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
1.8: Degree Distributions
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From the course by Stanford University
Social and Economic Networks: Models and Analysis
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From the lesson
Introduction, Empirical Background and Definitions
Examples of Social Networks and their Impact, Definitions, Measures and Properties: Degrees, Diameters, Small Worlds, Weak and Strong Ties, Degree Distributions
Meet the Instructors
Matthew O. JacksonProfessor
Economics
Okay, so let's talk a little bit about the distributions of degrees in a
network, and why, why do we care about this?
Well obviously, when we are trying to characterize networks in terms of basic
properties that they have knowing whether the most nodes have very similar degrees
or very different degrees can give us a lot of information.
So for instance you know, whether everybody has one or two links or some of
the nodes have eight and others have one, that's going to give us a very different
kind of network. And that different kind of network's
going to have different properties in terms of its difusion properties, its
path links and other kinds of things. So, you know, what we looked at in terms
of the GMP graph had, most nodes had order of magnitude, similar degrees to
each other. let's look at this in, in a little bit
more detail. So when we look at say these Erdos-Renyi
GNP random graphs. well the, the chance that a given node
has exactly d links, is just follows what's known as a binomial distribution.
Right? So the chance that I have exactly d links
present and d the rest of them not present, well I have chance p of a given
link being present raised to the dth power, then 1 minus p raised to the n
minus 1 minus d power. And how many different ways could I have
actually picked the, the other d nodes that I'm connected to?
well there's n minus 1 factorial over d factorial times n minus 1 minus d
factorial. So basically this is what's known as n
minus 1 choose d. that's the combinatorics of how many
different ways I could have exactly d neighbors out of a population of n minus
1 possible other neighbors. Okay, so that, that's a binomial
distribution. And these networks are also known
sometimes as Bernoulli because you know, you're flipping coins, binomial or, or
Poisson random graphs. Why Poisson?
Well, if you want to approximate this if you want to approximate this formula for
large n and relatively small p. the, when you look at 1 minus p raised to
the n minus 1 minus d power, this looks roughly like e to the minus n minus 1, p.
And this combinatoric choose the expression looks roughly like n minus 1
to the d over d facotiral. So when you, when you bring these two
factors together it's roughly n minus 1 to the d that you're left with, and this
is well approximated by e to the minus, n minus 1 to the p.
So, you know, the binomial approximation here, by a Poisson distribution means
that we can represent approximately the probability that a given node has exactly
d neighbors by, this formula here which is known as a Poisson distribution.
Okay, so lets have a look at some of these random networks, lets, this is
actually a random network I drew on 50 nodes with a probability 0.02 of having
any given pair of nodes connected. And this is one realization that was
drawn in what's known as UCINET, you can find that on the web if you want a
network package. It's one of several that we'll talk about
on the course. so I drew this using UCINET.
random graph 50 nodes 0.02. So let's have a look at the actual degree
distribution, compared to a Poisson approximation.
So if we have a Poisson approximation, where you had an expected probability of
links of 0.02. Then you would get the square dots here,
the realized frequency is by the diamonds.
And actually, these two are very difficult to tell apart.
So the frequency distribution of how many nodes have degree 1, degree 2, degree 3
and so forth. There's a couple of to few nodes that,
that had degree two, but everywhere else we're pretty much on the graph.
you know, a couple things to note, out of our picture that we had, when we go back
here there's a bunch of isolated nodes. it looks pretty much, the rest looks
sort of tree like, there's no cycles in it.
so we see several components, no component has more then a small fraction,
we're just starting to see one large component emerge.
if we go and instead bump this probability up to 0.08 on 50 nodes, then
we get a much more connected graph. We still have one isolated node, and we
end up with y'know, more nodes having many more connections.
What's the actual distribution look like? Well, now when we look at the actual, the
Poisson approximation, we have a curve here.
And then when we look at the actual realized frequency, we end up with
something which is not exactly on target. the piece is a little larger, so the
approximation's not quite as great, but it's still fairly close.
And if you had more nodes and looked at a random graph on a larger set, then you
know, these things would tend to coincide even more.
But again, the poisson gives us a reasonable approximation of what what we
would have expected to be realized. so this is a poisson distribution.
