 All right, thanks everybody for making it out this week. Today we have our own Professor Lin-Wan Lu who will be talking about Richie Curvature and Richie Flow on weighted graphs. Go ahead and take us away. Okay, so thanks to join us in the seminar. Today we'll talk about the Rich Curvature and Rich Flow on weighted graph. This is a joint work with a bunch of courses and I'll explain to you. And these are two papers. The work was done by two visits at the Harvard University in the 2019. The first paper is during the May of 2019 when I attend the first conference of STER and the second one is another visit in summer to the Harvard University. Now everybody knows that STER, so I don't have to explain him. Surya and Tsu-Yu are my former PhD students at the university. Surya was a post star at Harvard University at that time. Now she is a tenure check assistant professor at the University of South East. And Tsu-Yu now is a post star at the Georgia Tech. Ah-Yang Hua is an assistant professor at Brandy's University in the post area. So it's very close to the Harvard University. Yunlin is a professor at the Tsinghua University. He is one of my long collaborators and he's also visiting Harvard University in August of 2019. So this is how the work will be done. All right, so let me give some background. Why we consider the, choose the weighted graph. So let's start with the big topic. So everyone knows that this is really most significant in the two decades. So the paramount in 2002, 2003 and last, is proof of the Pankani Conducture. The Pankani Conducture is stated at every single magnet across three manifold in a homophict of three spheres. Now there's at least three groups of people contribute to make the skills of his proofs and include the trainer, loft, and the child and Jew, and Mogi and Ken. It's quite big control, so about his proof and who can get it created and so on. But I'm not a geometer, so I'm a commandorist. So we don't need to talk about that. So the whole proof is, the whole idea is the right, are the heavy things to reach for ideas. So let me try to draw and see what we can do for graph. So suppose that you have some manifold, okay? Let me just do a drawing. Okay, so I'm probably new too, okay? So yeah, so suppose that you have some bestian shape, you know, manifold, okay? And the way you want, so you creep with, you know, re-measure, I'm gonna try it. And you want to continue the morphism, finally you get a sphere, okay? And it's through the rich flow. So that's why, you know, rich flow is so powerful right now. And it's also very popular. And what we want to know is that suppose I give you a graph, okay? Right? So, and here what we should do, okay? And we want to say, okay, okay, give me a graph and I'm gonna try and I try to morph it into a standard form. Now, the sphere you can think about is how part of, is how the constant curators. So maybe we could get a graph with the constant curvature and some format, right? So this is the topic, then why we want to study at a written graph because I want to change distance. If you're continuous or shrinking, moving, you want a lot of distance can be changed, nothing for the one, okay? So let me give you some background of the rich curvature on graph. Okay, so here is the background. Let M be a manifold, X is a point, Y is another point. And V, the vector in tiny bundle, okay? So suppose I have two ball, BX and BY, centered X and Y with Ypsom radius and I want to move the mass from the ball BX to BY. And what is the area distance between the ball BX and BY? That can be compute, oh, so maybe probably I just, okay, so that would be better on the high DSA. So it is approximately the DXY, the center distance between X and Y, but with additional terms, the rich curvature and the dimension will appear. And this is indicated that can be used, this idea can be used to define the rich curvature on graph. So there are some many variation of the, you know, coaches. Oh, sorry, so I accidentally, all right. All right, so I want to talk about the coaches based on L1 distance, okay? So early in 1985, Bakery and Emory have found a way to define so-called low-rich-coach ball. And that works all the magic space. Now, Fentra and Yale, they define a class of graph called the rich-coach thread based on frame. And they said from that, you can have very good, a lot of subtlety quality. A storm and a lot of the brineus, they throw the components of the curvature dimension bond. And all of the coaches are in the Markov chains, such as graphs. And in 2011, and Lee and I and Yale, we kind of adapted all of us ideas and to give a definition of the crucial graph. But our definition is slightly different from all of us. I will explain later. So let's give a notation, okay? So the width of graph we consider in this talk are slightly different from all, you know, the width of graph. We actually have two widths. So given NRI, a simple graph, and then