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
2.4: Centrality – Eigenvector Measures
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From the course by Stanford University
Social and Economic Networks: Models and Analysis
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From the lesson
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
Meet the Instructors
Matthew O. JacksonProfessor
Economics
Hi folks. So we're back again, and now we're
looking at eigenvector-based centrality measures, and and again, as, in terms of
what we've looked at for characterizing the position of a node in a network,
we've looked at things like degree. or, you know, local clustering, distance
to other nodes and then in terms of measuring the centrality, influence and
power, one difficulty is that when we've looked at things like degree centrality
it doesn't necessarily capture the importance of the node's friends, so you
know, when we look at this picture, for instance...
the, there, there's sort of two aspects to it.
One is that we're, you know, this node might be much further, from a lot of
nodes than this one. So, things like de, decay and closeness
centrality can capture the fact that it's sort of out in the outskirts.
But another thing is that when we look at the friends of this particular node.
So if we look at the connections of this node they each have degree 2 themselves
whereas this node's friends have degree 6 and 7.
And so in some sense they are better connected than than this other node's
friends. And so the idea of,of eigenvector
centrality is that your importance comes from being connected to other important.
Nodes, and in particular here, if we look at a centrality notion where the
centrality is proportional to the sum of the neighbor centralities, then we're
defining eigenvector centrality. So the node i's centrality is
proportional to the centrality Of the nodes to which I is connected.
So that gives us a, a, an equation which says that CI is dependent on the sum of
the CJs, where what's going to happen here is if GIJ is equal to one, then we
get. The c, j being counted and if it's equal
to 0 then it's not counted so we're just counting the centrality's of the friends
you're connected to. Okay, what's what's interesting here now
is that this is a system of equations and a system of unknowns.
So we have That's in order to figure out what Ci is we have to figure out what all
the Cj's are and so this is a set of equations and a set of unknowns and so
basically what we have is that the vector C is equal to sum a times the matrix g
times the vector C. And so this is what's known as an
Eigenvector. Right?
So it's vector where you multiply the matrix by that vector and you get back
something which is proportional to the original vector you started with.
So this a is just some proportional constant which tell you how you're
scaling and that's what's known as an Eigenvector.
Now we'll say a little more about that in a moment.
but what we can begin to see is that if we calculate the eigenvector centrality
of these, so solving that system of equations and unknowns and looking for
non negative solutions what we'll find is this one gets a 0.31.
This one get a 0.11. This one is essentially three times as
important according to this Eigenvector Centrality measure of importance.
Okay, so what's happening here is you're getting value from connections to others,
it's a self-referential concept. So, we have to figure out how to solve
that. And in particular what we have is we have
this equation which we just wrote the Eigenvector centrality then of a network
g is proportional to itself times g. And one thing about Eigenvectors is
there's many possible solutions. So if we have n equations and n unknowns.
generally there can be upton N different solutions.
And what we tend to do is look for the one with the largest eigenvalue and
there's a theorem in matrix algebra known as the Perron-Frobenius Theorem, which
says that if we're dealing with a non negative matrix.
Then the eigenvector associated with the largest eigenvalue is going to have
nonnegative values. and in particular in a lot of contexts
we'll be looking at the other eigenvectors wont make sense that might
have some negative values or even nonreal values.
Use. so the conventional beats, so look for
the one which has all, non negative real values.
That's going to be the one with the largest Eigen value.
And, you know, there's a variety of programs that will calculate Eigen
vectors for you, so if you have a matrix...
representing your, your network. You can plug them into different programs
and at least get an approximation of the largest the eigenvalue.
the eigenvector associated with the largest eigenvalue.
then we can sort of normalize the, the entries to sum to one so any eigenvector
if, if you multiply it by two then that's also an eigen vector.
So, so these are all going to be taken up to.
re-scale things and so if we normalize these things and sum to 1 and then, then,
we'll have a well defined definition. Okay so if we begin to look at you know
the Eigen vector centrality of different farmers and the peasant nation data.
Actually these are not quite normalized. but here, you know, we see that again the
Medici come out higher than other families.
Not as strikingly high but that's going to give us a different picture of,
of what's going on in this, in this kind of setting.
now when we talk about eigenvector centrality, there's a number of, of
different concepts that are related to it.
