 Thank you very much for the introduction and thank you very much for this invitation to this nice event that I'm very pleased and honored to be part of. So okay, tonight I'm going to talk to you about some consideration about social dynamic models. So let me first start by introducing myself and then we will switch to science. So first I did a scientific baccalaureate in Orleans, where I'm from, then I decided to go to class préparatoire, where my specialty were mathematics and physics. And then I get into the Ecole Normale Supérieure of Lyon, so by, not by the competitive exam, but by applying, by sending a file. And once I was there, I got an opportunity to obtain a scholarship, so to do my first year in Lyon and my master in Nice. So this is how I arrived at Nice. I obtained the opportunity to do a PhD. And so when I did, I fell in love with, with, with research and I decided that, that's what I want to try to do. So I tried to obtain a position. So first I, I obtained a temporary position at Inria-Rennes. I, I stayed there one year and then I did the competitive exam to obtain a permanent position and I obtained one at Sorbonne University since I, and I'm there since 2017. Okay, so let's move to, to science. Okay, so I, okay, there is a little there. So, okay, I'm going to talk to you about interacting particle systems. So first let me start in the context of gas dynamics, because actually this is my original background. When I started my, the, the equation I was studying was in the context of statistical physics. So for instance, in, in that room where there is a gas and the, the atmosphere is made of a lot of a huge amount of particles. And so one way of mobilizing it is to say that, well, while the particle of the gas, it's sphere and they have a position and the velocity. And basically, as you can see on, on this video, well, they move in the, in the direction of their velocity until they collide and when they do, they bounce. Okay, so this is one way of representing it. And the thing is, okay, so this is the, the video that I've shown you. This is words. So how do you do math on that? Well, you have to write an equation in that case and the type of equation that you will be interested are what we call differential equation. So basically you need the notion of derivative of a time. So actually all of you are familiar with this notion. For instance, if you're interested into the position of a time, the derivative of a time, it's nothing else than the velocity. So basically, if you know your position, your velocity, in one hour, you know where you will be, okay? So this is a notion that we will be using in the following. Okay, so what I, when I started, I was in the context of gas dynamics, but actually what might surprise you, it's what I've been showing you, the fact that you have particles who moves. And once they interact, it changes their behavior. It can also be, you can use the same approach to describe humans and humans interaction. And this is what I'm going to describe precisely today. So in the context that I'm going to talk about, and it's no longer the particles of your gas, but now it's the number of agents, the number of persons. And the quantity you will be interested in, it's the opinion of a person. So what we write as xi, it's nothing else than the opinion of agent i. So you are interested into how the opinion of a person evolves over the time. And so one quantity which will be important is what we call the interaction coefficient. So what is it? Basically, what we denote aij, it's how the interaction, when j will interact with i, how it will impact the change of the opinion of i. Okay, so basically, now if you want to put that into an equation. Well, here is the derivative I was talking about, which lets you know how your opinion will evolve over time. And now the equation that you are able to wrote here. Well, it's saying that basically when agent one will interact with agent i, it will make the opinion of agent i evolve. So, and this is through this interaction coefficient, which is right here. So you are interested into binary interaction. And so basically, you do the same for all the agents. So agent one will impact the evolution of agent i with this interaction. The same for agent two, the same for agent n. And so basically, the fact that I'm summing over all the agents, this is summarized by this science here, which is just saying that, okay, I'm doing the same for all the agents. So basically, this is the kind of equation we are interested in. And let me show you a particular case, which is what we call the Excelman-Kraus dynamics. Well, it's the case where you are saying that your interaction coefficient has a particular form. It looks like that. And what does it mean? It means that basically you are saying that your interactions, they only depends on the distance between opinion. So for instance, you are just saying that, well, I decide that if two agents have two opinions which are too far away, they do not interact. Or for instance, if they discuss, it will not impact their opinion, it will not make it change. Which makes sense in real life. If you're discussing with a person which has opinion, which are very too far away, you do not share any value, it's less likely that they will manage to make you change your mind. So that's the spirit of this one. Okay, so here I will talk about system on graphs. So okay, let me first state that I'm not a specialist of graph theory, I'm just talking about them because basically this is the structure which naturally appears when you are interested into this type of systems. So okay, a quick introduction. A graph, this is a mathematical structure that you can use to model pairwise relations between objects. So of course it's natural with what I have just explained. So basically, you have two notions which are important when you are interested into a graph. You have the vertices or the points, the nodes. So here you have the blue circle and the edges which are the straight lines which connects or do not connect some vertices. So for instance, here you have a connection between those two vertices. But you do not have one between those two, okay? So this is the natural structure which appears. So why am I talking about that? Because for instance, let me show you an example of a system I will be interested in. This is what we call the L nearest neighbor interaction. So basically we're saying that when we are considering agent I, well it can only interact with the L agent to its right and the L agent to its left. Okay, so for instance, all of us in this room, at first I put all of you in a straight line and I'm telling you, okay, you can only interact with the two people to your right and the two people to your left, okay? And you cannot interact with anyone else. So the equation, if you write the equation, it looks like that, okay? So this is nothing else than the fact that you sum over all the agents that you can interact with, so the L to your left and the