 OK, thank you very much for inviting me to give a talk here. Snezhana, special thanks. Today I'm going to talk about anomalous super diffusive random walk. We call it levy walk. And the key word is anomalous. First, let me give you a brief outline of my today talk. At the beginning, I will remind you the basic idea behind levy flight and levy walk. This is just a reminder. And then I will concentrate on levy walk because levy flight is not a good model for super diffusion. So that's why I concentrate on levy walk. And my purpose today is to present a new equation for levy walk, so-called single inter-differential wave equation for levy walk. The next part of my talk involves anomalous transport on network. And I will discuss scale-free network, metapopulation transport. And the key thing today is axiom of cumulative inertia, which leads to so-called anomalous aggregation on network. Great. So it's a widespread phenomenon in nature, in biology, in physics. We have the following random walk. So the particle moves straight line, and then turn isotropically. Then again, move straight line, turn. So we have a specific random movement. And the specific things about this random movement that this length path, path length L, is distributed according to Paolo. And what is interesting is that alpha varies from 0 to 2, which means the second moment is divergent. And this is everywhere inside cell, the particle, in turbulence moves that way. Sometimes the human travel, animal foraging, especially bacteria moves in that way. What is the problem? The main problem is because second moment is divergent, we can't define diffusion coefficients anymore. So standard diffusion equation doesn't work. Again, we can't write down Focke-Plunk equation. And that's why we call it anomalous transport. And two basic models for this kind of movement is Levy-Flight and Levy-Wolk, with heavy tail probability distribution. Now, what is the difference between Levy-Flight and Levy-Wolk? Because of application in biology nowadays, biologists change words quite easy. For them, it's just the same things. Levy-Flight, Levy-Wolk, and a lot of publication nowadays. But what I'm going to tell you that Levy-Flight is an example of Markovian process, easy and very unrealistic. Levy-Wolk is more realistic, but of course, more complicated. So Levy-Flight. Levy-Flight actually involves instantaneous jump. So if we are here, we start from here, then we just jump with infinite velocity at this position. As a result, we have Markovian process. And if we consider long time limit for this kind of random walk, write down master equation, consider proba, large time scale, we end up with this very nice equation. So instead of Laplacian, we have a fractional Laplacian. And clearly, it's Markovian model. But the problem is that for Levy-Flight, if we consider the position of the particle at the time t, square it, and find the expectation, the answer is infinity. So it's really unrealistic model. It's beautiful mathematically. That's why it's used all the time to describe this phenomenon with length path with a power law distribution. And of course, mathematically, it's nice because this equation has a very nice alpha stable distribution. In fact, when we can't define diffusion, so central limit theorem breaks down, now we have so-called generalized central limit theorem. And that equation provides this stable distribution solution. But again, because of this property, it's really unrealistic. Now we have to turn to Levy-Volk. Levy-Volk involves a random movement with finite velocity. So if we start from here, we just smooth the long straight line with a constant velocity. Of course, it's more realistic. But the problem is that the whole process become non-Markovian. Because of non-Markovian nature, of course, this fractional Laplacian is not enough. And the question is, what is the governing integral differential equations for the PDF of the position of the particle at the point x at the time t? So until now, that equation was not available. So usually, random-Volk or Levy-Volk formulated in terms of two integral equations. But the differential equation for PDF was not available. And that's the purpose of my talk, to present this new equation for Levy-Volk. Basically, there is a two approach for Levy-Volk. One is based on continuous-time random-Volk. The main idea in terms of random-Volk theory is to introduce joint probability density function for running times and lengths of displacement. The idea is almost trivial. If we have finite velocity, of course, the length, the particle moves along straight line is trivial as speed and time. This time is random. Of course, this is also random. And in order to create a heavy-tailed distribution for the length, we just need to create heavy-tailed distribution for running time. So this is psi. And then we can formulate the whole problem in terms of continuous-time random-Volk. But there is another alternative approach. That's what we use now. It's a structural density approach. Since Levy-Volk is non-Markovian, we can introduce extra running time. And then the model becomes Markovian. This is standard precision stochastic process when we extend phase space and makes non-Markovian process Markovian. That's what we did. And we introduce a structural density. So this particle moving right and left, the position x is the time t. And then we introduce extra running time. And then we introduce the rate of change. So if the particle moves, there is always probability that the particle changed direction. And now the rate of change depends on running time. No, no, now I'm going to discuss 1D. This is just illustration. But it can be extended very easily for 2D. The key things. So the rate at which the particle changed direction depends on running time. And it's inversely proportional to the tau. Because for levifolk, we have a different behavior. If we consider the position of the particle at the time t, squared it, and take expectation, then it's proportional to the t to the power gamma, where gamma greater than 1 and less than 2. So this is an example of anomalous persistence. So if I move in one direction, then I keep this direction. And you see, the longer I stay in one direction, less likely I change this direction. And this creates anomalous persistence. So you see, gamma is greater than 1. So it's faster than diffusion. But at the same time, because gamma less than 2, it's a ballistic movement. So it's still random, greater than diffusion, faster than diffusion, but less than ballistic movement. How to create this super persistent movement? That's a very simple assumption. So the longer I stay in one direction, less likely I change that direction. And we consider the rate instead of joining PDF for many reasons. Of course, in linear case, it