What we'd see if we saw links formed indipendently at random.
what thing, what sort of interesting is that when you look at some networks they
tend to have what are known as Fat tails. And one of the early looks at this in the
network context was by Price in 1969. And he was looking at citation networks,
and so, look, looking at, at papers that cited other papers, and articles that
cited other scientific articles, and then looking at how many sites given articles
have, and then you, you know, you could just put down sites at random, or you see
what you, you actually saw in the data. And he saw what were known as Fat tails.
And [COUGH], what that means is, effectively you saw too many articles
that had lots of citations and too many that had very few citations, so if you
look at the extremes you saw those being over represented.
And then numbers that were close to the average, you saw fewer having sort of a
middle amount of of citations, you either got very highly cited or very low
citation. this is related to a lot of other things.
There, there's a lot of things that have Fat tails.
these distributions go back to Pareto in the 1890s so things like wealth
distributions distribution of cities, populations around the world, words
usage. There's a whole series of different
things that have these kinds of Fat tails.
And in particular there's a well know paper by Albert, Jeong and Barabasi in
the late 1990s where what they looked at was webpages and connections to webpages
in the Notre Dame part of the world wide web at that point in time.
And what is plotted here, is a comparison between the in, in the blue are the
actual data of the world wide web connectedness, and in the pink color here
is what we would see if, if the links had been formed uniformly at random.
And the idea of a of a Fat tails here are the idea that when we look at the actual
representation, we've got, so this is log of frequency, log of degree.
we have a higher frequency of really high degree nodes, and a higher frequency of
really low degree nodes, and then the middle degree nodes are under represented
relative to things. So we have this Fat tails, and in fact
the distribution in this log log plot is almost linear.
so, you know, a line is a reasonably good match for what's going on in these data.
we'll talk about that in more detail as the course pers goes along.
But the important thing here, is that there are the distribution is a little
bit different than what we'd have gotten if we had just thrown down the lengths
uniformly at random. Kay, so this is known, roughly is what's
known as a as scale free distribution. the probability that a, given link has
degree d, can be thought of as proportional to the degree raised to some
power. And, part of the reason that this is
known as scale free is if we, you know, double the degree, then we, we, we scale
up, the relative probabilities, scale up so that.
the we'll still get the likelihoods, will be proportional to each other as, as we
are increasing degrees. And in particular if you take log of each
side of this what do we end up with? We end up with the log of, of the
probability, the frequency of having a given degree is going to be proportional
to, some factor times the log of the degree.
and so it, it, if it was fitting this, this distribution exactly, we should have
a line. And indeed, when we look at what we saw
here and, and you know, we get a reasonable approximation by a line in
those data. Okay, let's have a look at a completely
different kind of network and see whether we also see Fat tails or not.
So this is, Bearman, Moody, and Stovel's data, the high school romance data, that
we talked about just, a few moments ago. and here, we're looking at one high
school, nodes are colored by, gender, male, female.
And a link between two nodes means that they had a romantic relationship during a
time period that was captured in, in the surveys in, in this high school.
And, in particular, you know, here we see a different kind of structure.
We've got a bunch of, of, you know, 63 different diads, so just two people
connected to each other. We see one fairly large component which
has a couple of cycles in it, but looks almost like a tree.
you see some other different shapes here, a number of, of different configurations.
So, a bunch of small components, one larger component.
And actually, there's some isolated nodes that are omitted from the picture.
let's look at the deg, degree distribution of this one, and compare it
to the uniformly at random. So, here again, log of degree, log of the
frequency. And in this case when we look at the
actual data compared to the uniform at random, the fit of, if we actually try
and match these up, the fit in terms of matching up the data is a very close
match from just having. The uniform at random compared to a power
distribution, doesn't fit this as well, so this doesn't seem to have the fat
tails that the other one did. What does that tell us?
It tells us that degree distribution's of different networks can have different
properties. And if we're looking at some, they might
have something that looks much more uniform at random, what this tells us
about romance in high schools. but it it's saying that this is looking
much more like a a purely at random graph whereas the other looked like it had
something else going on in terms of its degree distribution looking more like a
power distribution. So, that gives us a a brief look at
degree distributions. We're going to be visiting those a lot
more in the course, they're [UNKNOWN] a very important way of characterizing a
network. And understanding how dense the network
is and also what the sort of variation is in degrees across the population.
so let's have a look at some other definitions that are going to be
important in networks.
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