Stochastic geometry and telecommunication networks
Jahnel, Benedikt
Spatial device-to-device communications are expected to play a key role in future communication systems. Their often high complexity can, at least in part, be modelled with the help of a probabilistic approach, where the network components are considered to be a stochastic point process. In this talk, I will introduce some of the most important ingredients in the theory of stochastic geometry, and present examples on how we use them to study for example malware propagation in device-to-device networks or bottleneck behavior for the connectivity in such networks.
 Thank you for the introduction. My name is Benedikt Gjarnel. I work at the Weierstraße Institute. It's a pleasure to be here and present some joint work within my group, DICOMnet. It's a small group at Vias in Berlin and we are all probabilists and we use stochastic geometry to model and analyze telecommunication networks. So I also realized that this is a diverse audience so I tried to essentially do my talk in three parts. Some motivation, some general ideas, but I don't want to spare you from some theorems at the end of my talk. So you have an idea of what we're actually doing every day. So the motivation, as I said, comes from telecommunication systems. I guess I don't have to emphasize this too much. There is a strong increasing demand for communications on all different levels. This is just some slides that I took from some picture that I took from a big telecommunication systems provider, Cisco, and they make predictions in their white papers and all these, they indicate that we have an explosion of data transmissions, an explosion of subscribers. There's the large field of Internet of Things where small units communicate with each other, machine to machine is a bus word in the field. So we have this just this tremendous increase in mobile communication, but also in communication of small devices like sensors. Exponential growth essentially is what we see right now and in the near future. And I want to address a very particular part of this development and how operators try to cope with these challenges. And the part that I want to speak about today are spatial networks. So not social networks in the Internet. These are spatial networks. And in particular, an aspect of spatial networks that is also already to a small extent implemented in the current 5G mobile standard, but it will be playing a more stronger role. Many people predict that it will be playing a stronger role even in the future. And this aspect is device to device communications. So as these simulated pictures should indicate, we have maybe an environment, some kind of edge system here, some kind of urban topology maybe, and we have devices that sit on in this environment. You're indicated by blue particles, blue points, and they're supposed to communicate in a peer-to-peer fashion directly with each other. And this should bring some key benefits to the systems that are listed here. It provides more robustness to the system because there's multiple paths that data can be transmitted between components of the networks. It can come with much lower costs. We can get rid of heavy infrastructure, billions worth of infrastructure, and our current days mobile phones have a lot of power themselves, for instance. Security issues come decentralized. This is a benefit, but this can also be a challenge. And we can get rid of a lot of latency if you think of peer-to-peer communications. We don't have to go to the backbone of the system. We don't have to go through base stations. The devices can communicate in a peer-to-peer fashion, makes it faster and more robust. Many more aspects that I have listed here, and of course challenges. One of the main challenges is that from an operator's perspective, you cannot control the system. It's an ATOX system. So the network components building this network, and you have to deal with the fact that this is somehow not controlled by some operator. It's actually hard to read from here. Applications. I think I've already addressed some of them. It already implemented our main sensor networks, where, for instance, in large solar fields, the different sensors communicate with each other. It makes no sense to build up some centralized base station system. The sensors communicate. They accumulate the data, and they bring it to some exit point. So you see this implemented already in a few situations in the real world, but we hope and we try to support by our research more applications in the new future. For instance, by creating networks in areas of the world where there is no infrastructure available yet. So our a priori belief is that we can only understand these systems theoretically by using probabilistic methods. As I try to convince you, there is an intrinsic randomness in the system, because we cannot control how the individual components behave in the system. So I want to use the lenses of a probabilist to model and analyze these systems. Randomness enters on various levels. There is an intrinsic randomness on the behavior of the components. From an operator's perspective, you have a lot of uncertainty how they behave. We have stochastic algorithms that are partially used, but can still be developed. And when we deal with randomness, then we can essentially look at the system from