leads to the same equation. But if we explicitly introduce running time and the motion becomes Markovian, we can take into account a lot of things. And if we have a lot of particles, moving crack, levee walk, we can take into account proliferation. This is important for microorganism, bacteria, cancer cells. Also, we can change the rate. That can be nonlinear. So if the rate of change depends on running time, this is standard linear model. But if we have a lot of particles moving in one direction, it might change the rate of change of direction. So the point is that structural density approach allows us to take into account a lot of different things, including nonlinear effect. And no consistent methodology to include this effect if we start from standard levee walk approach involving PDF for the jump and running time. Right. Now, if lambda is constant, I hope everybody knows, this is a classical persistent random walk model. So in order to extend trivial diffusion equation, it was happened in the past by Taylor, by Kaz, but many other people, they introduced finite speed of propagation. So the particle moves with a constant velocity and then change direction. Moves in another direction and then change direction. If the rate of constant, then we all know that it leads to the classical telegraph equation. So that's a classical telegraph equation. And of course, when lambda, the rate of change tends to infinity. And velocity tends to infinity in a such way that the ratio is a constant. We can introduce diffusion coefficient. So central limit theorem works. And of course, this is just a trivial extension of classical diffusion equation. But you see lambda is a constant, the rate of change. What happened? From a probabilistic point of view, what does it mean? It means if the particle moves in this direction with a constant velocity, the running time is exponentially distributed. As a result, we have trivial Markovian model. But what happened if the running time is power law distributed in a such way that second moment is divergent or first moment is divergent? What happened? What kind of equation we might have in this case? And that what I said at the beginning, that equation was not available. So everybody knows telegraph equation. It's in books, in everywhere. But if the running time is generally distributed, extension of classical telegraph equation was not available. So I'm not going to give you technical details. I just give you final equation because I don't have time. So equation looks like this. So extension. Of course, we still have standard wave operator because we have finite speed of propagation. But instead of this simple term of relating to friction, we have a very complicated operator involving the memory kernel. And memory kernel can be defined through the Laplace transform of the running time distribution. And again, I'm not going to give you details. If you're interested, then this is a paper in which that equation solved in particular case. Of course, we have that property of super diffusion. And recently, we extended this equation for several dimension. And also, we managed to take into account proliferation of random walkers. This is just one example, a very recent example. Just imagine we have many particles. If particle moves in this direction, but then I've got around you a lot of other particles moving in the same direction. What do you think would be implication for the particle? How the rate of change of direction change? I guess if we have a lot of particles moving in one direction, let's take into account public opinion. If you're all your friends moving one direction in terms of particular opinion, what happened to you? It's very unlikely you change your opinion. You just follow others. This is very well known psychological effect. So the same for particle. If the particle moves in one direction, then the rate of change drastically change. And that's what we're taking into account. Also locally and known locally. As a result, we managed to find non-linear equation which generates levee walk. So it's not just postulated from the very beginning. We have power law running time. No, we don't have that. As long as we have a lot of particles moving in one direction, that might generate this ballistic, no, sorry, subbalistic super diffusive movement. And that's what I've done recently. So this is a new paper in which we managed to show that due to non-linear effect, we have emergence of levee walk as a result of dynamics. And this is just illustration of the structural approach. Right. OK, now I'm going to turn to transport on a network. Well, we can see the transport of particles on network. And this is classical scale-free network in which the node with k links had that probability. So it's a power law distribution for the number of connection. So if we have nodes, then, of course, every node got some connection, let's say, k connection. And that connection distributed according to power law. And this is well-known, of course, network. We know the dynamical mechanism, which leads to this scale-free network due to preferential attachment. Now, let us consider the transport or levee walk. Sorry, no, just random walk on network. We have a lot of particles that stays particular node, then jump to another node, another node. In general, it's quite a complicated problem. But if we consider everything in terms of mean transport equation, then this is the key. If we introduce the mean number of particles in a node of order k, let's say here, we have that amount of particles. Of course, it's a function of time. And we can write down a simple balance equation, mean field transport equation. So the particle escape from the node and coming from different nodes. And this is a very trivial equation. And it has a stationary solution. And this is really important. So can I have your attention, please? So in long time limit, the number of particles at particular nodes proportional to the k, the number of connection. So the aggregation of particles happens in the nodes with a lot of connection. So that's why we say it's good to be well connected. And well-connected nodes are more popular. So this is a very well-known result in network theory. Of course, it's many things based on this idea, like page ranking and many other things. But the standard assumption is that the escape rate is Poisson process. So they just introduce the rate of escape, lambda. And of course, if you want to understand what is the rate at which particle escape from particular node like this, this is just a product of lambda and the number of particles in this node. But it's well-known the human activity is not Poissonian. So now, again, I'm going to use the same law. If you are in some position, what is the rate of escape? For human, it's not