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Okay. So, let's have another look at centrality
measures in the context of this diffusion process in rural India we were just
talking about. And the idea here is going to be to come
up with a centrality measure that might even do better than any of the ones we
looked at before, any of the standard ones.
And I guess, you know, once your in this business long enough, if you've looked at
enough networks, you'll find some need to invent a new centrality measure.
so, what's the idea here? the idea is we'll define something what
we can call diffusion centrality. And the diffusion centrality of the given
node i is going to depend on two different parameters.
so again, this is out of the project with uh,[UNKNOWN].
And what we're doing is now looking at a given nodes are informed about
microfinance. They, they're going to tell their friends
and eventually we want to see how that diffuses through the villages.
but in general, we can think of, you know, there's some node which is
initially informed, and how influential is it going to be?
Well, we can do a calculation where each node is going to talk to its friends with
a probability p in any given period. So, I talk to my friends with some
probability p, that goes on, then they talk to their friends, and we continue on
in this manner. And this is going to run for T periods,
some number of periods. And at the end, we'll find out how many
people have heard the news by the end of this process, okay?
So, so one way to do this would be actually to simulate it.
You could just start with a network and form a node flip coins, so that if they
tell, talk to their friends with probability p, and then it goes on, and
then just keep track of who heard. another way in approximation of this,
would be just to multiply the adjacency matrix by p and raise that to the T
power, and of, multiplied by 1, so that keeps track of how many people we, we, we
have heard at each point in time. And then what we can do is sum this from
1 to T. Okay.
So, it is, this is going to be a situation where it's going to keep track
of basically how many walks of different length multiplied by times p we're going
out. So, in situations where, where T is just
equal to 1, then this becomes proportional to degree centrality, so
it's just calculating, how many people do I reach directly.
And so, it's just going to be proportional to degree.
if we look at a situation where T becomes large, then this is beginning to look
like the Bonacich Centrality calculation. And in particular, if p is smaller than 1
over the largest eigenvalue of the matrix, the adjacency matrix, then this
thing will converge and indeed, it, it will converge to Katz-Bonacich Centrality
where the p is playing the role of the weight.
So, this sort of doing Bonacich Centrality but with, with finite T.
in, in contrast, if p is larger, so if it's larger than the 1 over the
eigenvalue then as T becomes large, this begins to approximate eigenvector
centrality. So, this is a, a measure which for large
T, can look either like eigenvector or Bonacich, some variation of, of Bonacich
centrality. For small T, it looks just like your
immediate neighborhood. And for intermediate T, it's going to
capture some repetition of numbers of, of communications that you might have.
so, let's have a look at what this does in the Indian data that we talked about
before. And, in particular what we've got here is
a situation where we've got the diffusion centrality compared to eigenvector
degree, closeness, Bonacich and, so forth that we did in our regressions before.
And what we find here is diffusion centrality is significant and significant
at a 99% level, so highly significant. Diffusion centrality here is defined by
setting p actually equal to 1 over lambda 1, so equal to 1 over the largest eigen
value of the matrix. And so, it's and then running it for some
number of periods which was actually equivalent to the number of trimesters
that a village had been exposed to information.
So, some villages had 8 trimesters, some had 3 and so forth, but that gave us some
numbers of rounds of communication that might go on in the village.
So here, we see that this basic measure of diffusion centrality does fairly well.
And, you know, the other horse that seem to be running pretty well is eigenvector
centrality. So, we can say, well, it, it's hard to
compare coefficients, because the larger coefficient just indicates that the units
are different. And so, if we re-normalize things, we
could make either coefficient larger or smaller.
So, in order to figure out which one might be doing a better ex, explanation,
we can sort of put them together in the same regressions, where we keep track of,
of you know, things like the degree central of the leaders, the number of
households, self-help group, participation savings fraction in the
cast and so forth. So, if we keep track of all these
controls, then if we do the diffusion centrality it turns out to be
significant, eigenvector centrality is also significant.
When we put them together, then the diffusion centrality remains highly
significant and, and doesn't change much. The eigenvector centrality part of it is,
is disappearing there. And the idea here is diffusion centrality
is keeping track of the fact that communication is not going on forever,
it's only going on some finite number of times.
And if you have some feeling for that, then this is a very practical measure
which is designed to actually figure out how nodes are, what's they're position
in, in a network as designed to spread information, and that particular measure
seems to do very well in this, this sense.
So, once we've looked at a very specific process, that can also suggest how we
should measure the importance of nodes. We can, we can weigh them directly by the
process that we think might be governing the communication or whatever it is that
we're examining in a particular network and that will give us a new idea of, of
centrality measures or other kinds of measures of nodes.
Okay. So, that's a little more about position
in networks. There's a, a lot of ongoing research in
this area, it's a fascinating area for study.
but now, what we're going to turn to is, is beginning to understand network
formation processes.
Matthew O. JacksonProfessor