two sides. What is the expected behavior of the system? That's important from a design perspective. And something that is not yet developed so good, and we want to make contributions there. We do make contributions there is to also understand the system in its behavior, in its unexpected behavior, which is also critical. You want to understand situations where the system behaves really bad and understand why it behaves bad. This is typically also very hard to simulate if you have a model, because maybe these bad situations are very rare. So to make a statistics on the typical behavior within the rare event is difficult. We have some tools to tackle this problem. The undots we use is stochastic geometry. So that's maybe roughly 40, 50 years old field within spatial probability. And in essence, you can make a mapping between notions in stochastic geometry and the notions that come up when you speak with people from engineering in big telecommunication companies. So for instance, the device locations can be seen as point processes. So point processes would be the equivalent notion within the community of stochastic geometry for the locations of the devices or the components of the networks in space. For instance, we can deal with an environment. And in this picture, the environment is a street system. And this can be seen as a tessellation process in space, which is also a quite fundamental notion in stochastic geometry. Mobility is a big topic. How do the components move in space? And there's a large array of models within the probabilistic community that try to model this. For instance, the random waypoint model, which is it's different from a random walk, but it also has some similarities. There is medium axis. Basically, what I try to convince you that there is a, you can, this is really the right framework. We can find the appropriate notions within the stochastic geometry world for a variety of, for the models. What we want to do, and this is our domain, we want to make rigorous analysis. So by solid mathematical theorems and deductions. So this should enable us to make predictions for the future by understanding these models. With a particular interest in the critical behavior. So do we see maybe faith transitions? Do we see situations where this model dramatically changes its behavior when you just vary a certain parameter a little bit? So discontinued newities with respect to the parameter space. And many of these instances are also supported by simulation. So I present to you now three different applications and associated methods that we use. And the first application is connectivity in mobile urban networks. And just stick with the picture. Maybe there is, there is a city topology, maybe some street system. And we have mobile devices in blue. And we have some base, additional base station infrastructure. And now we want to augment the base stations or the system by the possibility to have intermediate peer to peer connections. So the green base stations, they come within certain interaction radius where they can directly be connected with devices. But there are lighter green extensions of these areas that by adding one, two, three additional peer to peer hops. And the base stations, they don't move in space, but the devices they move maybe along the streets, maybe away from the streets. And, and a characteristic of the network that we want to analyze, I put here in the display in mass symbols, don't worry too much. Maybe later I will be more precise with the notation. One, one important quantity is the connection time. So how much time does a typical device connect to an infrastructure using at most k hops. So if you build such a network, this is very important to understand the quality of the network, how much time do you actually spend connected to the infrastructure by at most k hops. Second, maybe how often do you have to reconnect to new base stations, all these quantities play a major role in the design of these networks. Let me also say that in many instances, our analysis will be performed in some, some kind of limiting regime. It's, it's rather hard to come up with, with vigorous results. If you fix parameters, so we want to understand essentially the edges and corners of the parameter space by, for instance, letting the number of hops become large, letting time become large. So the actual analysis is typically performed in some kind of limiting regime. Another application is data routing. So here is an, here's an, the image or the picture is as follows. One has a central base station and a number of randomly placed devices around the base station, base stations in the center. And now the idea is that every device has a certain algorithm that tells them, okay, I want to connect to the base station, the center, but I cannot go there directly. So I use I look for instance for the device that is closest to me in the direction of the base station and therefore creating a tree where messages has through via other devices towards the base station. And then for instance, the question can be how is the data, how much data does actually a point close by the, by the base station has to handle. So how much data is actually passed through nodes that are close to the base station. And also here, the system can be analyzed in its expected behavior. But what is even more interesting is to understand