 So everybody should be in. Thanks again for being here. So what we have now is, well, let me remind again the rules. So please, if you have a question, post it in the chat or raise hand using the Zoom feature. And let me introduce the next speaker. So the next speaker is Arsham Gavazie, who is a PhD student at the University of Trento and from the Center at Bruno Kessler, who works on network science, statistical physics, and complex systems in general. And today is giving a tutorial on complex networks. So please, Arsham, thank you very much for being with us. And yes, great. Hello and welcome to this tutorial on complex networks. This would be a very brief introduction to network science, a science that has application to a broad range of disciplines. So no matter you are from a sociology background or physics, biology, economy, ecology, probably at some point in your future career or academic life, you will encounter networks. And it's better if you have the basic backgrounds to understand what's going on in network science. So first, I would like to thank the organizers for inviting me as a tutor. I'm Arsham. I'm a PhD student at Trento University. And I really hope that this lecture would be something of value to you. And also, you enjoy this time that we are having together. Let's go a little bit into the details of this talk. This is the content. First, I'm going to talk about complex systems with you. You all know complex systems. Complex systems are everywhere around you. You just need to learn how to identify them and how to characterize them by their important properties. Then we move to structure and discuss the structure of these wonderful systems. And after that, we can talk about the basics of networks because we model the structure of these systems as networks. We talk about the features of real-world networks. We consider only two or three important features for this short tutorial. Although there's a lot more to learn about the common features of these large-scale networks. After that, I elaborated on the concept of node centrality, which is just to answer the question that which node is the most important one in the network. So you have a network. You want to rank the nodes according to their importance. How to define importance? We will talk about it, but it's ill-posed question. So you can define the importance in different ways based on what problem you are going to solve with these definitions. After that, we move to network formation. What processes are underlying the network growth, network formation? How these networks emerge from the smaller-scale interactions between the nodes? And at the end, I will provide a very short glimpse at the problem of network robustness, which is a very important problem. Because, as you might guess, all systems in nature are in danger of damage. And it's important to know if a system is robust or not. So this is a very brief look at the contents of this talk. I'm going to introduce two references for this science, for the network science. These books are loved across the world. So it really depends on your taste, which of them you would choose. Network science book by Barabasi is probably easier to understand. So if you are from a non-mathematics background, I mean, if you are not a physicist or computer scientist or mathematicians, perhaps you will prefer Barabasi book. Also, there is a network science introduction by Professor Newman from Michigan. This book is a little bit harder to grasp and contains a lot of details that are probably more interesting to people from mathematical backgrounds. So it really depends. It's a preference to choose between them. We are going to talk about complex systems. So complex systems, by definition, are large collections of entities that interact in non-trivial ways. And they are characterized by their emergent properties, which means that their properties cannot be understood if you take the system into parts and study their parts. You should always consider the system as a whole and take a systemic view to understand what's going on in these systems. And these systems appear across disciplines. So you can see them in sociology, economy, physics, or biology. And perhaps that's why famous scientists like Stephen Hawking has said that this century is the century for complexity science. So probably you are learning something really important. Some examples of complex systems are shown in this slide, like the brain that is probably the most miraculously complex system in the world. And it contains a collection of neurons. So there is a large collection of neurons that are interacting in different ways using different neurotransmitters, electrochemical signals. And from this system emerges consciousness, memory, and other properties that you cannot learn by studying neurons in isolation. The same holds for our societies. So we are exchanging information in forms of text and voice messages. And our larger-scale behavioral patterns are not yet fully discovered. We cannot claim that we understand human behavior. Also, there is transportation systems, biological systems, like protein-protein interaction networks or treated or prey interactions in nature. And also, map of airlines, they all show complex systems. And you should have really a systemic view to be able to say something about these systems and to understand them at some point. All these systems exhibit structures. What do I mean by structure? So if you take a look at this network of neurons, this bunch of neurons connected together on the left side, you see that the connections between them are not random. So each neuron is connected to a specific set of other neurons and they are exchanging information. And there is something like a structure that is determining their relationships with one another. The other example is, for instance, Twitter. You can see in the middle a network of people that are