L to your right. And so here there is something that is interesting because actually this is significantly different about what I've been talking about before. In the previous case, at the beginning, actually, everyone was able to interact with everyone. The only things that mattered was to know the distance between two opinions. As soon as your opinion were close enough, you were able to interact. So everyone could interact with everyone. This is significantly different in that case because actually, well, you can have two opinions which are the same. Still, if at the beginning you were not seated two seats close to the person, you will not be able to interact. So basically, we're saying that not everyone can interact with everyone. So, well, basically there is a network, there is a graph underlying the interacting system, which describe the interaction which are possible or not. So this is how it naturally appears. Okay, so I will write what I call the classical opinion dynamics models that I've been describing so far. Now I will show you a variant which actually corresponds to the one that I particularly study. So this is a variant of this model, which is the following. So we are still interested into the opinion of each agent, but now we are introducing a new variable which is what we call mi, which is the agent's weight. So basically, this represents the carousel, the charisma or the popularity of a person. Okay, so the bigger your weight is, the more charismatic, the more popular you are. So now you have two variables of interest. And so the equation that you will study now is the following. Well, we are saying that the interaction coefficient aij, so the interaction of j on i, it is nothing else than its own weights. And it makes sense because basically we are saying that, well, the more charismatic, the more popular someone is, the bigger it has an impact into making the opinion of the person change. Okay, and the second equation of our model is the following. So it's just to say that we decide that the charisma, the popularity of a person, it's not fixed over time. So basically, it can evolve. So you can start being popular and then lose popularity and regain popularity, et cetera, et cetera. And it only depends on, well, the opinion and the weight of all the other agents. Okay, so this is the model that I've been particularly working on. And so, okay, let me say another word about the graph. So what I didn't say here before is that actually there exists two types of graph. There is undirected graph and directed graph. In that picture, this is actually undirected graph, meaning that when you have a connection between two vertices, it's actually on both way. It means, for instance, if it's vertices one and two, that one act on two and two act on one. This is symmetric. But sometimes you have directed graph where you have to take care to the direction. And for instance, it's not because one act on two, that two act on one. So this is the difference between directed and undirected graph. First point, second point is that actually what I didn't say, what I also didn't say, is that here it's what we call unweighted graph. So basically, you are just interested into knowing if you have an interaction or not. Okay, it's a binary, one or zero. But with weighted graph, you are not only interested into knowing if there is an interaction or not, but you are interested into the weight of this interaction. Because basically you have interaction which matters more than other ones. Okay, and so in my case, I will be in the case of directed weighted graph. Meaning that I rewrite my equation which is right here. And now I represent the graph associated to my equation, which is, for instance, this one with three agents. Well, as I explained to you, the action of one on two, the weight which is associated is nothing else than the weight of one, which is M1. And the action of two on one, the weight which is associated, it's M2. So you can ask yourself two types of question when you are interested into, well, there are more than two types of question, but I have an interest of two type of question. So the first one are self-organization. So basically you are wondering what happened in long time behavior, okay? So for instance, in the case of excelman and chaos dynamics, it has been studied. And we know that actually naturally what we obtain is what we call consensus. So basically the agents, they are trying to reach the same opinion naturally. And other case which appears, it's what we call clusters. So the same, but actually at the end you have a finite number of opinion. So the population divide into different groups. And at the end, they have reached common opinion but a finite number of opinion. Actually, this is not the type of question I've been working on. Yes, this is the case where everyone interacts with everyone, absolutely. So actually this is not the question I've been working on. The question I've been more interested in are what we call large population limits, so basically we are wondering what happens when n goes to infinity. So what is the interest of this type of question? Well, it's the fact that as you have seen, I have one equation by agent. So if I have a huge amount of agent, I have a huge amount of equation. So if you want to, for instance, solve it by computer, well, it will cost you a lot of energy, okay? The interest of this type of question is actually to trade this huge amount of equation to only one equation, not on the same quantity. But only one equation, which happens to be what we call partial differential equation, and as it was said in my introduction, this is my specialty, this is my field, okay? So this is the type of question that I've been dealing with for the special class that I've shown you, the special social model that I've shown you. Okay, and to finish, I will talk to you about two examples of social dynamics. So the first one is the case where you decide to divide your population into k groups. And into each group, you have two type of people, you have leaders, or you have followers, okay? And you cannot switch levels. You are either one or the other one. And for instance, we can decide, so this is a model that we built to do some simulation. You can decide that, well, the weight of a leader will increases proportionately to its own weight, and to the total weight of the followers of this group. And the weight of a follower will decrease proportionately to its own weight, and to the total weight of the leaders of this group, for instance. Okay? And, okay, so this is what I just wrote here. Well, this is what this formula means, nothing else, okay? This is the dynamic that describe exactly what I've written here. And if you do some simulation, well, here, this is my position, which evolved over time. And here, this is my weight, which evolved over time. And you can see, if you start, for instance, with one group of 20 agents, and 10% of leaders, so you have two leaders and the rest are followers. You