nodes and their centralities.

Â if we look at the degree centrality, then node 3 is a, is the most important in

Â some sense. Its got three connection as supposed to

Â two for the others. Closeness centrality, node 4 is actually

Â the closest. So here it wins out in terms of being

Â able to reach all the other ones in, in shorter paths.

Â Decay centrality depends, if we do 0.5, then these two are, are basically equal

Â to each other. If we raise it to 0.57, then node four

Â ends up doing a little better. If we drop it to 0.25, so that more

Â immediate connections matter, then node three starts to do better, so you can

Â begin to see that these different, Definitions are going to give different

Â positions in terms of importance to different nodes, depending on which kind

Â of centrality definition your looking at. you can normalize decay centrality, by,

Â dividing through, by you know the, the lowest possible decay you could have to

Â each one of each node. So it's n minus 1 times delta, is the

Â lowest possible. and, you know that gives you sort of the,

Â the numbers, we looked at, before. So, you know, normalizing, you can get

Â different numbers here in terms of, you know, what these numbers would be, so

Â that's just readjusted by a normalization.

Â Okay so looking back at between the centrality that we looked at before, so

Â now the formal definition of between a centrality due to Freeman.

Â So the idea here is that when we look at two nodes, i and j, we can keep track of

Â the full number of shortest paths, the short, the number of geodesics between i

Â and j. And then for any k that's not equal to i

Â and j, we can ask what's the number of those shortest paths that k lies on,

Â between i and j? Right, so if we're looking then for the

Â between the centrality of a node k, we can look at all pairs i and j that aren't

Â equal to k. And look at what's the number of shortest

Â paths between i and j that k lies on compared to the number of shortest paths,

Â that exist between i and j. And then we can normalize that by the

Â number of alternative pairs of nodes that we can be considering, and how, you know,

Â the, the most you could be is, is to have b on the shortest paths of all of those.

Â So we're going to normalize by there's n minus 1 times n minus 2 over 2 or this is

Â n minus 1 choose two other pairs of, of nodes that are out there.

Â Okay. So when we look at that calculation,

Â we're basically saying what's the fraction of shortest paths that k lies on

Â between other nodes. And then when we look at that again what

Â we saw was that the Medici now have a much higher number than the others other

Â families in terms of their centrality. And between the centrality captures this

Â idea that you know, at this point in time, if other families wanted to deal

Â with each other they might have to go through somebody that they were connected

Â with. So if it's difficult to enforce

Â contracts, then maybe you have to go through somebody you know in order to

Â deal with somebody you don't know. And then the Medici could be powerful

Â intermediaries connecting other families on pairs between them.

Â