exchanging information on Twitter or Facebook. You are friends with a certain number of people and you are disconnected from the rest of the world. There is a structure and a pattern of connectivity around every individual. And this structure brings people together and restricts the flow of information between them. The third example is a network of animals. This is the predatory pre-relationships between different animals. As you can see, Fox doesn't eat mice. It eats rabbit. And the rabbit eats carrots. But it's not that everyone is eating everyone. So there is some patterns and regularity there. And this system has a structure. These structures are often modeled by networks. So just to tell you briefly, neural networks or neural networks just show you how these neurons are connected together. Social networks are just representing the structure of a network like Twitter. And also, food web is there to represent the asymmetric way animals are interrelated in a predatory pre-relationship. From here, we are safe to move into the basics of the networks, get a clear view of the structure of complex systems. So a network is basically a set of nodes and links. Nodes are those blue circles that you see. They represent the entities or units or components of a complex system. For example, in brain, they show neurons or brain regions. In social networks, they show people. In food web, they show animals. And then links show how two nodes are interrelated. So if node one is in some way connected to node B, then you put a line between them and it defines a link. There is different terminologies in this domain because the science is old and there is social networks, there is graph theory, there is network science, there is complex networks. So depending on the background, people use different terms. In physics, we almost always say no-down-link. In mathematics, people usually say vertices and edge. In social science, people say actor or connection. People use these words interchangeably, but they are all referring to the same thing. So they are really the same thing. I'm putting it there because many people get confused by reading texts on network science. The important point of this slide is that networks represent the structure. Without networks, we miss a systemic view. And also the number of links attached to each node is important and we call it degree. Degree is the number of connections each node has and it will come important in the next slides. One distinction you need to be able to make is between asymmetric and symmetric networks. So in an asymmetric network, the connections between nodes can be asymmetric. It means that maybe I am your friend or but you are not my friend. So there is a one-sided relationship between us. And this is an example for a social system, but in many other domains, you can see asymmetric networks and you need to use directed networks to represent asymmetric connections. As you can see on the left side, section A, we are observing arrows going from nodes to nodes. So for example, let's take a look at node D and B. Node D is connected to node B, but node B is not connected to node A, D. So this is a type of asymmetric relationship between the two nodes while in the network represented in section B, you can see that there is no arrows, there is only lines. So every node is in a bi-directional relationship with every other node to each it is attached. So we have directed and undirected networks. And another concept is that as I told you in the last slide, the number of connections each node has defines its degree. Here, we can have two different definition of degree. You can take a look at the connections inward and call it in-degree and you can take a look at the connections outwards and call it out-degree. For example, let's take a look at node B in the directed network. As you can see, there is two arrows entering node B. So the in-degree for node B would be two. And there is one arrow emanating from node B, which defines its out-degree. So the out-degree for node B is one. Of course, for undirected networks, we don't have out-degree or in-degree, there is only degree. Examples can be food web. In food web, animals often are either prey or predator. So their relationships between every pair of animal can be better shown by an arrow rather than a line. But then in WhatsApp, you are either in contact with someone in a bi-directional way or you are not. It's very low probability that you are connected to someone who doesn't respond at all to your texts. So in WhatsApp, probably you want to consider the structure to be undirected. There is weighted networks and binary networks. And this is important to learn the distinction. It really depends a lot on how much information you have about the system. So if you know that, for example, in a social system, Fred and George are friends, but you don't know how much friendship there is. You can't compare the friendship between Fred and George with other people. You only know that they are friends. You define the network to be binary, so you make a link between Fred and George and every other people that you know to be friend. And for sure, for people that are not friends, you don't draw the line. There is no link. It defines a binary network. Invaded networks, in contrast, you know how much friendship there is between two people. So for example, you measure how many text messages are exchanged between Fred and George. You see that it would be like a 10 message per day. You compare it to other people and you decide whether the weight of connection between Fred and George is high, or is low. And then you get a weighted network in which there is more information compared to the binary version. To represent them, people often use the thickness of the links. So a link with higher weight is represented thicker and the tinier links show the weaker connection between pairs of nodes. This is