you have two widths, one is called the distance function and the other one called the width function. And what did they do? One of them was for Ws, okay? So let me just try to draw. So suppose I had two vertex, okay, X and Y. The way we think about, we have Y is called WXY, okay? Y is TXY, okay? TXY, you can think about, TXY is the distance between X and Y and that indicates, you know, if you want to send information from X to Y, how long to take, right? And WXY is kind of capacity after the information you send X to Y. So that way, if you have information you want to send out to the neighborhood, if you have a large capacity and then you will receive more information. So that defines the random work. So the random work, the transient probability, okay, from U to V, you give it just the width, W. But the distance is the D, okay? Now, if we go back to the simple graph, the simple graph, we just said all the distance of U, V and W, U, V equal to one. And so that's all the definition that I come back to simple graph. Okay, so with these annotations, let's talk about probability distribution. A probability distribution is simply just function from vertex to zero one, and such that the summation is equal to one. And if we are, you can see about there's some mass distributed on vertex set. You have one distribution to other distribution, one to change with some mass to move from one to another distribution. And what's the distance? The copy is the way you can move from one with others. The copy of your definition is a function from A, from V cross V to zero one, such that if you send one index, you get the first distribution, you send another index, you get the second probability distribution. So the transportation distance is simply just the distance in optimal distance, and it's a conference, or possible conference. Now here the cost of mass from X to Y is the distance of X and Y. So if you take all the information and then it will just play distance, it is a real existence of probability distribution. So satisfy the travel if you call it. This is the solution of the optimal problem. So any linear open problem have a dual problem. So you can define another way using the dual problem. So dual problem is defined as the Lipsch's functions. So what's the Lipsch's function? It's a function of vertex to R, such that the difference between X and F, X and F, Y is that it's a most of times constant times D, X, Y. Now usually we put the C for the Y and we put the C for the C for the Y. So the transport distance also can express as a supreme among the FX times M, Y, X, M, M, or two X, that take the supreme among all one Lipsch's functions. So two definition is quite useful to appear. So one could give you the low bound, advocate up bound. If both of them achieve the same values and that is very much the optimal. So it's very useful to compute the transport distance usually. So now let's give you a definition of rich cultures or graph. So we firstly, what you find is so-called alpha rich culture. Alpha is the idonis that tells you you do random work, you do a really random work. So you start from X, you move to Y of the neighborhood, right? So with the probability of alpha, you just get at the vertex. We probability that one minus alpha, you move to Y of the vertex with probability proportional to the width of that edge, WXV. So here, WXV appears. And then we alpha equal to zero. This is the usually a random work. And that culture is very obvious or in the version culture. So we kind of interest idonis and then we can define so-called alpha rich culture here alpha. And finally, we take the limit. So this is some probability. Okay, just give an example. Here is the graph in the C full and the WXV, you have one alpha center at the X. And so you have alpha at the X and then you have one minus alpha here is one neighborhood. And then you want to move to this notation. Okay, this is the situation. And what is the optimal way you can do it is just, you move the one minus alpha over two units from U to V and move the alpha minus one more alpha unit to from X to Y. And then the calculator tells you that the kappa alpha is given by the piecewise linear function. And it's always like this. All right. So, oh, I forgot. All right. So now, because the n lips function, you can extend from service space to the whole space. And the transparent distance only depends on the distance and model vertex in support of M1 and support M2. So in other words, it's a local property. So, rich culture is really local. And it's a concave function in alpha that can be done and it's bounded. And so then we can really take the divider by model alpha to the limit. This limit always exists. So this is called, now people call it the Lin-Lu-Ya rich culture. And that is one of the popular rich culture on graph. And here's some examples. If the culture on the triangle, you get two. And if it's C4, you get one. C5, you get one