Google page rank, so didn't got for little page, was the system that sort of
helped Google become the search engine that, that really dominated the market
early on. And the idea was that, that when you
tighten something and it had a bunch of pages, that it could show what it have,
whatever term you to type in on it. it had to pick one and the way that it
sort of picked the ordering originally was that it would look for the ones with
the largest eigenvector centrality. So it was trying to look for pages that
were connected to by pages that were important and that was a sign of its
important. And no it's that.
Algorithm is now much more complicated and is evolved over time to do all kinds
of other kind, you know, things people are manipulating the algorithm.
It also could take into account what it thinks people might want and so forth.
but the basic idea here was eigenvector centrality to begin with.
Something known as the random surfer model as well, which is.
You know, suppose you hopped on the web and you just started picking pages at
random. And then so you go to a page and click on
one of it's links at random, go to that page, click on a link at raondom and so
forth. And then look at the fraction of time you
would spend on each page as you went through this process over time.
That's going to be related to an eigenvector calculation.
So eigenvector calculations are ones that encapsulate this idea of, of important
connections to other nodes and then being connected to by big important ones helps
bring more traffic to you. we can go back to the the network we had
before and you know we can put in eigenvector centrality measures, and now
node 3 actually comes out ahead of node 4.
so, so you know depending on which way you do these things, you get different
measures. One last measure which is going to be
very useful and is related to eigenvector centrality is what's known as Bonacich
centrality or Katz- Bonacich centrality. It actually builds on a, a definition
originally by Katz that was later extended by Bonacich, and the idea is
that you give each nodes on base value. And the base value is some some number a
times the degree of the node. Okay, so, so if I have degree 10, and ,
and a is, is a half, then I start with a value of 5.
And so everybody starts with some value proportional to their degree.
And now you add in all paths of length 1 from i to some j times b times j's base
value. So I get some fraction of my neighbors
value. So let's say that b is a third so now
I've got a base value, plus I add in 1 3rd times my friends space values.
Then I go to length 2 and I add in b squared times those peoples values.
So this is you know 1/2 times my friends or sorry 1/3 times my friends values now
this is going to be 1/9 Times friends of friends values, right,.
So I can go out length 2, blocks of length 2.
Some of them might come back to me. so it, does sort of counts these things
doubly. But what it's doing is just counting, so
I get a times my degree which is just g times the vector of 1.
Then I get b times walks of length 2, remember g squared gives us walks of
length 2. so I'm getting b times the connections of
length 2, I have times their base values. Then b squared times length 3 and so
forth. So what we end up with is this
calculation where we're sort of raising g to successive powers, and then weighting
that by some level b. And if we normalize the a here to 1, it's
just multiplying the whole thing. Then basically, you know, this thing is
going to be like a sum of a series, and I'm getting more Value from inderect
walks times the value of those nodes. If you want this thing to ber convergent
so that actually gives you a number, b's going to have to be relatively small in
order for that to be finite. it's going to have to depend on the
magnitude of the matrix. the, the largest Eigen value.
but a small enough B will make sure that this thing converges.
So when we look and Bonacich Centrality, then we can go through and calculate this
for different values. and you know go through and for instance
then, the Medici again appear more important that some of the other
families. so, you know, this kind of calculation
gives you something like Eidenvector centrality, and in fact for b being close
enough to 1 over the, largest eigen value of the matrix, then it's going to begin
to approach eigen centrality, but more generally it can diverge from them.
And depending on what bs you work with you are going to get different notions,
so here if we normalize a to 1 and work with different bs we can get different
values of Bonacich centrality out, and it's going to be another measure of
centrality. Basically we've go a whole sequence of
different measures of centrality. Some of them are going to work better in
some contexts than others. Exactly how we do that and which ones are
going to be right for what contexts depends on, on what we're studying.
And again this sort of four different ideas that we've gone through degree is
just getting basic connectedness of a node.
How, how many neighbors does it have. Then there's things like how close are
you, how easily can you access other nodes.
Then there's connector based definitions. So how important are you as an
intermediary. And then there are these sets of
influence, prestige, and eigenvectors sorts of things that are capturing not
only who you know, but weighting by their importance and doing sort of iterative
calions of calculations that will capture importance of the value of your friends
as measuring your own importance. so what we'll do is, is now we'll have a,
a quick look at one application, which will distinguish which one of these
centrality measures does well. And which ones don't, in a particular
setting. and then we'll start to move on and, and
try and understand a little bit more about formation of networks.
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