can see that, well, you reach consensus in that case. And what is interesting is that the consensus that you reach is actually a position which is not so far away from the initial position of the leaders of the population. So basically, the population is trying to reach an opinion which is not so far away from the one of their leaders, okay? And at the end, the total weight, the total popularity is divided, split between the two leaders, and in the model we have built, well, the follower, at the end, they have zero popularity, okay? Okay, with the second example, now this time we have two different groups. So we have 10 agents in one group, 10 agents in another one. And we have one leader by group, okay? And here you can see that in that example, there is one leader who take more weight than the other one. It's actually because this is a leader which is popular among people who have a big weight. So basically, this is a leader who is popular among popular person. And so this is why he's taking more of the weights. And this is why the consensus is reached at a position which is close from the leader or the first group. And we actually do not care of the leader of the second group because this is a leader in people who are not so much charismatic, okay? So this is what we observe. So basically it corresponds to the intuition that we could have for this model. And let me finish with the following model, which is the second example that we called the List Influence, Gain Influence. Okay, so we're interested into this quantity, the influence of j on i, that I did not like that. And we're interested then into EI, which is the sum of all the influence on I, okay? So I'm interested into all the influence that I will get. And now we, so we have that for each agent. Each agent, he has a total influence on him. So we can do a mean and average with this quantity, okay? And we want to compare those two things because in our model we're saying that, for instance, how your charisma popularity will evolve. Well, basically if you are less influenced than the regular one in the population, well it makes you gain popularity. Basically I'm saying that if you are reluctant to conspiracy theory, if you are harder to convince, etc, etc, it will make you more popular. On the contrary, if you are more influenced than the regular person on the population, it will make you lose popularity. So basically, if you're very easy to convince, it will not make you a popular person, okay? So this is what the model described. And this is a simulation, in that case, well actually you observe clusters. So at the end there are three different opinions in the population. And this is an example of how the weights of the different agent can evolve. And I think I will stop there. Thank you for your attention. Thank you, Natalie, for your wonderful talk. Now there is actually a bit of time for some questions. So please raise your hand so I can reach you with the microphone. Thank you. So if I understood correctly, opinion is measured on a one-dimensional scale. So that applies for say politics, are you of the left or are you of the right? Exactly, for instance. How about, could we generalize this to two-dimensional or n-dimensional? For example, you like jazz, you like classical music, you like rock and roll. And you know, it's not just left and right. It's not on a plane, at least one in a space. Actually, in our paper, our results are in RD. I just showed you pictures in one dimension, but it works in higher dimensions. Any more questions? How did you measure the stability of each opinion? What do you mean? Well, there is a variation of the opinion according to time, according to the surrounding. Yes. So, do you have a kind of parameter which characterizes this evolution both in time and both in space, I would say. So, if you didn't localize the people in space, you didn't do that, you did not do that. To change the... I mean, each people as a localization in the free space. Yes, for instance. And so, do you take into account of this localization? Yeah, we're really tracking the position in the space. So, yes. So, I didn't see that. Okay, we're good. So, I'm interested when you have this convergence, we see like two convergence of opinions and what is the stability of this? Is there a way you can get out of that? Okay, so, maybe this will answer your question, I don't know. So, let's first say I didn't work on the emergence of patterns. So, maybe I'm not sure about everything I'm going in to say. But the thing is, for instance, what I can well explain is how in one case you have consensus and how sometimes you have cluster, for instance. Well, the cluster, they appear when, so we are in the context of Excel Man-Cross Dynamics. They appear when basically your interaction function is compactly supported. As soon as it's compactly supported, you will... Basically, you are people from the population which will never see each other. And so, this is why you have different opinions. But as far as I understood, if you're not compactly supported, the more... So, you have some decreasing or non-increasing assumption to have. But as soon as you're not compactly supported, the natural behavior, it's consensus. You observe the same pattern, for instance, with the second order models. So, with the birds, actually, where you are not tracking the opinion, but now you are tracking the position and the velocity. The natural behavior pattern that you will observe is that they align their velocity. This is the equivalent of consensus. So, this is the same type of model. So, this is very stable, I guess, if you answer your question. This is what you naturally observe as soon as you have the right assumption on the interaction functions. I'm not sure I answered correctly the question. Okay, one last question. But have you oscillation, like in chemical oscillation? To the convergence? Permanent oscillation. Permanent oscillation. Like in chemical. I don't know. I don't know if you converge through oscillation, but all I know is that you have a limit. Okay, but I cannot characterize the way you converge to it. So, do you have a concrete example in life where you tested your model on some particular thing? Yeah, that's a good question, actually, that I've been asked a lot. This is one of the things that I would like to do. I would like to, I have an interest in social medias, and so I would like to develop the media too. For instance, typically our model where you are taking into account charisma, this is something that you can parameterize on social media. It means you can only take into account the number of followers, the reaction to your post, how much people interact with them. This is concrete measure that you can use to feed your charisma parameters. So, this is something that I would like to do, but so far I haven't started. All right, thank you for the questions and Veronica. Thank you Dimitri. Thank you very much to Natali for her exposure and for sharing her experience with us.