half. And if the cycle is greater than or equal to six, the culture is zero. Somehow the culture will depend on the triangle, four cycle and the five cycles. All right. So this is the Lin-Lu-Ya culture. So now, this culture can be easier to return to their background. Just very recently, not five years ago, so it's 2017, almost like six years after our paper published. And then they give us Mutual and Trotsky first gave a definition of a limit free version. So it's very clever way that you define two operator, one for data. It's basically the average operator operation. And otherwise, neighbor X, Y is just the gradient vector, you know, the direction to reach some kind. And it turns out, it can define this way. So you are taking the if them among all the neighbor X, Y, F on this data Y, F, and among all the one lips of function such that data X, Y, F equal to Y. This condition is the same as the distance F X minus F Y minus F X equal to distance X minus Y. So yeah. The original version is really just for simple graph, but we can easily extend to the weighted graph. Now, every linear optimum problem, you have to have your problem. So that's why we ask that, you know, what's the dual format of this formula? And in 2019, and we figure out, so we just call it the stock copy. Stock copy is like copy, but it's a match this, okay? So basically, let me just draw and tell us what that is. It's a match this. So index and I will X and then it's neighborhood. Okay, let's say for X, Y, X, two and so on, okay? And they have Y and then Y one, Y two and so on, okay? And between the X and Y, this is actually in the party, okay? All others are negative, okay? Some way. And if you sum up the row sum, one is negative of the probability version, other one is sum up with the column, you get the column sum is, except for the first one and it's another probability version. And summation of all edge together should be for zero. So this kind of very special coupling function we call the stock copy. I would like, this is the analog of like the combinatorial or the plastic versus edge is kind of relationship, okay? So you have one part of the entry or diagonal, but I think it's only one entry and all entry are negative. And then we prove that indeed, the little yaw furniture can have right as a supreme of the BXY times the cost of BXY and they are divided by the distance between the U and the V. So this format, you have no limit and it's just due to the previous formulas. So it's very useful because one give you, this one give you the lower bound. If you find a stock coupling, you immediately get the lower bound of the curvature. The other one give you an up bound. So if the up bound and lower bound, you pick equal and then you calculate this curvature. All right, so now back to the first part, the problem. So total curvature, what's the total curvature? Total curvature is simply the sum of the curvature of all edges. I have a quick question. So in the previous two slides, you have this info, I mean, one more. Info and soup, yeah. Previous slides, this one, this one looks like a continuous, you know. Yeah, yeah. So I think so, well, yes. So that's why, you know, there's some connection. It's nice. There's a geometry background, like every distance and average operator and also this gradient vector, you know, those are really geometric terms. Operators. So I think if it is continuous level, I will compute this operator, I will do this inf. I will end up at the curvature tensor. I think so, yeah. The Ritchie tensor, okay, something interesting. And I probably the second one also have like a continuous version, you know. This looks more complicated, but some says, you know, if y exists, the other one should be also exist. Okay, so now let's talk about the problem of total curvature. Total curvature is the sum of all the curvature on edges. Let's forget about the tree like, let's just think about this problem. So it's probably the way I would attend the YOS 70th birthday conference at the Harvey University and I meet Anhua. Anhua told me the problem because he's a physics collaborator, told him that they can show by the experiment. If they choose the weight to be the function of one, so if they choose the weight, it's like, let me get around here. So if you choose the weight of, if you pick the weight, okay, W equals to like a function of one over D squared or something like this. And then they found the total curvature is minimized at a simple graph. So they say, oh, they try it experimentally. This works and this asks why. So that's how the problem comes from. So now here's the solution. Let me explain what the tree like means. The tree like is this way. So I had vertex X and Y and X kind of a neighborhood, okay. I let's call it the K and Y has neighborhood equal to L, right? Now, it's basically say the distance is the distance of the path. If the super graph, this condition really said the gross is at