[MUSIC] Hello, so welcome back to our class simulation modelling of natural processes. So now I'd like to start the third module of the first week, which is about modeling space and time. So of course, you know that natural processes they take place in space and time. And typically, we need to integrate this dimension in our model and there are actually some difficulties about that so. All the system, they evolve in time and the state of the system may change in place. So for instance, if you consider the atmospheric pressure or temperature, it's different from one place to another. I mean, if you move from one town to the next or the temperature also changes. And of course, it changes during the day and for the next day. So you see that everything is space and time dependent. Another example is that if you are on the road, [COUGH] you can see that the car is moving, hopefully and then it changes position over time. So again, there's a time and space which are involved in this process. Sometime, we don't need to consider both space and time in a model. So for instance, you may be just interested in global quantity like how many individual of a given species I have in an area. So, it's basically counting the number of individual in a population. I'm not really interested in where they are, especially. We can also be interested in the total amount of CO2 in the atmosphere. So in some model, you can remove the spacial dimension and sometime you can remove the temporal dimension if time doesn't evolve. The process is steady. Maybe you can get rid of the time and just consider the evolution in space. So I give an example, for instance, the temperature in the room, you maybe interested whether it's cold near the window or warm in the middle. But if you have a good heating system, probably doesn't change over the day and it's something which is steady. So if we focus now on the time dimension and would like to see how we can describe it in a model and we'll see that there are several solutions for us. First, we know from physics that time is a continuous variable. So time can be as detailed as you want. You can have microsecond, nanosecond. So every real value is potentially possible, although some physicists claim that maybe at some scale we should stop and fill the gaps. We'll assume that time is a continuous variable and this is very difficult to describe in the computer. So, only mathematical model like a differential equation that can deal with the continuous time really with a real variable. So most of the time when you do a model and it goes to the computer, you have to change this continuous time. For instance, by discretidizing it, so you basically take your time into value you wanna describe and you split it in several time step and then you look at your system at every of this time step. So for instance, if I call delta t the time stamp can be one second, one millisecond, one hour depending on the model or even one year depending on the time scale we wanna capture. Basically, you look at your system at time zero, at time delta, and so on until you reach the end of your simulation. So you discretize the time, but basically, you follow your system in a continuous way. You are always able to know what your system is like. But alternatively, you can have another approach to say that maybe most of the time my system is not interesting and like to concentrate on one thing's happening. So that's the interesting moment of a system. For instance, if you have a queue in a post office, you don't really care to describe the evolution of the system when nothing happen. If you wanna just capture the moment when your customer comes? Or when one done in the booth and you have availability in between. It's very static in terms of, for instance, studying the time it takes for the customer to get served. In that case, if you focus on the event, the time t can be any value. It's just the moment at which the event takes place. It doesn't have to be a discretized value. It is the value you get from your model with, of course, the accuracy of your computer. Which is I mean, this day rather good. So in that case time is not discretized, because you can potentially have any value for the time for the event you're interested in. But basically, you are not looking in a continuous way at your system, you're just. Basically, focusing on some event. So basically, the history of your system is broken into this event. So we'll have a special chapter on this approach, which is called Discrete-Event Simulation. So to summarize in this graph, you see the three ways to reproduce or to represent time in a model. So the first is this continuous line while basically, you know everything, every time within the interval between zero and capital T. And again, only math can do this with this. In a computer, you have to go to one of the two of the solution, which is either you split in regular interval, usually smaller or you just focus on the events that are interesting somewhere along the timeline. So for the space, there's also some different approaches. And we'll consider, for instance, the case which is known as the Eulerian approach. So the idea of the Eulerian approach is that the observer is just sitting somewhere in the system, of course, virtually. But he just observes what happens at this specific position over time. So he is basically at a fixed position acts in space and determines the state of the system at this point. It could be, for instance, the atmospheric pressure. At this specific position at given time, but it could be also in the traffic example. Simply the number of cars that have crossed some line on the road that an observer can just count. He can see every minute how many cars is so passing in front of him. So this idea of the Eulerian description, which