Learn how to model social and economic networks and their impact on human behavior. How do networks form, why do they exhibit certain patterns, and how does their structure impact diffusion, learning, and other behaviors? We will bring together models and techniques from economics, sociology, math, physics, statistics and computer science to answer these questions. The course begins with some empirical background on social and economic networks, and an overview of concepts used to describe and measure networks. Next, we will cover a set of models of how networks form, including random network models as well as strategic formation models, and some hybrids. We will then discuss a series of models of how networks impact behavior, including contagion, diffusion, learning, and peer influences. You can find a more detailed syllabus here: http://web.stanford.edu/~jacksonm/Networks-Online-Syllabus.pdf You can find a short introductory videao here: http://web.stanford.edu/~jacksonm/Intro_Networks.mp4
2.3: Centrality Measures
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Social and Economic Networks: Models and Analysis
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Background, Definitions, and Measures Continued
Homophily, Dynamics, Centrality Measures: Degree, Betweenness, Closeness, Eigenvector, and Katz-Bonacich. Erdos and Renyi Random Networks: Thresholds and Phase Transitions
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Matthew O. JacksonProfessor
Economics
Okay, hello. So, now we're looking at Centrality
Measures. We're going to talk about positioning
nodes inside a network, and understanding how they're positioned.
And in terms of the way we've been going through our definitions and trying to
understand structure of networks and so forth.
what we've done so far as we talked about, you know, basic patterns of
networks,degree disrtibutions, path lengths things like that get to raise
works. we talked a little bit about homophily
and effect that there can be segregation among nodes.
we talked a little bit about Local patterns, things like Clustering and
related concepts of transitivity. We'll talk a little bit about, you know,
how, how many of links are actually supported so they have friends in common
and so forth as we go forward in the, in the course.
so these are things which characterize the network itself and we'll also be very
interested in understanding how different nodes are positioned in a network.
And so how we can talk about whether a node is important or not or influential
or central, powerful, etcetra. And so the idea of how to describe a
position in a network you know, there's different aspects of individual
characteristics, some of which we have already talked about.
you know? Just, how, how connected it is.
How clustered its friends are. distance to other nodes.
but more generally, trying to capture centrality, influence, and power are
going to, you know, build on, on specific definitions which, keep track of, of a
node's position. So, in terms of looking at nodes
centrality. the most basic measure in just trying to
figure out how important a node is, or how influential it is and so forth.
It's just how connected it is and that's captured directly by degree.
So degree captures some notion of connectedness of a, of a node.
And you know, in order to make it a measure between zero and one, we can just
keep track of dividing through by n minus 1 the most possible links I could have
and then what fraction of people am I connected to compared to how many I could
be connected to. so if we look, you know, for instance at
the Medici, the, the Florentine data we had before.
here, what do we see from, from the different families we see that the Medici
here have a degree of six the Strozzi, degree of four, Guadagni, degree of four,
Albizzi's, degree of threes, so some of the most important families, the Medici
were better connected in terms of having higher degree.
It's not an enormous difference, but there's some difference there.
so degrees capturing some of what goes on.
[BLANK_AUDIO] but degree is actually going to miss a lot of what's going on
in, in terms of a network. And you know for instance here, degree
you know node three has the same degree as node one or node two, and in some
sense we might, just looking at the network think if three as being less
central than some of the other nodes. And how do we capture that fact that, you
know, degree isn't really gathering all of position.
It's just saying, you know, how big is your local neighborhood.
It's not saying where you are positioned in the network, or how central you are in
a, in a deeper sense. So in order to get at things like
centrality, we'll have different types of things that we can think about capturing.
So what I've done here is, sort of, break things down into four different
categories. And so degree is really just capturing
basic connectedness. Another thing we might worry about, is
how close you are to other nodes. So, closeness centrality measures, and
what we'll look at is in terms of decay, is sort of an ease of reaching of other
nodes. So, how far am I on average from other
nodes? between this, we talked about very
briefly, we'll look at that in a little more detail.
Role as an intermediary or connector. So are, do other pairs of nodes have to
go through me in order to reach themselves?
That's a very different concept then, then thinking how close I am to somebody
else. This is saying is, am I as important as a
connector of, of other individuals. then there will be a whole series of
influence or prestige or eigenvectors kinds of, of notions, which we'll try and
capture the idea that your are important if your friends are important.
So being well connected is something which depends on the connectedness of
one's friends. And so you know this is the old it's not
what you know, but who you know it's not necessarily important to have more
friends but to have well positioned friends, we'll take a look at the
definitions which capture that. So we have sort of four different
concepts of, of centrality or power and we'll try and incorporate these into
different definitions and see, I don't know, the differences between these
things. And one thing to emphasize here; there's
lots of different measures, and not one it, it, it, there's not one which is
best, in a sense that it dominates. These things are capturing different
ideas, different aspects of a position, and some of them are going to be more
important in making predictions in one setting than another.
And so, what we really do in terms of, of using one of these things, it's going to
depend very much on the context as to which one was important, most important.
Okay. So let's have a look at Closeness
centrality. So, Closeness centrality one basic
definition of it here is just to look here, this is the length of the shortest
path between two nodes i and j. And we can sum across all j.
So how far am I away from all the other nodes?
And then look at n minus 1 over this and it keeps track of sort of relative
distances to other nodes. And so the idea here is that if, if these
are very large numbers, then my closeness centrality is going to be a very small
number. so I'm dividing by larger numbers, it
makes this small. So how close I am to other people the
closer I am if, if I'm a distance 1 from everybody, then this thing normalizes to
1, and otherwise it, it's going to become further and further.
this scales directly with distance so twice, twice as far from everybody makes
me half as central. right, so if I double all these things,
I'm going to get half. if I quadruple them, then I'll get a
quarter. And so forth.