the distinction between weighted and binary networks. Another important definition is the definition of path. So imagine that you are on a certain node of a network and you want to navigate your weight to another certain node. So let's take, for example, node A and node B and to do that, you need to jump from the node on which you are, go through a link and land in a neighboring node and continue it. So you move through a sequence of links and you eventually reach node B. So paths are just sequences of links and they can be of different length depending on how many links you have passed to get to your target. So you can choose a very long path, like a huge number of links to reach node B or you can just go through the shortest possible path that connects node A to node B. Perhaps one can argue that the shortest one is the most efficient, whether it is really a transportation network when you want to change the flights to get from New York to Sao Paulo and you want to get to your target in the fastest way or it can also be neurons in human brain that want to exchange information and it would be probably much more efficient if they choose the shortest path to communicate. So finding the shortest path, connecting every pair of nodes becomes something of importance and there's a lot of algorithms to calculate and compute the shortest path between every pair of nodes in a network. It's not easy, so you need to find an efficient algorithm because especially if your network is very large, finding shortest path has become very time consuming and computationally costly. Another important concept is transitivity. So as you can see on the right, there is a network of unconnected nodes. The second picture from the right is showing a connected pair A and C while B is isolated. The open triad example is where B is connected to A and C but A and C are not connected together and there is the closed triad. This is basically a triangle and this is very important because again in a social network example, you can think of yourself being friend with Fred and George to other person and you want to calculate the probability that if you are friend with both of them, they are friend with one another. So you are shaping the triad, the closed triad. This is very important because it shows how densely connected people are around you and to measure it on a large network, you only need to count the number of triangles or closed triads and divide it by the maximum possible number of triangles in that network. So it gives you, for example, 60% or 0.6, then you say that my transitivity in this network is 0.6 and you can compare it with another network and obtain that the other network has 0.8 transitivity. You can conclude that in the second network, the connections around people are more dense. They are more likely to be in closed triads like three body interactions and physics. So we move to probably one of the most important yet very simple concepts, which is modularity. This is a network you can see that there is two groups of nodes, two communities that are almost separated from each other while they are very connected inside within each of these communities. The red community represents the people with more tendency towards conservative parties. This is America, so it is showing the Republicans. The blue party, of course, is Democrats and you can see that people inside the party tend to make connections with similar people, with people of similar ideas and similar political thinking. So as you can see, there is two modules formed in this network and there are algorithms that allows you to find the modules in the network, find the communities, find how strong these communities are, meaning that how separated they are from other communities. This is not compulsory that a network has two communities, so a network can have a lot of communities. And this is just an example because political system in America is, has two poles generally, so that's why you see two communities here. Again, there is a lot of algorithms there is no consensus that which of these algorithms are better, but you can use these algorithms to find the community structure of your network. Now, can I ask a question? Sure, sure, let me just share the video and stop sharing here. Okay, so I'm here. Yeah, so I wanted to ask you about the difference between transitivity and connectivity. And what? The difference between transitivity and connectivity. So how do you... Okay, so connectivity is just the number of edges you see in the network, the number of links divided by the maximum possible number of links, okay? Yet the transitivity is about the triangles. So connectivity is about the interaction between a pair of nodes while the transitivity is about a triangle, three nodes. So this is of course in a very dense network, they too are high simultaneously, but then in a sparse network, it can be different. So in a sparse network, you can have high transitivity by low connectivity. So how would you interpret high transitivity but low connectivity? So how do I interpret it? I'm not sure if I correctly get the question, but anyways, around every person, there is a community of densely connected people, but perhaps globally, if you think of network, it's the connections between different communities is low compared to a new model or a typical network. So I have to probably... Dr. Greeley, should I answer all the questions or should I put it under? So if you want, you can ask, you can answer now. So I remember the participants that if they want to ask the question... Okay, okay. So probably I will move ahead, but at the end we will have 15 minutes to discuss. There is a question about the slide that you just showed. So if you want, I can ask it now, but... Okay, okay, let's go ahead. It's really, I think, a tough question. So you showed this plot about the two parties in the US and there were some yellow links. What