what are the typical configurations of the system. So for instance, a large proportion of devices cannot connect to the base station because maybe some, there's too much throughput in a certain region of space. So this is, we also do again, an analysis with respect to the expected behavior, but also the unexpected behavior. This is, this is very important. Third application is malware propagation. And here, the idea is that there is wanted data in the system, there's unwanted data. And this is a decentralized situation. So, so there is maybe a malware at the, at some typical node at the center of the network. And how does it propagate through the network? Obviously, this is, this is very much connected maybe to two models for epidemic spreading. So, so I think we make there quite a strong connection towards spatial epidemic modeling. So here you see a picture where you have susceptible devices in space, they're connected whenever they're sufficiently close to each other. And we have, we have an initial malware at the center, it's spread through the edges step by step in continuous time in this case. And, and red infected devices appear. But here comes a speciality of, of our model, which we designed together with people from, from orange, by the way, I should mention this, we have a major collaboration with, with industry. So just to make sure that this is not purely academic. So there is, there is, there is regular contracts. I think we have seven contracts already, where we analyze and simulate these systems for them. So, so there is, there is some, some grounding there. Anyways, the speciality is, is now, so how can you, how can you counteract this, this, this spreading of a malware over, of an infection through the system. And what they imagine, and already in part have, have, have prototyped is something that they call a white knight, which is, which is also a device in space. It's, it's essentially indistinguishable from, from the regular devices. And it can patch when, when it can patch an infected device whenever it is attacked. The problem here seems to be that you cannot simply patch any device because of legal reasons you have to wait until you're attacked. And only then you can patch. So this, this is, this makes the system actually quite complicated, highly non-monotonic. And what you see in the picture here is a later point in time. Initially, there is an infection at the origin. There are white knights in green in space sparsely. And then the infection grows. And whenever an infection meets a white knight, it can retaliate. And therefore, on the set of in set, it's just infected devices patch and they become green. So you see a kind of a chase escape dynamics here. The infection has a slight advantage because it can go to all the infected, the susceptible, susceptible devices, but the white knights has, they have to chase because they can only use the infected devices. So this is, this is an analysis that we do. And let me also say that, that one of the main inputs here is that we deal with a random network of positions. So many of these things have been done actually quite, quite recently on, on non-random, let's say grids or non-random topologies, geometries. In our case, we always deal with these random, random point configurations in space. So these are three applications. Now, let, let me address briefly methods that we use from stochastic geometry. One key method is called continuum percolation. This is closely linked by the way to statistical physics. And, and this is one of the, one of the key concepts that we explore and use. And it goes like this, for instance, in this picture, you have some kind of environment in green where you see, you see a certain intensity, density of points in blue, and they all come with, with a certain interaction radius that might be device dependent. So we can model this also by, by assigning individual IID areas where, where people can, where, where the devices can, can form connections. And then the question in, in continuum percolation is if you now tune the intensity of the devices, the expected number of devices you see in a unit volume, does when, if you tune this up, so you see more and more devices in expectation, is there a point where the system dramatically change from a locally, a local behavior where you see only local clusters of connected devices to a global situation where all of a sudden with a positive probability, a typical device can actually connect to infinity. For this is, for us, this is a, this is a rough estimate of a, of a breaking point of the system where it jumps from a purely local behavior to an global behavior. So, so for us, this, this is, is, is some kind of a, a first indication, how many people do you actually have to attract to such a system, have to be part of the system. So you see long range, the possibility of long range connectivity. The second method that is quite underdeveloped in the field is the, is the method of large deviations. So as I mentioned, briefly large deviations is a, is a, is a highly developed machinery methodology within Socastics in general to deal with rare events. And the rare event is, for instance, this, this random variable Xn, it wants to be, it has an expected value where, where, where it wants to be, there's always a running parameter and I will maybe later explain more. And