This is a high school dating network. So, the red ones are the girls and the blue ones are the boys. And this comes about from interviews, asking with whom did you date or who are you going to date. And then, they did a lot of those kind of interviews. And then, they found connections between individuals. So these kinds of ties in between the cell just tell you that this boy was, at some point, dating two girls. Now, have a look at it. That's a very happy boy here. There's a lot of interesting things here. That's why I like this slide so much because directly, it gives you the feeling, what I mean, with the network. Right? It gives you the feeling okay. There are notes, which we call agents, which we call individuals. And their connections between notes, which tell something about how they are related, and that could be many types of relations. In this case, it's just dating relations. There's also a couple of happy girls. I don't remember where. I think here. Yeah. I think I was around here myself. But here in Baltimore, I mean. And you see all kinds of intricate structures. You see structures that are kind of that. And that you see there is this big loop in there, so of there's something going on in that society, will go on and you actually hit yourself again. Think about it first. I'm not going to explain it, but you can think about that. There are all kinds of interesting processes happening. From this very simple graph, very complex network, as we call these things, we come to learn a lot of things. And it's been an area and it is still an area of very active research. Mathematicians, physicists, you name it, a lot of people have been struggling with trying to get a mind about these things. Can we quantify those structures? When we measure some things, can we predict how those clusters are really for, how this network really evolve, etc.. There's a question here. Sure, of course. I see there is differences in distances. Okay. The question here is that there are differences in distances like this one might be smaller than that one or so. That's not relevant in this case. No, that's just the way we visualize this. That's one of the drawbacks actually of those networks. They all look pretty cool, but they all look pretty cool. That's it. If you really want to understand what's in the network, you need to quantitatively analyze them. Sometimes, the things get so difficult and so, Harry, I will show you a couple of hairballs later on. Mark Newman, one of the guys who was quantifying all these things, he called those things hairballs. And at some point, he said, "If you really want to get your paper in nature, you should at least show two hairballs". But it's all about also how you visualize these things. And this is a very relatively simple network, but as networks get more intricate, it's very hard to visualize. There's a question there. You talked before about [inaudible]. Well, that's an excellent question. Yes, of course, it will matter. The question is if we were talking about the heterogeneity versus homogeneity. And now, if there are different behavior, for instance, in subgroups of here, that will then definitely drive. The topology might still be the same. Thank you very much for the question. The topology might still be the same. So the network, if you look at it, might still be the same, but dynamics on the network can be totally different depending on the kind of heterogeneity. That's why we always talk about dynamic on and the dynamics of networks. And, as always, you need to combine those things. It's not just one or the other. You need to look at both. You need to look at the dynamics of and the dynamics on those things. Now, those complex networks, you find them anywhere, everywhere. And, like I said, I've been at the field of research. I'm not going to go into the whole history. It's just a few remarks. We want to measure your those things. We want to measure things in those networks. And we, as the human race, we also form a network, right? So, to some extent, you are now all connected to me and I'm all connected to all each of you and you are connected to each of you other and some of you were never connected before, but now, we are because you are now part of this group that do this complexity course here and to you. And so one question you can ask yourself, "Okay, if I want to reach out to someone in the world, how far would people be away from me? Well, how far would that person be away from me? How many handshakes would that person be away from me?" So you are more or less one handshake away from me now, and then if I pick someone else in the world, it doesn't really matter where, let's say in Kamchatka. I don't know how I've come at that, but the same Kamchatka, how many handshakes is that person away from me? Now, what you find if you look into these things and there's lots to say about that, but that's about six. Six handshakes away, I can reach everybody in the world. Six handshakes, we're all six handshakes whereas we are seven point something, so many billion people scattered across a huge globe. We're all six handshakes away. There is always a person you can reach, any person you can reach through these six handshakes. And so this is me at some point in time. In 2010, I shook the hand of this fellow. I'm still recovering from that. That's [inaudible] from Russia and he shook the hand just a day after that. This fellow, and so I am two handshakes away from Barack Obama. And for a cup of coffee, I'm happy to shake your hands. So then you're handshakes away from Barack Obama. But pay attention, you're also to handshakes away from him. Okay. But gives you the flavor of what we mean with these kinds. So that puts a number to huge cup. I mean think of the hairball that all those individuals that we have