do they represent? This is a question. Okay, so there is other parties in America and I think they are libertarians. I'm not sure, but they belong to other parties. But the density of them is really low compared to the two big parties. Yeah, great. I think that answered the question. So please go ahead and again remind the participant to use the raise and back to... Okay, okay. So I will go ahead with the talk. Real-world networks. Because in real-world we are observing networks for two decades and now we can say that these networks are showing unusual and extremely interesting characteristics. They are similar in one way or another. I'm going to discuss three features that are common among a lot of networks. So the first feature is a small wordness. As you can see on the left, there is a network with regular pattern of connectivity. So you can really see that there is a symmetry between every pair of node. Each node is connected to its neighbor and to its second neighbor, but not connected to any other nodes. This is a regular network. You can get the pattern by a loop. One point is that there is no difference between curved links and straight links. They both are telling the same. It's just a matter of representation. On the right side, on another hand, there is random network. As you can see, there is no pattern of connectivity. Everyone is randomly connected to everyone and it's chaos there. In the middle, you see the small word network. So there is the pattern, there is the regularity, but there also you can observe some irregularities, some randomness. And this randomness is represented in terms of long range connections, connecting these 10 nodes of the network. This is the topology of the small word networks. They are between regularity and randomness. And they have a lot of important properties that are good for complex systems. For example, they have high transitivity, which is related to the number of close triads, as I told you in the last slides. And the average shortest path connecting pairs of nodes is relatively low. So this would be easy to navigate your way from each of the nodes and reach another one. And that's why these networks are often considered as efficient. They are observed in multitude of systems, from brain networks to social networks. And the model of a small word has been used to justify many observations people have done on real world systems. So small wordness is a characteristic of real world networks. There is heterogeneity. It means that in real world networks, often there is a lot of nodes having only a few connections. As you can see, most of the nodes in this network have only one connection or two. And then there is a very tiny minority having a lot of connections, as you can see. This is a node A and C in this example, having each of them have, okay, about eight or even more links connected to them. So they have high degree. And that's why we are calling them heterogeneous networks or a scale free networks. So these heterogeneous networks are reminiscent of what we observe in economy, as you can imagine or you know, probably the distribution of wealth between people is somehow power law, meaning that it's heterogeneous. There is a tiny minority having a lot of money while most of people in the world are poor. So the distribution of degrees in these networks are important. They follow a power law distribution. If you are a physicist or a mathematician, you probably know what a power law is, but it's a signature of a scale free behavior in the system or a scale free systems. That's why we call these type of heterogeneous systems scale free. So a network can be heterogeneous, but not be a scale free, but let's forget this distinction now. And from now on, I will call heterogeneous networks like these scale free networks. The other element is hierarchy. As you can see here in this picture, there is five communities. Each of them consists of five nodes that are interconnected in a dense way. And these communities are just all of them connected to the community in the center. So the community in the center has somehow access to all other four communities, but the communities on the peripheral part are not having the same access. That's why we call this network a hierarchical modular network. And there is the element of control in this network. So the top module probably can control some properties or some information processing going on in the lower modules. So this is the third feature of complex networks that I was intending to share with you. I hope that you have now an idea of the common properties, of some of the common properties of real-world networks. And now we are probably safe to go to the topic of centrality. So what is node centrality? You have a network and it's a mess, believe me, when you look at the networks in your computer, you cannot really say many things, but what is the most important node in the network? Or what is the most important connection or link in this network? And you want to find a ranking of nodes based on their importance. This is a very abstract network represented here. It can be a network of neurons, individuals, proteins, and you want to know which of them are the most influential, important, or whatever different definition of being critical for the system you have in your mind. Also, you might ask which synops in a neural network is the most important or which social connection or which biological interaction is the most important link, defines the most important link in my network. And to answer this question, you first need to define the relative importance. How do you define it? So this is something that's worth thinking about and it really depends on the type of the system, the type of question you have in your mind, but I'm going to give you some of the most important, yet simplest definitions of node centrality. Probably you have guessed this one because it's quite