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.