we, we are interested in what is actually the probability that the system behaves away from what it wants to be, where, from, from its expected behavior. And in many instances, if you have enough independence in the system, this goes exponentially. And there is a certain speed, rate function I, and this rate function can be used to understand the typical behavior within the atypical event. And, and this, this can be, for instance, employed to, to, to build an important sampling algorithm. For instance, in this picture, we were interested, okay, here, here, all these, these many, many devices, they want to connect to the base session, there's interference. So the problem is, there is a cocktail party problem. If, if a person wants to speak to another person, but there are many, many people around, it makes it very hard to form this connection. And we're interested in what is the typical behavior of the system in a situation where you see a certain atypical number of this disconnected devices. And in this case, you even see a face, a symmetry breaking phase transition. So it's also from a purely mathematical point of view, an interesting theory and interesting applications for, for large deviations within the realm of space-time point processes. The third method is interacting particle systems on random graphs. This is a quite old theory where you look at Markov processes, or maybe beyond Markov processes on fixed configurations. You have a configuration of, of, of, of particles, let's say on a grid, they come in different local states. And now we, we, we devise exponential rates, interacting, depending on the behavior of a device. And one point depends on the neighbors, for instance. And then we look at the, at the long-time behavior of these Markov processes. And for instance, in the, in the case that I gave you with the white nights and the, and the infection spreading, we can, we can, we can model this in the, in the framework of interacting particle systems, but now on random graphs. And in some cases, for instance, derive face diagrams of global extinction and global survival rigorously in the framework of interacting particle systems. How much time do I have? When did I start? Oh, two minutes. Okay. So I was a bit slow, but that's okay. Just, okay, maybe I actually, two minutes, including questions. Okay. Just, just to give you a flavor, quick flavor of how we, how we deal with these things. So we start with a random process, so catholic process. Typically, if we have no statistical knowledge, it's a good idea to start with a so-called Poisson point process. This, you should essentially think of a uniform distribution of points in space. We can create a graph G, by just connecting two points that are sufficiently close to each other. And there is a famous phase transition of continuum percolation, where I see an infinite cluster emerging. And then we can look at such a Voronoi tessellation. These are all the areas of points that are connected to the infinite cluster. And on the second level, now, now introduce a time process, where we associate to every edge a certain passing time. And then we can look at the, the set of, of infected devices, when we initially have an infection at the origin, we can look at all the devices for which there is a pass, where the passing times, the sum of the passing times does not exceed a certain time t. So this is the set of infected points at time t. And this is a, this is a recent result, where we, which is called a shape theorem. So I will also move. We're interested in the limiting shape of the infected region. This is the infected region, we scale at time t, we scale by t. And what is indeed true, if you put some moment conditions and, and zero conditions here on the passing times. So that we see a limiting shape, which is in fact the ball, where the parameter phi is speed. So here's a picture. Between two time points, there is t1, you see a certain region infected and at some later time point, you see additional infections in the, in the red area. Yeah, some, some rescaling results, maybe I'll skip this. Thank you for your attention. Thank you very much for this really interesting talk questions. Let me start. You always mentioned random networks. For my little experience, I never saw in reality a random network. So you have all some structures. Do you mean Erdisch-Renie or also other kinds of random networks? Well, here the randomness, I mean, the simplest case, the randomness enters, enters via the position of the devices in space, the points. And randomness here is, is I mean, we can have a philosophical question about what is randomness, but enters from an operator's perspective, I have no control on where, where, where you move. I mean, I cannot control where you move in space. So I have maybe some knowledge that you have a typical path to your office. But, but if you look at from a global perspective, I have no clue what you're doing. Okay. Then, then this is a different perspective. I call this topology, whether it's randomness or not. And this is a bit different in your point. I think, I think it has some structures in topology. Yes. Yes. But okay, good, good. Then it's clear. Thank you. Further questions? Hi. So if you have such an attack network, how do you then solve the network routing problems? Because I guess you cannot solve them in a centralized