trivial. You want to know what node is the most important. You calculate the degree of each node and you would say that the node having the most number of connections or the highest degree is probably the most important one. Here in this example, I highlighted the node with red. So you can see the highest degree centrality node is node J and you can really make the case that these nodes are very important. Like in social networks, imagine someone having millions of friends and connections so they can have really an impact on how society thinks about different things. The same example happens in protein-protein interaction networks and brain and also any other complex system that you can possibly think of. So degree centrality is a way to rank the node and say that, okay, this one is more important than the other one. The second definition of centrality, I'm going to discuss here is close and centrality. So you calculate the average distance of each node from other nodes. To do that, you need to compute the shortest pathos connecting each node to any other node and average the length of these shortest pathos. And you will find one of the nodes that has a very low average compared to others or the one that is simply closest to the rest of the networks. And you would say that this node is probably the most important according to closeness centrality. In this example, this is node P. To really calculate it, you need a computer or you need to really write pages of calculation for this network, even though the size of the network is relatively small. So there is algorithms, as I told you before, to calculate the shortest path and there's algorithms to derive the closeness from the shortest path. Another way to define the centrality is really to find the middle guide. So imagine that you are working in a physics department in a university and you want to have some collaboration with biologists from another department. And to do that, you usually find the middle guy, the guy who works between the two departments. Maybe he is a physicist, she is a physicist or he or she is the biologist. It's not important. The only important point is that he is the middle guy. He knows everyone from both departments. He can introduce you to many people. He can speak the language of both departments. So the middle guy is really important. Also in brain, if you think of two brain areas that are trying to exchange signals, there probably is a middle region that the signals should travel through it and it should pass the signal from one region to another. So the betweenness of nodes in network is important. And as you can see here, it's quite graspable usually that node H has high betweenness. It's connecting or less two patches of nodes. And this is another way that you can think of the importance of nodes in a network. So here we are moving to the subject of network formation. I'm going to discuss two theoretical models of how a network can be formed. What are the mechanisms underlying the growth of networks in nature? They are very complicated and probably much different from these two theoretical ones that I'm going to introduce. But the theoretical models are providing some insights into the system. This is the random connection. This is a very simple model. You can see on the left side, there is a lot of nodes that are totally disconnected then you add links with probability. So you add link with probability zero. There is no links added. So the nodes are disconnected. In contrast, on the right side, you can see that all nodes are connected to all other nodes. So you are in a regime where probability of connectivity is equal to one. Then moving from the left side to right side, you are increasing the connectivity probability and you will see that there would be different patches of nodes that are connected inside but they are not connected outside. They are not connected together. And the number of these patches would decrease as you increase the probability of connectivity. And at some point, there will appear a large giant component, a component that is very big and it's integrating all the network together and the network will be formed after that point. The point at which the giant component emerged is called the critical probability, has been shown by PC. And the size of the largest connected component in the system is a measure of how this phase transition from disconnectedness to integrity happens. As you can see in the plot on the bottom, this is the size of the largest connected component you are increasing the probability of connectivity. But firstly, there is not a real change in the size of the largest connected component. The size of the largest connected component is basically the number of nodes you can see in the largest connected component. So this is not really zero, it fluctuates but this is approximately zero before critical probability and then after critical probability, you can find that the giant component appears and then integrates the whole network. So this is only a formation model based on adding links with probabilities and from it, you see the interesting behavior of the giant components. The other model is preferential attachment and this is very similar to the idea of reach gets richer. So you start from one or two nodes, you add nodes, each of the nodes that is added in one specific step gets connected to the existing nodes in the network with probabilities. It is more probable that the newly arrived nodes get connected to the node with the highest degree. So you can see that the nodes that has already the highest degree will increase its degree very fast and you end up in a heterogeneous network where there is a tiny fraction of nodes with a lot of connections while most of the nodes in the network are having only one or two links as I told you in the last section. In this network, in this visualization, the size of nodes is taken