this six. Oh, actually you can show the gross should be of this graph should be at this six, if the tree line. So locally like a tree, but the gross itself is not getting T. So you have, this condition is slightly stronger, but it was simple graph in the equivalent of this six. So we use the formula we have returned, you know, in previous slide, we have the, you know, you have the name and you have supreme. And then we can use to calculate a tree like graph. So we have to get this very nice curvature bond. So remember that if we want the lower bond, we can use the formula of a stuck-up ring. Use our stuck-up formula, we calculate a lower bond of the curvature. And we summarize it here, this is formula. It's a very nice formula. I know it's a two-v gradient Hw and Hw is with function, okay. It's a very nice functions. And now we can study, you know, when this functions achieve a maximum, achieve a minimum, and that should give us the result. So I will show you the next slide, okay. So the next slide, here is our mirror dot. So what, that's the result. So let's, G is this weight of graph. We have two weights, distance weight and weight for the random work. If we are weight and distance have some relation, given by any function F, F is a non-increasing function. As long as non-increasing function, then the total distance at the curvature, at least, two v times v minus two times v. And you call the holds, if and only if you need to treat it. And all this weight function F is just one number. So because the scaling that done that, the curvature. So that tells you essentially, it's a simple graph. And that's also the question, you know, asked by Huang. So, and then we realized, we also get, what if the F is a decreed function, not decreed function. And then you can turn the, you're quite a run, but this time we need a condition, if it's a tree line, then you can turn the encoder around. Now, as run F is a decreed function and the encoder at the most, two times the number of edges. And you call the holds, if and only if once again is just a simple graph or the weight as same on the graph, but distance could not be same. So this is the more general, because obviously this is a simple graph, but now it's right at the game with each edge equal weights when you do the random work and that gives you the total closure reaching the minimum. So this kind of I'm surprised with that class. All right, so this is the part of one. I probably the good time to ask the question, do you have any question so far? Okay, so let's move on and we'll talk about part two. The second part is about the rich flow. I have quite a quick question. For this kind of total curvature of this, do you compare this with the, compare the curvature with the entropy? But I mean, the entropy? No, we didn't think about the time. We had no real problem. We have this problem is really motivated by Hua Fonsi. His physicist friends, and as they did the experiment, and they found this and they saw that because maybe the function in the one over the square is very special. You see the very special function for the physicist. So they asked the question and we gave a solution and if you have a very general solution, when you call it holds, what condition is sufficient to set with the total curvature is maximized on minimum. Yeah, yeah, very good. That's a little bit. Okay, yeah, sure. So let's move on. The sixth topic about the rich flow, okay? So rich flow manifold is just by the simple equations. So here's GID is the real magic and RIG is the rich culture. So you solve this differential equation by the time T of the GID equal to negative two times the rich culture. And then, so that will change the rich culture or the remanifold and remanifold. And slowly, when to stop, it will become a constant culture and it became a sphere. Of course, the difficult part is you don't want to say, oh, you have to say, I'm security, you grow into two spheres instead of one sphere, but this is a really hard part. That's why those are working there. So, so important also significantly. In 2019, a group, actually there are two papers. And one paper is made by me, Lin, Luo, and Li Kao. They wrote a paper, trying to generate the rich flow to the graph. And they consider this iterate a discrete time and update the weight and the distance. And also they assume the weight and the distance is read by some fixed functions. And they have some special functions, they try all different functions. And then you say, oh, and some functions works well to be able to detect communities. So this is kind of interesting. And the paper is almost all the experiment dot instead of one mine, you know, straight or not. So that you didn't say why this, you know, the rich flow could exist. But just that it looks, works well. So we saw their papers and we want to give some mathematical theoretical result and what it can prove, whether the flow exists and if it exists, what