way. Who passes messages to women on which path? Oh, we don't solve them. We, we, we make, for instance, we evaluate expected behavior. I see. Okay. So the, the routing takes a random path to the, it's a random path depending on the configuration of the networks. It's a random configuration. We can say something about the past in this configuration. And then we have a probability distribution and we can make analysis of, for instance, the expected behavior, the unexpected behavior. Okay. Thank you. Yeah. Okay. Maybe a quick question. You talked about motivating people. So how can you incentivize participation in a talk peer to peer network for mobile communication? I mean, you, you have to have your GPS on. You have to use data all the time to relay calls or data. How do you incentivize and how do you make sure people don't just make their own system? Like, you know, what do you call Uberization, I believe? This, this is actually an interesting story. One of our own older projects with, with this major telecommunication company, they're, they're a bit scared that something like Uber happens to them in the sense that you have, for instance, only in the city, you have this company and they say, well, we don't have infrastructure, but we all sign up for this contract and you can use your cell phone so you can, other people can send their messages, messages via Uber. Actually, such apps exist. For instance, Firejet played a role in the, in the protested in Hong Kong, where the government shut down the infrastructure and then via this app, you can, you can send messages. So for them, really, it's a bit of a threat that all of a sudden this company comes up and they, they basically exist via an app and they, they, they, they provide good network. So people will sign up for them and leave there. There are billions of worse of infrastructure, it's useless. So, so they're actually interested in what is the critical number of people you have to, you have to have in your business. So this actually works. You can have long range connectivity, for instance. So the attraction comes from low cost, I guess, essentially. Yeah, also a bit in the direction of the two questions. If I understand correctly, you try to avoid, of course, percolation transitions. And if you speak on good or bad messages, I mean, percolation is good if you want to have connectivity. Okay, good. And then you can give conditions for the companies whether to avoid or to, to reach. For instance, if it's a one parameter model, where the only parameter is the expected number of devices per unit square kilometer, then I can rigorously, but also by simulations, tell them there's a critical point where the system jumps from only being local to being global. Okay, okay. Okay, that's interesting. Yeah. Yep, last question. I'm wondering about bandwidth problems, like I'm really having the mobile phone users in the back of my mind. So if I order a file from a central server, then I expect the server to have a large bandwidth. But if I kind of take this route through all small devices, then maybe somebody just doesn't have a good know them and then it will kind of stop. Yeah, that's true. Okay. Cool. Then it's not a question. Was a statement or was it a question statement that I could answer? Okay, but it's not a right question. Yeah, I don't I am not at the point where I can maybe join lunch. Yeah, you should discuss. Yeah, thank you. Okay, then thank you very much. We are really in the beginning of a discussion. What is the best situation to start the break? Thank you very much. Okay.
 Thank you for the introduction. My name is Benedikt Gjarnel. I work at the Weierstraße  Institute. It's a pleasure to be here and present some joint work within my group,  DICOMnet. It's a small group at Vias in Berlin and we are all probabilists and we use stochastic  geometry to model and analyze telecommunication networks. So I also realized that this is a  diverse audience so I tried to essentially do my talk in three parts. Some motivation,  some general ideas, but I don't want to spare you from some theorems at the end of my talk.  So you have an idea of what we're actually doing every day. So the motivation, as I said,  comes from telecommunication systems. I guess I don't have to emphasize this too much. There is a  strong increasing demand for communications on all different levels. This is just some slides  that I took from some picture that I took from a big telecommunication systems provider,  Cisco, and they make predictions in their white papers and all these, they indicate that we have  an explosion of data transmissions, an explosion of subscribers. There's the large field of  Internet of Things where small units communicate with each other, machine to machine is a bus  word in the field. So we have this just this tremendous increase in mobile communication,  but also in communication of small devices like sensors. Exponential growth essentially is what  we see right now and in the near future. And I want to address a very particular part of this  development and how operators try to cope with these challenges. And the part that I want to  speak about today are spatial networks. So not social networks in the Internet. These are spatial  networks. And in particular, an aspect of spatial networks that is also already to a small extent  implemented in the current 5G mobile standard, but it will be playing a more stronger role. Many  people predict that it will be playing a stronger role even in the future. And this aspect is device  to device communications. So as these simulated pictures should indicate, we have maybe an  environment, some kind of edge system here, some kind of urban topology maybe, and we have devices  that sit on in this environment. You're indicated by blue particles, blue points, and they're  supposed to communicate in a peer-to-peer fashion directly with each other. And this should bring  some key benefits to the systems that are listed here. It provides more robustness to the system  because there's multiple paths that data can be transmitted between components of the networks.  