So let's continue through our search for an explanatory model. Consider the following situation. Suppose we are given ten nodes. And we want to connect them in what's called a random graph. So I'll explain the process now of random graph formation. So with the random graph we take each pair of these nodes, so each of these ten nodes for instance. We can line them up in a circle maybe one, two, three, four, five, six, seven, eight, nine, ten. And we consider each pair of nodes. So, let's say, we'll consider this node and this node first, for instance. And, we're going to establish a link this a certain probability between the two. So we'll take some fix probability, whatever it is. And then we'll establish the length of a certain probability and the way to think about this, if the probability was say, 10%, so say there's a 10% chance we'll establish a link. You think of a, a biased coin, right? Which has a 10% chance of lading on heads and a 90% chance of landing on tails. And if it lands on heads, every time it lands on heads we establish a link. So, suppose in the first trial, we'll say, okay there was no. Nothing. So, it land on the tails, so we don't establish a link. Now we consider this pair. And again, we, it doesn't establish a link. Okay? Maybe then when we consider this pair, it does establish a link. And when we consider this pair, it doesn't. This pair, it doesn't, and so forth. And you just continue that, and you'll end up with a certain amount of the nodes connected. Right? So, I'm just randomly, connecting nodes here. So, with a higher probability, we're going to expect to have more links. And the reason we say, expect is because whenever we're dealing with probabilities, we always have to think of what's the expected outcome of that situation. So, the higher the probability is, the more likely that each of these trials that are running of success-failure, link establishment, or link non-establishment are going to succeed. And therefore, we can say that we expect to have more links with a higher probability, because link establishment is a random process. So, this could be an example, for instance of 10% link establishment. Okay. So now as we increase it to 25%, we could regenerate that and do the process again with 25% probability and we see generally, we have more people connected and here's 50%. So we see there's even more and as we went more and more obviously, in the extreme cases of zero percent we'd have no one connected and a 100% everyone would be connected to everyone else. But you can see that increasing this probability. It just makes us have we, as we increase the probability we expect to see more links and a higher density in the resultant graph. Now this doesn't sound like the way most networks form, right? So, this would be somewhat equivalent to saying, okay, we have ten people in a room randomly, and they don't know each other. Now, we put them in a room and we tell to decide randomly, whether or not to be friends. Right? That's probably not going to happen. Now, the question is, what small-world properties are needed? So, even though it doesn't sound like most networks, would it be able to satisfy the properties of small worlds? Well, for small worlds, we need small average shortest distance on the one hand. And random graphs do actually satisfy this, because there's no change in the probability between a very, between two different links depending on how far they are. It's some fixed probability, we said p, here. So whether the link [COUGH] is here to here, or the link is all the way over here, the pair. This pair and this pair are going to have the same exact probability of being connected by a random graph. So therefore, we are going to get or we're going to expect to get long range links, as well as short range links. So, it does satisfy this property, of having a small average shortest distance. But the other thing we need, is we need homophily in the graph. But, it doesn't satisfy that property. So it does satisfy small average shortest distance, but it doesn't satisfy homophily. And the reason it doesn't have this property is that, to precisely for the reason we said here. For homophily, we need to have some triangles or some triad closures in social relationships. Right? But, just because this person's connected to this person, and this person's connected to this person, suppose they are. Then it has, there's no increased chance of having these two people and be connected to make that trial closure happen. So in general, it's not going to have the homophily property that we need of friends of friends more likely to be friends. So therefore, the random graph is not going to satisfy the small-world property. So, it does not satisfy, small world. And so, what we really need to do is figure out a way to be able to quantify this homophily, which is what we're going to look at next.
