Remember what a component is. Again, so a component is when things are connected. We have the strong and the weakly connected component. But a calm component is when the nodes are like connected in one group, and a giant component is basically the largest components. So we can ask when are most nodes are actually connected? When they are no isolate groups anymore, no separate groups, when they're all in. That's a very important question because everybody is connected if there are no isolated groups that are not connected. For example, a disease could spread to everybody or innovation could spread to everybody. So when do these giant components emerge. Let's go into that question a little bit deeper because it tells us something interesting how these random networks behave. Often, they have threshold function. That means nothing happens for a long time and then there are these tipping points and suddenly something happens. So are these tipping points threshold function? So we can ask at what average degree does a giant component emerge? So that's an emergent phenomenon. It happens. So for example, to make it more concrete, how many friends do your friends need to have for information or a disease to spread quickly throughout most of the networks. So that's actually what I'm asking. So the giant component is when most are connected then a disease can spread, would be a negative thing or innovation could spread, could be a positive thing or political opinion can spread or that depends and if you like if that's a good or bad thing depends on if you like public opinion or not or rumor or lies are fake news could spread through the entire network if everybody is connected. So when do we have networks where everybody is going to and how many friends do you and your friends on average, how many friends does everybody need to have in order for us to have most people connected in one giant component that's the technical term of it. So let's do a numerical simulation and just by simulating a lot of random networks see that on average, what do we find, how many numbers of friends do my friends need to have, what's the average degree that we need in order for this giant component to emerge, very straightforward. We just simulated many times. I used a software here called NetLogo, a software that we will use a lot today and also in other lectures. That helps you a lot to simulate as it makes it very easy to simulate these things. So here I have my random network, I've 50 people that are just threw on that and don't have a network. I've 50 people, unconnected people. Now, I randomly pick two of these people and I connect them. So randomly, out of these 50 people that are created here, I connect two of them always or I could also add them. So here for example, I added one to appear that already existed and I have that's my giant component that is red. So that's the largest connected subpart of this component. Now, you saw here. Now, I have five, a group of three and a group of two merged into this giant component and my giant component is also increasing. The likelihood of connecting with anybody is uniform, here it's a random network. But the giant component already big. Sometimes, people connect to somebody of the giant component in the giant component is increasing with that then. So here, I have a pretty large giant component and already and you sometimes see subgroups start to emerge into it. So that's the idea of what's happening with the giant component. At the end, you see like most people are now connected to the giant component. What you see here on the side of this little graph over there is it measures the fraction of people in the giant component and the connection per node. So here for example, we have four people in the giant component now and you see by the little red line, not a lot happens. We have two connected with four and now both of them connected and what happened is now a big fraction of 60-70 percent of the people are already in the giant component, so that made a big jump. But still, the connectivity is pretty low, the percentage, the fraction of people in the giant component is pretty low as we can see. Now, we have two components, one that's a little bit bigger and the other ones that not open but it also can switch between them and check out what happens now. Now, these two big groups merged and that gave us a big jump in the fraction of people that are in the giant components or what basically happens subgroups build up and at one point all these subgroups start to emerge. That gives us a big push in the number of people that are in the giant, in the largest components. So see again down here, nothing happens about 12 percent, 13 percent, 15 percent in the giant component and then at one point it starts to go up. It's pretty flat and then at one point it starts to go up. The question is when that happens. Well, as you can see here indicated by this line that happens if you have an average one connection per person. That's tends to happen. Let's do this again, let's run it again to see if there's something to that. Yeah, nothing happens a lot. The fraction in the giant component is pretty low and suddenly after we get to this threshold of one connection per node, it starts to go up. There's this phase transition, this tipping point, this threshold function after which things start to change. So there's a qualitative difference, more is different. You can add and add and at one point, well, it becomes qualitatively different. So emergence habits. Now, why is it one connection per node, what is special with one connection per node. Well, think about it. If you have your group of friends here. You have yourself and then you have a friend, and then you have a friend,and then you have a friend, if there's minimum of one connection, yeah, we can all be connected if we've less than that we might not or we cannot be all connected. After we have one connection per node, we can all be connected. Then, what you saw in the simulation as well, what often happens is we have these groups and then these groups start to merge. Then in subgroups in this giant component and we get really a giant component. Most people want these groups start to merge happened to be in this giant component that's why you get this tipping point function here. Mathematically, you can, if you think about it more and if that's what you're into, you can see when that happens if the probability of Meko connection is larger than one divided by the number of nodes. Then you get this giant components and cycles by the way. Remember what cycles were? You get them too after that threshold and you can have other things. For example, if you have a probability equal to the logarithm of the number of nodes divided by the number of nodes, you will get a fully connected network. So these are different things that you can say about the network with almost certainty. So almost surely these things will happen almost surely as you saw we run this simulation a few times. Every time it's a little bit different but there's something we always saw after it hit this threshold, something qualitatively different happened. Now, when you use a random network hypothesis since you know that a random networks almost surely, these properties will emerge if your empirically observed network, the network that you scraped on your favorite social media sites doesn't have these properties. You can say that with less likelihood of that it's very unlikely that this network that you have is a random network. At least with a specification that you gave it. You can also give this network other specification. If you do an exponential random graph model, you would specify some other things