It can come with much lower costs. We can get rid of heavy infrastructure, billions worth of  infrastructure, and our current days mobile phones have a lot of power themselves, for instance.  Security issues come decentralized. This is a benefit, but this can also be a challenge. And  we can get rid of a lot of latency if you think of peer-to-peer communications. We don't have to go  to the backbone of the system. We don't have to go through base stations. The devices can communicate  in a peer-to-peer fashion, makes it faster and more robust. Many more aspects that I have listed  here, and of course challenges. One of the main challenges is that from an operator's perspective,  you cannot control the system. It's an ATOX system. So the network components building this  network, and you have to deal with the fact that this is somehow not controlled by some operator.  It's actually hard to read from here. Applications. I think I've already addressed some of them.  It already implemented our main sensor networks, where, for instance, in large solar fields,  the different sensors communicate with each other. It makes no sense to build up some centralized  base station system. The sensors communicate. They accumulate the data, and they bring it to some  exit point. So you see this implemented already in a few situations in the real world, but we hope  and we try to support by our research more applications in the new future. For instance,  by creating networks in areas of the world where there is no infrastructure available yet.  So our a priori belief is that we can only understand these systems theoretically by using  probabilistic methods. As I try to convince you, there is an intrinsic randomness in the system,  because we cannot control how the individual components behave in the system. So I want to  use the lenses of a probabilist to model and analyze these systems. Randomness enters on  various levels. There is an intrinsic randomness on the behavior of the components. From an  operator's perspective, you have a lot of uncertainty how they behave. We have stochastic  algorithms that are partially used, but can still be developed. And when we deal with randomness,  then we can essentially look at the system from two sides. What is the expected behavior of the  system? That's important from a design perspective. And something that is not yet developed so good,  and we want to make contributions there. We do make contributions there is to also understand  the system in its behavior, in its unexpected behavior, which is also critical. You want to  understand situations where the system behaves really bad and understand why it behaves bad.  This is typically also very hard to simulate if you have a model, because maybe these bad  situations are very rare. So to make a statistics on the typical behavior within the rare event  is difficult. We have some tools to tackle this problem. The undots we use is stochastic geometry.  So that's maybe roughly 40, 50 years old field within spatial probability. And in essence,  you can make a mapping between notions in stochastic geometry and the notions that come up when you  speak with people from engineering in big telecommunication companies. So for instance,  the device locations can be seen as point processes. So point processes would be the  equivalent notion within the community of stochastic geometry for the locations of the devices or  the components of the networks in space. For instance, we can deal with an environment. And  in this picture, the environment is a street system. And this can be seen as a tessellation  process in space, which is also a quite fundamental notion in stochastic geometry.  Mobility is a big topic. How do the components move in space? And there's a large array of  models within the probabilistic community that try to model this. For instance, the random  waypoint model, which is it's different from a random walk, but it also has some similarities.  There is medium axis. Basically, what I try to convince you that there is a,  you can, this is really the right framework. We can find the appropriate notions within the  stochastic geometry world for a variety of, for the models. What we want to do, and this is our  domain, we want to make rigorous analysis. So by solid mathematical theorems and deductions. So  this should enable us to make predictions for the future by understanding these models.  With a particular interest in the critical behavior. So do we see maybe faith transitions?  Do we see situations where this model dramatically changes its behavior when you just  vary a certain parameter a little bit? So discontinued newities with respect to the parameter space.  