constant. I'll give you an example from our academic life. If we have postdocs, they move from one university to another. And the rate is usually constant. But what about professors? As long as you find a permanent position, you're trapped. The longer you stay in your position, less likely you escape. I just look at you. How long you spent in your permanent position? For ages, 10 years? I'm personally, I'm trapped in Manchester. Already, I didn't expect. When I turned up in Manchester 20 years ago, I didn't expect I would stay. But I'm trapped. Why? Of course, there are a lot of reasons. Of course, promotion, kids, mortgage. But the whole idea is the longer you stay in a position, less likely you escape from that position. And it doesn't matter. Residence time, employment, state of mind. If you are leftist for a long time, believe me, you will stay liberal. The same for conservative. So this is quite universal law which we explore. And that law called axiom of cumulative inertia. So the individuals escape probability from a node decreases with the residence time spent in this node. This is a well-established fact. It's just the same. If I move in one direction, the longer I move in this direction, less likely I change the direction. The same things if I stay in the particular point, the longer I stay, less likely I escape. Of course, mathematically it can be formulated like escape rate depends on residence time, and it's inversely proportional. Of course, this rate, it's very easy to show, leads to the power law distribution. And the question is now, let's say we have a node with a lot of connection. And now we have another node, let's say with two connection. But the escape rate obeys this law. What happened? That's a key question we put in this paper. So standard theory tells us the aggregation happens here because of a lot of connection. Now we have a competition with a node with a lot of connection, but with a standard Poisson law of escape, and then with anomalous behavior. And of course, you can expect what we showed that this anomalous node in long time limit completely dominates the process. So it's not important the number of connection. It's important how much time you spend on that particular node. And if the residence time is power law distributed with infinite first moment, that would be dominant node. That's what we observe in marketing. You see, let's imagine the street with a lot of shops. You're doing shopping, get in, get out. But then if the manager managed to attract you and trapped you in a shop, and you spent a lot of time with the power law distribution, you spent a lot of money in that shop. Because it's a nice correlation, nice data between the time you spent in particular shop and the money you spent. So it's very important to be. And again, the whole idea, as I mentioned, what about attractiveness of the place? What this means? Bad university or good university? In a good university, you are trapped because of different reasons, of course. But this is just universal law. And that's why my PhD student, Helena Stage, they've got a lot of review. They ask the member of staff in our university how much you already employed in this university. And this very unusual behavior, that's what I'm going to discuss right now. So main result is, but what about main result? How to formulate this mathematically? We managed to translate this axiom of cumulative inertia, which is famous law in social science, into a very nice mathematics. So that's the difference in terms of mathematics. So for the classical escape rate, we had this law, which is a standard first order reaction. Now, if we apply axiom of cumulative inertia, instead of constant lambda, we have a operator. And that's Riemann-Lübild fractional derivative. So this law of cumulative inertia leads to this fractional derivative. So fractional derivative is not just postulated from the very beginning. It just derived. And of course, this fractional derivative generate this phenomenon, anomalous aggregation at the node with this behavior corresponding to axiom of cumulative inertia. Again, this is the result I already described. Now, what happened? So if all particles, at the end, aggregate in this particular node, of course, how to prove this is real phenomenon, then that's what we did. We introduced a structural density. It's a function of time and also the residence time. And when time tends to infinity, this has a very beautiful structure. And that's what is here. It's not clear if you look at how it looks like. But in terms of picture, it looks like U-shaped distribution. So we found the structural density in this anomalous node behaves like this. And this is ratio of residence time over total time of observation. So in terms of academic life, it's a very simple interpretation. If you look at the member of staff in a good university, their eyes are all professors on the right and young assistant professors. I'm not saying nothing is between, but that will be a distribution in a good place. That's what we are going to do. My PhD students, of course, ask the Human Resource Department and Oxford and Cambridge they refuse to provide the data so far. But again, we strongly believe that would be because from my particular experience, as long as I speak to professors, they always stay in one place, usually. Of course, some staff professors move around. But most of the people, as long as they're in a good position, they stay for 10, 20 years. Because of different reasons, of course. About, oh, no, no, that's what I put in this category, post-docs. Yes, yeah. But again, this is like a joke. But in fact, it's not a joke. In the past, I described this as like a joke. But then when we got this anomalous aggregation result, again, I asked my brilliant PhD students to find the data. And that was not easy. And she managed to find data in American Midwest. There's a statistics available of residence time in some particular five minutes. And then she managed to find that this is your shape distribution. So if you found a good street in a good place and do a survey, I am sure you will get that result. Because either residence from the very beginning or newcomers. And of course, something in between. So to finish, of course, everything is here related to non-Markovian process. It's not easy to deal with. And again, it involves fractional derivative. And those fractional derivatives, in terms of time, is not just again postulated. We managed to translate this social law into this fractional derivative. Now we have an extension of that model. But generally speaking, of course, description of non-Markovian reaction transverse in network is not available. It's just at the very beginning. And in this book, we can find the theory on non-Markovian process. OK, thank you very much.