the limit of it was like. This kind of things. So here is the approach. We'll pass the so-called normalize rich flow on the weight graph. So the first part is very similar to the second one, try to realize the normalization. So with this limit, you know, but that's too difficult, you know, we have to be take care of. One is that at some point, because the distance are changing, right? The weight are changing. And at some point, maybe the age is no longer the shortest distance between the U and V. So it's what I have 80 U of V. And because this distance will be increasing, increasing and no longer the, so this age is no longer needed. In that time, we'll remove this age. This is the time to not. And then we'll consider a sub-graph and it will continue this system. That another difficulty with some distance became too short. They're probably not for them to get zero. You cannot have a natural distance. So at that time, we'll just control that age. So U and V became one point. So we'll keep doing this and we'll hear the result. If we keep doing this and then this system will have a solution. The solution always exists for all the time from zero to infinity. So eventually we have some kind of a rich flow. But we don't know about this, whether the limit is equal to infinity, the limit exists. If it exists, if it converges and it will converge the constant curvature with graph minor of G. Why is the graph minor not graph itself? Because in the process, you might give some age, you might control some ages. At the end, you get a graph minor. And that graph minor eventually will be have a constant curvature. This is not hard to see. So if I give you, if it converges, okay? What happened is, so as a tickle the infinity, okay? Okay, tickle the infinity. And this will become zero, right? If it converges, right? And you get here. So here you can solve that the curvature of K e of t equals to, you can solve this one, it's just summation of the other things, okay? And here is K of h of infinity, okay, it's infinity. And then the edge infinity. So support the limit exists, okay? So you can see it's a constant. That's why once the limit exists, you give it a constant curvature object. Okay, so this is kind of nice. Okay, there's some, you know, exists. And we also use a, if the limit converges, what it looks like. All right, aha. So this is after a page. I just want to give you the ideas after you run something. So I see online, you know, the web, you really change the way we present, we present, sorry. Okay, so I need to give it, okay, so, okay. So I want to give you the ideas, okay? Why? So the way you can think about it, I have captured off, you know, so width and the function, they are width, bottom functions, okay? So let's consider the way I have correction of this, all this, you know, the, okay, so here you are to this power, okay? So this is, give you, this is just the correction of all this, the ideas. D, U, V, okay? U, V, H. And the way you can think about, so something goes around. Suppose I have one cycle, okay? Your graph, okay? This is cycle C. And this is my U and V, okay? And this D, U, V might be longer. Sometimes, you know, so let's say this here is the U, 1, U, 2, U, 3, U, let's say U, A, okay? So there's something like, sometimes the time T, you know, D, U, V is the same as D, U, U, 1, plus E, U, U, 2, plus E, U, A, U, V, right? So that times, you know, this, this, this ages, it just is so long. That is no longer needed, right? So that's, there's a better situation. Another better situation is those D, U, V equal to zero. And those clenching, so think about it as the whole space, the whole partition by, you to feel, you know, okay? Just some readings, the connected readings. Now, you have a rich culture, okay? So rich culture is a flow, it takes the curve in this space, okay? So they go somewhere, okay, they grow here, they hit as well for the boundaries, okay? Either this or this, okay? And then, they will look at some space and they will approach on this one, and then you do it here. So somehow, this is not a criterion, I will say, okay? So suppose I have, you know, the high dimension, okay? So maybe this is a three-dimensional parameter space, okay? And then you have a rich culture, you move a sum of this, you know, on the boundary, okay? This is kind of boundary, halfway. And then you study to another rich flow inside this flow, and you keep doing it. Maybe at the end, you hit another boundary, and then you go to the low dimension, you keep doing it. So in that way, I can find a piecewise, you know, piecewise differential test, you know? And that's the solution we're talking about. This is the rich flow culture, rich flow we are talking about. Okay, so then help you to understand what's the rich flow we are talking about. Any questions? This is just in the sense, you know, the system has a solution. And that's why we also have an underlying graph also changing. Could