And many of these instances are also supported by simulation.  So I present to you now three different applications and associated methods that we use.  And the first application is connectivity in mobile urban networks. And just stick with the  picture. Maybe there is, there is a city topology, maybe some street system. And we have mobile  devices in blue. And we have some base, additional base station infrastructure. And now we want to  augment the base stations or the system by the possibility to have intermediate peer to peer  connections. So the green base stations, they come within certain interaction radius where they can  directly be connected with devices. But there are lighter green extensions of these areas  that by adding one, two, three additional peer to peer hops. And the base stations,  they don't move in space, but the devices they move maybe along the streets, maybe away from the  streets. And, and a characteristic of the network that we want to analyze, I put here in the display  in mass symbols, don't worry too much. Maybe later I will be more precise with the notation.  One, one important quantity is the connection time. So how much time does a typical device  connect to an infrastructure using at most k hops. So if you build such a network, this is  very important to understand the quality of the network, how much time do you actually spend  connected to the infrastructure by at most k hops. Second, maybe how often do you have to  reconnect to new base stations, all these quantities play a major role in the design of these networks.  Let me also say that in many instances, our analysis will be performed in some,  some kind of limiting regime. It's, it's rather hard to come up with, with vigorous results. If you  fix parameters, so we want to understand essentially the edges and corners of the parameter space by,  for instance, letting the number of hops become large, letting time become large. So the actual  analysis is typically performed in some kind of limiting regime. Another application is data routing.  So here is an, here's an, the image or the picture is as follows. One has a central base station  and a number of randomly placed devices around the base station, base stations in the center.  And now the idea is that every device has a certain algorithm that tells them, okay, I want to connect  to the base station, the center, but I cannot go there directly. So I use  I look for instance for the device that is closest to me in the direction of the base station and  therefore creating a tree where messages has through via other devices towards the base station.  And then for instance, the question can be how is the data, how much data does actually a point  close by the, by the base station has to handle. So how much data is actually passed through  nodes that are close to the base station. And also here, the system can be analyzed in its expected  behavior. But what is even more interesting is to understand what are the typical configurations  of the system. So for instance, a large proportion of devices cannot connect to the base station  because maybe some, there's too much throughput in a certain region of space. So this is, we also do  again, an analysis with respect to the expected behavior, but also the unexpected behavior.  This is, this is very important. Third application is malware propagation. And here,  the idea is that there is wanted data in the system, there's unwanted data. And this is a  decentralized situation. So, so there is maybe a malware at the, at some typical node at the  center of the network. And how does it propagate through the network? Obviously, this is, this  is very much connected maybe to two models for epidemic spreading. So, so I think we make there  quite a strong connection towards spatial epidemic modeling. So here you see a picture where you have  susceptible devices in space, they're connected whenever they're sufficiently close to each other.  And we have, we have an initial malware at the center, it's spread through the edges step by  step in continuous time in this case. And, and red infected devices appear. But here comes a  speciality of, of our model, which we designed together with people from, from orange, by the way,  I should mention this, we have a major collaboration with, with industry. So just to make sure that  this is not purely academic. So there is, there is, there is regular contracts. I think we have  seven contracts already, where we analyze and simulate these systems for them. So, so there is,  there is some, some grounding there. Anyways, the speciality is, is now, so how can you,  how can you counteract this, this, this spreading of a malware over, of an infection through the  system. And what they imagine, and already in part have, have, have prototyped is something  that they call a white knight, which is, which is also a device in space. It's, it's essentially  indistinguishable from, from the regular devices. And it can patch when, when it can patch an  infected device whenever it is attacked. The problem here seems to be that you cannot simply  patch any device because of legal reasons you have to wait until you're attacked. And only then  you can patch. So this, this is, this makes the system actually quite complicated, highly non-monotonic.  