is extremely common is to attach a physical property or state to each point of the space. So space again, supposed to be continuous from what we know from physics. And again, only mathematical model like a partial and differential equation can describe that in a proper way. Otherwise, in computer model, we have to discretize the space into cells or little chunks and that usually makes a mesh, which covers the space you wanna describe. Now there's another way to describe space, which is called the Lagrangian approach and the idea in that case is to take the point of view of the object in this space. And basically, the observer sits on one of these object and moves with it. Then he can give the position of this object over time and the position, again is given with as much accuracy as your computer can give it. So then the space is not anymore discretized. So for instance, when you wanna describe the movement of the Moon, you will give its trajectory. So its position to the accuracy that you can over time and you are not just in spacing order. I've seen the moon at this position and now it's gone somewhere else, so you just give trajectory rather than Eulerian description of whether the Moon is down. So in a traffic model, of course, you can give the position of the cars over time, which is different approach than just looking at a specific position on the road and saying whether there's a car or not and what's the average density of car disposition. So as I told you, the Lagrangian approach is taking the viewpoint of the moving object. And if you want to represent these two ways of modeling the space, this feature can illustrate that. So on the left, you have part of the space, which is a square here, which is divided in little cells. So, I've discretized the space in the mesh. And for each of these cells, I give a color describing the state of this position. So it could be, for instance, the temperature. So when it's white, that means that it's cold. When it's black, it's maybe warm. And when it's gray, it's in between. So then that's the way you would represent a phenomena in space according to the value of quantity at each of the discretized position, the Lagrangian point of view is illustrated on the right So here, you see particles, for instance, particle can be anywhere in the domain and you can described them as their coordinates, Cartesian coordinates with all the accuracy you need. And of course, this can change all the time. Now maybe to terminate this module, I'd like also to mention that sometimes we have processes that do not really happen in space as I described before. So for instance, if you have a system of person interacting like, for instance, you take a social network or whatever, you may have very interesting phenomena. But it's not so much whether the people are physically close to each other, it's rather whether they interact or not. They can interact with the phone or with any other way. So I would say, the spatial relation is not really distance in the real space, but more whether there's an interaction or not an interaction. So what really matters is really the link that relates the component of your system. As I said, this is the case in social economical model whether you interact or not. So if you would take the example of an economical system, you would say that two agents are Interacting. If they exchange information, they exchange money or goods or whatever. And that's what makes the fact that they are close to each other not in terms of distance, but in terms of action and the right way to represent this in a model is to use a graph and a graph means that you put a link between component or agent that are the relation. And of course, this graph can be also evolving in time. So it can be dynamical in saying that we can create new links, meaning that you create new relation or you remove old relations or links are destroyed, for instance. So as an example, I show here a graph. And in this context, people like to talk about complex network rather than graphs. But mathematically speaking, that's a graph, but they can be very big and they have very interesting properties. So this example is just an example of a community where you have this circle that represents persons and the color represents their opinion and the size of the circle representing the person is just a number of people they're connected to. So if you're a big circle, you mean that this person is connected to a lot of these person and so we could study how opinion maybe evolved in such a structure. And on the right, you have time evolution according to our model, which again, I have no time to discuss here. But you see that in that case, initially you have a lot of green person and some blue person. But as time goes on, you end up with having everybody convinced that blue is the right. So there are a lot of important problem that we can discuss on complex network. We won't have time in this specific class to do it. So I mean, it's just an important Information I give you that sometimes the spatial structure or the spatial relation is represented as a graph. It's a field, which is extremely important now. You find a lot of article and books on this. And what's interesting is the graph topology, it creates a rich structure. Which of course, has an impact on the dynamic of what happened on this graph. There are many interesting property of this extended space I would say, which is degree distribution. For instance, the clustering coefficient, centrality measure and so on so. Again, also that would deserve a full chapter that I'm just making very short, so that you know that this is also a way to represent interaction between component in a model. So, I thank you for attention. This is the end of this module on Modeling Space and Time. Thank you. [MUSIC]