[MUSIC] A very important group or often a subgroup of a network are these triangles that you've already seen in this examples. And that has to do that's often called clustering, triadic closure, and transitivity. Transitivity is the term actually taken from mathematics. So if you've done some mathematics you heard this term before. What it means here is basically how many triangles are closed or what fractions of my friends are also friends. So I'm here and I have two friends and then are they also friends, is this triangle closed? And it turns out that in social networks, we can predict future friendships with very high likelihood if it's in this kind of situation. Eventually your two friends will also meet. So I can now start actually now we're going almost in network dynamics already, be making some predictions about what will happen. Sooner or later your friends might also be friends and that often actually what happens. Now that's not the only, the closed triad, the triadic closure is not the only triangle that we have. Actually I can make an entire sentence of triangles, triangles are very important in social networks. And here you can see all 16 possible triad types. Which they are also called often motives or isomorphism in a directed networks and these motifs, we then look for them. So we go through the network and actually instead of looking at bigger groups, you just say, how many of these motifs, one of these 16 triad motifs can we find in the network. Kind of like make a triad census around the network and then you can compare the amount of triads that you find it in your network to some other network and say well in my network a find a lot of this kind of substructures. That's what you do in social network analysis. And it turns out that completely connected cliques in a triad, that means what is here called 3-0-0, appear more frequently than any other motif in social networks. So that's when I said before, right, triads, closed cliques among three people are very common. So that's called a clustering coefficient. So the clustering coefficient is the number of triangles. The number of closed triplets of the number of connected triplets of all the different nodes that you have in the network. So that gives you an indication that's that one number. It gives an indication of how many of these triangles are closed. The triangles, you and your two friends, and them being friends, that's extremely important and can tell us a lot about social network structure. On the other hand, if these triangles are missing or some other connections are missing, that can also give us some very interesting insights about our social network. And that's where this term, social capital comes from, or structural holes. So these are related to I mean, they're many shades of grays and we cannot go into the details. But a structural hole base, you can think about it is a separation between non redundant and redundant context. So here what you can see in this network, they are three structural holes, right? So, A, kind of like the bottleneck between these structure holes and it brokers, A brokers between them and this network here, it's E. So E is kind of like brokering these two these two partitions of the network, these two groups. Whereas here on the side we have structural holes. And that gives us information about about the network structure, about the importance of different nodes. And often structural holes also tend to close and if not, we would like to close them, right, especially in social networks. If you want, if you're going to diffuse an innovation. Sometimes, sometimes you want to create structural holes. For example, if you try to have a vaccination. If you try to stop the spread of something, of a rumor, or of a disease, well, it's good to have these structures because then it's very easy to control. In this case, I can control it just by controlling the node E or here I can control the node, A. So the structure, these brokers connect different partitions of the network.