Now, there's been two idealized versus, we had uniform connection, or preferential attachment to a vicious proportional to the number of connections that you already have, and we can also have hybrid models. Let's take our random connection, and our preferential connection, and preferential attachment, and mix them up a little bit. So hybrid models for example could work something like this, to grow networks. For example, we could take a fraction of the links, and connect them uniformly at random, and another fraction by for example, looking at friends of friends of the connected nodes, right? So it's random uniformly with equal likelihood, I just connect to some, and then I look at the friends of those that I randomly connected at. So for example, you meet somebody at random, and then yes, it is likely that you also will get to know their friends. It's more likely that you meet somebody else at random again. So usually especially because triangles is the triadic closure, remember that? The triadic closure and the clustering coefficient? What was all that about? Yes. So that it's very likely that triangles will actually close. So you meet somebody, and then you will get to know their friends, and you very likely that's what we often see in social science data, that you will become friends of the friends of your friends. Now, for example you come in here, and you meet a group of people, and you randomly connect to those two and then preferably, you start to connect to their friends, not randomly to anybody in the network. You walk your way through this network, and can get you know other people. Now interestingly enough, most of my friends are connected to highly connected people. Means if you randomly meet somebody, it's very likely that they are connected to highly connected people. Why is that? Well that's simply because, highly connected people have lots of friends, that's why they are highly connected. So you can turn it the other way around, people who are highly connected, are highly connected because they have a lot of friends, which makes it very likely that, that your friend is connected to a highly connected person because, this person has a lot of friends, that's why it's very likely to be connected to your friend, as it's very likely to be connected to many others because it is highly connected. So if you look at it from the other side, it becomes evident why that is. Now, the second part is preferential attachment. Why is that? Well, that's simply because, nodes with more connections are easy to find, and will grow more friends. So the ones who are highly connected, those are likely connected to your friends, they are easier to find as well. They are easy to find because, they have more connections. So almost naturally see. That's why we find preferential attachment so often in social networks, because they almost naturally happen. The rich get richer, the ones who have many connections will receive more connections because they're easier to find. Especially, if you make your way through their social networks, and get to know one person, get to know another person, it's very likely that you will find somebody who's highly connected, which with your getting to know you increases the rich get richer, they get more degrees, more connections, and so forth. In reality, if you mix these two models, so these are two mix of a random connection, and then a preferential attachment network, that often that fits for example very good to World Wide Web data. So in this study here, they have shown that the World Wide Web, that means the distribution of how web pages are connected on the World Wide Web is asserted random, in two-thirds preferential attachment. So on the World Wide Web you might think, well this is preferential attachment you try to connect to very connected websites, but that's not the case, a third is really random. You just If you set up your new website, you randomly connect to some, but it might seem random. To you it might make sense to who you connect, but there's upon the overall picture from the bird's-eye view, for us it would leak a random, why do you connect to these random pages while there may be, I really like these things, but then, two-thirds you are driven by this preferential attachment logic. You will connect to highly connected nodes as well. So that's an example of a hybrid model, now we mixed random and preferential attachment. Now, we can use also other models. For example, we could use a random network, and then the other faction connecting it by geographic proximity. Just by proximity of you encounter. For example, you start to stand in line to pay your lunch, and you are at the cafeteria, you walk out of the cafeteria with your lunch, you meet somebody at random. The other person you meet, might not be the friend of this person, but might be somebody who is just close in this case. So it might be also that, so you can make your own model, and find the mechanism that actually also theorize in theory satisfies the mechanism you are after, and then make hybrid models and see how much of the growth of this network is according to this mechanism, random preferential attachment, geographic proximity just to have three, and see where that comes from, and then grow, simulate in theory your networks. Then compare to the network that you found, and you can surely link, well that's I think how this network actually grows.