lose some edges, could combine some vertex, counter some edges to get a graph minor. Any questions? Okay. I have a quick question. Very, very, I'm sorry. So many questions. Yeah, this is very interesting question. Very nice. So, I know it's like eventually you're getting this constant curvature, you know, in this sense. So have you studied the dynamic of behavior, you know? Yeah, that's the next slide we're talking about. So that's all kind of, all kind of system will happen. Where the system could happen, okay? Okay. So that's the next one. Let's look at a very special example. It's just a P3. So just pass our three vertex and you put X, Z and Y. And AX, AY, you know, it's just give you the, from Z, you know, we're probably AX equals to X, you're probably AY equals to Y, okay? And XL, no choice. NMS, if it's X, it equals to Z, AY equals to Z, right? So, and so this ritual flow equation is very simplified, given by this line here. And we can actually study the behavior. Depend on what kind of function we choose. So the width depends on the distance. Support is identical functions. So gamma of X equals to X. So that means my width, WXY equals WXZ equals to DXZ. So is that a Y? It actually it converges. So this is the first example, it's really converges. And then just write something out. So this is why the limit object is like this. Just a simple graph of a piece of it. So each of the weight, you can see this, the D, which is one half and one half. And the culture is one. So it's just a standard, you know, piece of it. This is one scenario. And then the second scenario is more complicated. If we choose the square, okay? And then it's eventually claps, the three point. Eventually XZ, Y, right? So first it goes to XZ, Y and Z, what's the million? And then you get one point. So the whole graph just claps in one vertex. Now the most interesting part is the K3. So K3, we define this very special function. It's not trivial, trying to find this function, okay? And we define the gamma function like this, and the piece of Y is function. And with the initial value, you give the W of the weight is three eighths and five eighths. And then the result is spicy is a periodic solution. So basically, I tell you what it is, okay? So here is graph, X, Z and Y. And here is the distance, okay, D, X, Z, U, Y, Z. Now this is the focus, you can see what X, Y is fixed. And this Y is just oscillating. Z is moving from left to right, right, left. And this function is, I tell you right, this function looks like this. So I would draw this graph. So here, this is a line. This is, I think this should be a quarter. And then you add one in. So it's a sine function, and then the top is three quarters, okay? This is three quarter. You come in like this. So this is the one half, okay? So now I have a sine function like this, right? I have another sine function like this, okay? So then function, X, Y, okay, something like this. So it's a one part of a sine function, then followed by another sine function, and then go down. So this is what it looks like, okay? And then similarly, the curvature is also oscillating. Yeah, the period, the period is two pi. So that means, it's never converges, it's just oscillating to each other. So I guess this is dynamic, I think Wu Chun is asking, you could have some kind of oscillating function. You don't have a gym down there. Okay, so maybe we should move on. Okay, so now, we can have this discreet algorithm. We can, so let's just kind of describe it. We have a distance condition, this condition, okay, at some point, the UV distance is too large, larger than other shortest thing along another path. And the multi-composition is that distance is too small, that's close to zero, and you have some actions. And the terminal condition is that, oh, you know, it's converges, the curvature, each ratio, let's say some arrow term. So we think about the stable. Okay, so now here is the average. So we calculate this AI, this is sum of all the weight, and then with the adaptation according to this formula. So, and then we'll check all three conditions until one company is certified. If we, the distance function in the virus, then it means that the DL-UOV is too large, and then it will continue. If the distance is too close to zero, we will merge the UNV, and it will self-continue. If the curvature is stable, and it will stop, we will output the graph. So that's no kidding, since every step, because like the example we show here, might be oscillating, they're not equal to any limits. But if it does, they'll give you something, okay? They'll give you the graph minor, and which is, have the constant curvature. All right, so the height of structure, it's a very interesting, you know, this I can repeat, then just draw something, and I'll help you understanding what we're talking about. So what I have a graph, okay? It's a huge graph here, right? And now, after I do the rich flow, okay, algorithm, I