So, it's scaling, with the distance. When we look at the closeness centrality
back in our, Medici data again. ignoring the Pucci now because if we add
them to everybody and we think of everybody has being infinitely distant
from them, then everybody would have closeness centrality of zero.
So, if we ignore them and just look at the remaining network, then the, the
Medici are 14 out of 25, Strozzi 14 out of 32, Guadagni 14 out of 26, and so
forth. So here you know, closeness centrality
gives us some differentiation between different families.
It's not it, it, it doesn't sort things enormously.
it gives us some feeling for who's further and, and who's closer.
another measure that we could use instead, is what's known as decay
centrality. And this is, designed to capture the idea
that, what I, I might get is, is value from being connected or indirectly
connected to other nodes. So I might have some value from a friend
a different value from a friend of a friend, and so forth.
And so the idea is that there's going to be some delta factor which is, generally
less than 1 and bigger than 0. And the centrality then of a node i under
this decay notion, is going to be, look at the distances to other nodes, and
raise delta to that power. So if I'm a direct friend I get a delta.
If I'm an indirect friend, distance 2 from somebody, I get delta squared.
distance 3 delta cubed. So if this were 0.5 then we're going to
get 0.25 here. and, and, and so forth.
0.125. if this were 0.9 then these numbers would
be much closer to each other. If this was, you know, 0.05 then this
would be 0.0025 and so forth and so, so it would scale more dramatically.
So as delta becomes near 1, then this just sort of counts all the people that I
can reach indirectly. As delta goes close to 0 then this is
just going to become degree centrallity. All it's going to do is, is really
emphasise the direct connections and all the other ones are going to be much
smaller. And then somewhere in between it's, its
going to weight indirect connections compared to, to direct connections.
So you can think of varying this delta, as sort of how much do I think of, of it
being important to be close to many people, or, of how much do I get from
indirect connections, of different varying lengths.
[BLANK_AUDIO] . Okay.
So, you know? Here's a network, with, you know, sort
of, like, bow tie kind of network here. we've got, you know, node 4 in the
middle. Node 3 over here.
Node 1 over here. basically there is three different types
of nodes, so nodes two, six and seven all look node 1, node five looks like node 3.
So we're going to just keep track of these three nodes and their centralities.
if we look at the degree centrality, then node 3 is a, is the most important in
some sense. Its got three connection as supposed to
two for the others. Closeness centrality, node 4 is actually
the closest. So here it wins out in terms of being
able to reach all the other ones in, in shorter paths.
Decay centrality depends, if we do 0.5, then these two are, are basically equal
to each other. If we raise it to 0.57, then node four
ends up doing a little better. If we drop it to 0.25, so that more
immediate connections matter, then node three starts to do better, so you can
begin to see that these different, Definitions are going to give different
positions in terms of importance to different nodes, depending on which kind
of centrality definition your looking at. you can normalize decay centrality, by,
dividing through, by you know the, the lowest possible decay you could have to
each one of each node. So it's n minus 1 times delta, is the
lowest possible. and, you know that gives you sort of the,
the numbers, we looked at, before. So, you know, normalizing, you can get
different numbers here in terms of, you know, what these numbers would be, so
that's just readjusted by a normalization.
Okay so looking back at between the centrality that we looked at before, so
now the formal definition of between a centrality due to Freeman.
So the idea here is that when we look at two nodes, i and j, we can keep track of
the full number of shortest paths, the short, the number of geodesics between i
and j. And then for any k that's not equal to i
and j, we can ask what's the number of those shortest paths that k lies on,
between i and j? Right, so if we're looking then for the
between the centrality of a node k, we can look at all pairs i and j that aren't
equal to k. And look at what's the number of shortest
paths between i and j that k lies on compared to the number of shortest paths,
that exist between i and j. And then we can normalize that by the
number of alternative pairs of nodes that we can be considering, and how, you know,
the, the most you could be is, is to have b on the shortest paths of all of those.
So we're going to normalize by there's n minus 1 times n minus 2 over 2 or this is
n minus 1 choose two other pairs of, of nodes that are out there.
Okay. So when we look at that calculation,
we're basically saying what's the fraction of shortest paths that k lies on
between other nodes. And then when we look at that again what
we saw was that the Medici now have a much higher number than the others other
families in terms of their centrality. And between the centrality captures this
idea that you know, at this point in time, if other families wanted to deal
with each other they might have to go through somebody that they were connected
with. So if it's difficult to enforce
contracts, then maybe you have to go through somebody you know in order to
deal with somebody you don't know. And then the Medici could be powerful
intermediaries connecting other families on pairs between them.
So that's a different measure, captures different things.
And if we go back and look at our bow tie again, what do we see now?
Well, you know, now between this, these nodes out here they're not connected,
connecting anybody in terms of being an intermediary.
So they get a betweenness of zero. and here node 4 now becomes more
prominent. It's really an important path connector
between all of the nodes on this side and all of the nodes on this side.
And so it ends up on 60% of the shortest paths in this network, whereas node 3,
for instance, only ends up on 50% of the paths.
So, you know, depending on how you're counting these things, different
centrality measures, give you different, notions, and different measures of what's
going on. Different ideas.
we'll, next topic we'll be looking at is, is, looking at eigen vector-based
centrality measures, other kinds of measures, that you're going to capture,
importance of, of nodes and how well they're connected to other nodes.
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