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.