end up with a very nice, here's the graph minor, and it could out, okay? This is the graph minor. So each graph minor, it's a point, and it really corresponds to the radius, right? Okay, so this is, if you go back and get this, okay? So now, for this picture, so this is the V, or this is the V, this is the V on V, okay? It's the pre-image. And the pre-image itself is a graph, right? So this is an induced graph. And now we can, iteratively, we can apply this rich flow algorithm to the inverse of V. And we can apply here, we can apply here, right? And we'll continue. And then we'll get another graph minor, right? So, and then this actually, each one you will get to the community. And this kind of get the height of structure. So the right-of-way, just to deal with the algorithm that we can convert it as a graph into some standard drawing. So I will use the Google map as an example. The first you see the whole world, it's graph is so huge, you get the world map, right? And South Carolina is kind of like a one-dot, you know? It's not that big, maybe it's a small reading, just sort of in the United States. And then you can zoom in, because in South Carolina, if you look at it clearly, then you can see the cities, then you get the Crimea, and the one point. And then when you see the University of South Carolina in all kind of streets. So same as here, let's give it a natural way to present the information of the huge graph. This algorithm I can, each time I present you is a graph which has a constant culture, supposed to have very nice appearing, it's nice drawing. And then you can zoom in for each dot, and you get another subgraph, and then you get another map, another layer. So this gives a high, high, I could call left to Georgia. I think it's very useful. Okay, I hope I can explain this idea well. Any questions? Okay, so. Do you have any computational figures? Fast? Yeah, I think so. The ideas, yeah. Surya actually included some of the pictures in the paper. Okay. You can read it, he didn't, for this idea to work, so you have to have a huge graph in the very light. She wanted a text, a very small graph, and not really very good, but the way you want to really want to like, you have a hundred, so there's a lot of this, you know. And then you will study, see what kind of, different layers work, yes. But this is to give some, it's kind of like an idea about the rich flow in the geometry, right? You give any metaphor, you can get like, most of them into a standard version. Here, I give any graph, I give the most of them a standard version again, give like, you know, the constant rich culture graph, with a graph, and of course, the difference is that instead of already graph, you keep the project here to probably change, but you keep the graph minor. The one you got is a graph minor for the graph. All right, so let me just give some open problem. I think I'm down at the top. So we actually suspect that the chaos could exist if you have freedom to choose, you know, the function. So if I carefully design the function, I can, I should get some chaos behavior, like, you know, one initial value, you change the distance of weight a little bit, and the result is ending up in a very different or never converges, you know. We actually try some, try to embed a chaos system, the next system into this, but we didn't succeed. So we don't know. Even the example I show you, you know, the oscillating example, the function is not, not true, you know, it's a really need some time, but we think this chaos behavior should really exist. Now, if your gamma function is very nice, like an edit function, or let's say one over, you know, some part, you know, the D to some power or one over squares, for example, or just a constant functions. And maybe in that time, we can prove that this culture always corrodes. We don't know, but we hope we can get some of that. So, so this should start, you know, this paper is just starting, given mathematical foundation, so we, about the leech flow on graphs. There's a lot of question remains, we can out of them, I am sure some of the questions should be very deep. And then we, you know, we don't know how to answer them. Yeah, but anyway, I think this is all I want to say today. And thank you. If we have any questions, this is time for questions. Thanks, Lincoln. If we could all thank Lincoln in some way, and we'll go ahead and open it up for questions now. So do we have any questions for Lincoln? Yeah, if you wanna kind of see the body, so maybe I will exit and then they will move back to the room so I can see you guys. Any questions? Yeah, so the paper is available at outcome, you know, if you are interested in topics, you can choose to read the paper. Okay, and also you can ask about it closer, for them to be like here. Okay, well, it seems like there's no additional questions. So thanks again, Lincoln, and have a good weekend everybody. Okay, bye-bye. Bye-bye.