on the earth were formed. That's just mind-boggling, right? But there is a simple measure, dimensionality six. Now, unfortunately, life is not always that simple because if you look into those networks, we expect that if you look at the connectivity of those networks, if I would look at your connectivity in your social network, I would look at how many people are connected to you. So the k here is the number of the connections that you have. And then I look at how many people have the same kind of connections. Then I would expect this, as you expect everything in science, a boring normal distribution. Right? So you would expect that it will be an average of connections that people have for individuals or things have. And there are some that have a few connections and there are some that have a lot of connections, and that's it. But if we look into complex networks and we find them again everywhere, and this was first actually done by crawling the internet and looking out documents are connected to each other. So you see how easily this allows me to switch from talking about people to talking about documents to talk about how banks are connected. So that's a strength of complex network that you can make that switch very easily. But what was discovered actually that it's not this kind of normal distribution but it shows this scale free recall, the scale free redistribution that has this characteristics. So this has the probability of having k connections is k to the power of some kind of function gamma. So with one gamma, I can actually express that connectivity. But the interesting thing here is that here is k. This is on the lucrative scale as you can see. So the interesting thing is that there are a lot of people, or a lot of documents in this case, with few connections and are a few with a lot of connections. Right? And we find it everywhere. We find that in airports, the connection of airports where planes are done the link between the airports. We find it everywhere. And the important thing is that in the case, we would have a bell shaped curve, a normal distribution. We would see things here. But in the case, we have this logarithmic power law distribution. We do find a few hubs that really have a lot of connections, which are the super hubs, the super connector if you like. And that's one of the very typical characteristics of those networks. Now, like I said, you'll find them everywhere. Here's an example of the structure of the organization. It doesn't really matter how it was done, but again, you can lay out the networks of that. This is a recent paper last year. This is right here on networks in anthills. You'll find them in metabolic networks. We did some research ourselves on protein networks, how the protein networks of your body, of your immune system interact with the proteins of the viruses that could be in your body. So you find them everywhere. I'll just skip this one. And so the trick is if they are everywhere, they might not be a panacea to our problem but they might be a thing that we can use to study complex network. Here, actually, might be of interest to you. Here, we see an opinion network. So you could just say I'm in your social network and somehow, I'm trying to change your opinion about complexity science at this point of you. And then you can ask yourself a question. You've talked to someone else how and how much of my opinion will then percolate to the other one. So that kind of questions you can try to find an answer to. And, in the meantime, the network that you have and that I have will change over time and then makes things actually very interesting. But the thing I wanted to say here is because of this structure, to p (k) to the power of k minus gamma. With one variable, gamma, I can describe those networks, and that's an interesting observation. But, at the same time, if you still remember your high school mathematics, if I would calculate the average of that function p (k) to the power of minus gamma, if I would calculate the average of that one, that would be k times p (k) integrated over k. I see some people nodding. That's good. So we want to calculate the average. That would be something like k times p (k), dk. Integrate it. And that would give me k to the power minus gamma plus 1 because it's k times that and then integrate that thing, and that means that when I have gamma around 2, I get a divergence. So there's some kind of weird thing happening at certain points at certain values of gamma that actually make these characteristics very special. So I skipped this part because the movies I can not show. And now looking at it, you see we're actually making use of the network theory to go a bit deeper about things. Looking at that gamma, we find that for collaborating networks like here, for actor networks in movies. We can actually determine those gammas and we find that there is a region where the average of the gamma diverges and there's a reason when it comes more finite. Now, you'll also notice that's just the average. I can also calculate the variance. Then I need to multiply it with k square, and then things get hairy. They get a divergence of the function at gamma is 3. So this is the way we characterize those networks by using, by calculating that gamma. Now, of course, it's a very simple way of characterizing it, is way more complicated because this gives you the average number of connectivity. It doesn't tell you anything about the rest of the topology. I can create a billion different networks with one gamma. Right? So you need to measure more. You need to measure out of the clique's. What is the things that you will learn in the afternoon? What are in between the statistic connectivity, the structure, the topology of those networks? But that's a thing you will come across soon.