to be proportional to their degree. So the number of connections each node has determines the size of the node, as you can see, there is five very big balls in this network. It means that they are hugely connected and you can see a large number of various small dots representing how heterogeneous these networks are. From this preferential attachment, you can analytically derive the probability distribution function for degree. So you can really show that degree is distributed in a power law way. Again, another emphasize on how scale-free the networks in nature can be people who are familiar with power laws from a statistical physics or statistics will get what I say. It's not a hard concept. You can really follow it from Borobasi book on the topic of network growth and preferential attachment. You will find the full derivation of how the power laws are obtained mathematically and what they mean. But I suffice to this for the moment because I don't have a lot of time. So the last part of this talk is about robustness. And robustness is the study of how robust or tolerant the network is against failures inside or external attacks. So complex networks because they are always in touch with dismantling two pieces and the system will die. System wouldn't maintain its function and it's very important to see how robust network is. So the procedure considered to model the damaging networks is just to remove notes. So you either randomly or according to an algorithm you select notes from the network, you remove them one by one as you can see from the image on the top left, the network is whole and then you remove the note around which there is a green circle and then in the image on top right you remove another bottom left, you remove the third one and the bottom right shows the network after removing these three notes. As you can see, the network was connected in the first place but it is now dismantled in the fourth place into one, two, three, four, five disconnected components. And the size of the largest component here is just one, two, three, four, five, six. So the largest connected component contains only six notes in the fourth step. While in the first step it contains probably more than 20 notes. I don't take the time to count all of them but as you can see the size of the largest connected component shrinks as you remove the notes and as the network dismantles into pieces. So this gives you a criteria to measure the robustness of networks. Here is the example of a lattice, a regular lattice. So every note is connected only to four of its neighbors in a very regular way. And then the size of the network, I don't know how much it is but it would be probably around 300 or 1000. So, and then you remove the notes randomly. So you make a code to remove the notes randomly and you will study how many disconnected components there is in your network. What is the size of the largest connected component? Of course, in the beginning, the size of the largest connected component is equal to the size of the network because everyone is connected to everyone. There is no different patches, there is no disconnected components. Then the number of disconnected component grows and the size of the largest connected component shrinks. As you can see, there is a phase transition again around FC, the critical fraction. So you are removing about 0.4 of the total links. It means 40% of the total links. And you see that the size of the largest connected component goes to zero. So the network totally dismantles. There is no connections in the network after that. This is very important. So you can see that the lattice is not very robust against random failures, against random removal of notes. But in other types of networks, this is internet that has a scale-free topology. So the structure of internet is really scale-free or heterogeneous for sure. You can see that if you follow the green light, you can see that the network dismantles after removing 90% of notes. So this is very, very shocking. These networks are extraordinarily robust against random failures. They can just randomly remove all of the notes. As you can see, the largest connected component has large size even after removal of 50% of the notes. And this is weird. So the take-home message here would be the real-world networks are probably robust against random failures and random attacks. While I'm going to get a bit into the details and I'm going to introduce you to the targeted attacks. So you can use the notion of centrality that I introduced in the last sections. And you can rank the notes based on every centrality of your preference here. The degree centrality has been chosen, meaning that each note is important according to the number of connection it has. And then you remove the note according to that ranking. So if this is degree centrality, you are basing your attacks on, then you're firstly remove the high, the note with the high key. The note with the highest number of connection, then you remove the note that is second with respect to the number of connections. And you go ahead. The purple line shows the response of system to targeted attacks. As you can see, you are removing around 10% of the notes and the system collapses. If this mantles into pieces, there is no network after these attacks. This is very important. So this is again a heterogeneous network with parallel degree distribution. And we see the green light shows the random failures. So the scale-free networks are very robust against random failures, but they are not at all robust against targeted attacks. And it depends on the type of study that you are doing. You might want to consider the random attacks. You might want to consider the targeted attacks. You might want to base your attacks based on, that base your attacks on different types of centralities. So these are the ways in which you can explore the robustness of networks in different situations. I'd like to finalize the talk here. Thank you very much for listening to me. I will be ready for Q&A. Disclaimer, I'm not an ecologist. I'm not a biologist, but I'll be happy to answer your questions about network science. And also I will try my best to answer your questions from other domains from ecology, biology, sociology, but no promise. Thank you very much. Thanks a lot, Arsham. Thank you very much for this very nice, broad, but yet compact introduction of a huge field. Thank you. Perhaps I'll start asking some questions that are in the Zoom chat. And please, if you have any other, if you have any other, they can raise hand or ask them in the YouTube chat if they're following from YouTube. So there is one question which I think is very broad, which is, is hierarchy always associated with modularity? Okay, so you can basically imagine networks that are not at all modular, but they are hierarchical. And there are examples of that in nature, but when I'm thinking about complex systems like brain or societies, modularity is always there, hierarchy is always there. And it's common that these networks are not homogeneous. So that's why I picked these properties. So basically there can be networks with only hierarchy. Great. So there is another question by Deepak. Why is the central network, what is the central network called small world? Okay, so this is based on a very old experiment in the field of social networks. So long before physicists entered the complex network domain, there was Milgram, a scientist who did a very famous experiment. You can search it on the Wikipedia, but it basically showed that your distance from any other person in the world is very small. Like you are connected to every other person with only a few number of other connections. So this is basically the shortest path between you and other people are very small. And the model that I showed you, that was something between the regular and the random network. This model was there to show how this is possible, how the small wordness immerse in the word, how we are really connected and our distance is very short. Great, thanks a lot. So there is a question by Pablo. Please unmute yourself and you can ask it. Hello, thanks for this talk. It was really interesting. You mentioned a derivation of the probability of the degree of a node when the network is constructed with preferential construction. And this is in the Barabasi book, but I looked it up and I can't find the chapter on network formation. I don't know if it works. Okay, professor, is there any place I can share documents with people that are interested in something? Yes, I mean, you can send it if it is something that you have in your hand now, you can share it in the chat. Okay, no, I don't have it now, but of course it's there in the book because this is the central claim of Professor Barabasi that networks are parallel. So this is probably in that book, but for sure there is papers. Yeah, so we can share material, so links to paper, not the opinion, they are on the website. So we have the program. Great. So just send an email. Sure. Great, so there is, we have time for a few more questions. So there is a question by Dionessa in the chat. So what insights do we usually make from the distribution of between a centrality in the network versus the network robustness? And is there any analytical way to get the critical FC robustness? Okay, so the critical FC can be obtained for a specific topologies. And this is a very interesting topic to discover if you are a physics fan and you like the networks. And the first question, I didn't get it. So what is the distribution of betweenness? The first question is, what insights do we usually make from the distribution of between a centrality in the network versus as opposed to a network robustness? Okay, okay. If I get the question correctly, please tell me if I'm mistaken. But if I get the question correctly, it's because I didn't really explain how to calculate the betweenness centrality. So between the centrality, you again have to calculate the shortest pathos and you will calculate how many shortest pathos cross a link. That's how you generally say that the link is in between other nodes. So there is many shortest pathos going through that specific link. This is how you can say that the betweenness of a node is high or low. And then according to betweenness centrality, if you base your targeted attacks on betweenness centrality, this is one of the best strategies of attacking the network. It's probably the best among the at least classical ways of attacking the network. And it dismantles the network very fast, very quickly. Great. So is there any other question? So there is another question in the chat which I'm not sure but I'll read it. So it's, can we combine network analysis with Markov chain by taking into time? So I guess it's related to temporal methods but I'm not sure. Okay, okay. So I will take this time to briefly talk about the dynamical processes on networks because this is the subject I'm working on. So you can have these structures like imagine the neurons connected to each other but you don't know how information travels from one node to another. Okay, so you can calculate the shortest path but it's not really the way that electrochemical signals flow in the network. That's when you couple the network with the dynamical process. It can be a random one, it can be continuous diffusion, it can be synchronization and the nodes exchange information in terms of these dynamical processes. That's a very huge topic and I really enjoy it and I chose it to be my masters and PhD thesis. Any other question? You're welcome. This was a reply to our thank you in the chat. Yeah. Great, if not, I think I'd like to thank you to thank Arsham very much again for this very nice tutorial which will be available on YouTube for the next generations. And before that,