And what you see in the picture here is a later point in time. Initially, there is an infection  at the origin. There are white knights in green in space sparsely. And then the infection grows.  And whenever an infection meets a white knight, it can retaliate. And therefore, on the set of  in set, it's just infected devices patch and they become green. So you see a kind of a chase escape  dynamics here. The infection has a slight advantage because it can go to all the infected, the  susceptible, susceptible devices, but the white knights has, they have to chase  because they can only use the infected devices. So this is, this is an analysis  that we do. And let me also say that, that one of the main inputs here is that we deal with a random  network of positions. So many of these things have been done actually quite, quite recently  on, on non-random, let's say grids or non-random topologies, geometries. In our case, we always deal  with these random, random point configurations in space. So these are three applications. Now,  let, let me address briefly methods that we use from stochastic geometry. One key method is called  continuum percolation. This is closely linked by the way to statistical physics. And, and this is  one of the, one of the key concepts that we explore and use. And it goes like this, for instance,  in this picture, you have some kind of environment in green where you see, you see a certain intensity,  density of points in blue, and they all come with, with a certain interaction  radius that might be device dependent. So we can model this also by, by assigning individual  IID areas where, where people can, where, where the devices can, can form connections.  And then the question in, in continuum percolation is if you now tune the intensity of the devices,  the expected number of devices you see in a unit volume, does when, if you tune this up,  so you see more and more devices in expectation, is there a point where the system dramatically  change from a locally, a local behavior where you see only local clusters of connected devices  to a global situation where all of a sudden with a positive probability, a typical device can  actually connect to infinity. For this is, for us, this is a, this is a rough estimate of a,  of a breaking point of the system where it jumps from a purely local behavior to an global behavior.  So, so for us, this, this is, is, is some kind of a, a first indication, how many people do you  actually have to attract to such a system, have to be part of the system. So you see long range,  the possibility of long range connectivity.  The second method that is quite underdeveloped in the field is the, is the method of large  deviations. So as I mentioned, briefly large deviations is a, is a, is a highly developed  machinery methodology within Socastics in general to deal with rare events. And the rare event is,  for instance, this, this random variable Xn, it wants to be, it has an expected value where,  where, where it wants to be, there's always a running parameter and I will maybe later explain  more. And we, we are interested in what is actually the probability that the system behaves away from  what it wants to be, where, from, from its expected behavior. And in many instances, if you have  enough independence in the system, this goes exponentially. And there is a certain speed,  rate function I, and this rate function can be used to understand the typical behavior within  the atypical event. And, and this, this can be, for instance, employed to, to, to build an  important sampling algorithm. For instance, in this picture, we were interested, okay, here, here,  all these, these many, many devices, they want to connect to the base session, there's interference.  So the problem is, there is a cocktail party problem. If, if a person wants to speak to  another person, but there are many, many people around, it makes it very hard to form this connection.  And we're interested in what is the typical behavior of the system in a situation where you see  a certain atypical number of this disconnected devices. And in this case, you even see a face,  a symmetry breaking phase transition. So it's also from a purely mathematical point of view,  an interesting theory and interesting applications for, for large deviations within the realm of  space-time point processes. The third method is interacting particle systems on random graphs.  This is a quite old theory where you look at Markov processes, or maybe beyond Markov processes  on fixed configurations. You have a configuration of, of, of, of particles, let's say on a grid,  they come in different local states. And now we, we, we devise exponential rates, interacting,  depending on the behavior of a device. And one point depends on the neighbors, for instance.  And then we look at the, at the long-time behavior of these Markov processes. And for instance,  in the, in the case that I gave you with the white nights and the, and the infection spreading,  we can, we can, we can model this in the, in the framework of interacting particle  systems, but now on random graphs. And in some cases, for instance, derive face diagrams of  global extinction and global survival rigorously in the framework of interacting particle systems.  How much time do I have? When did I start? Oh, two minutes. Okay. So I was a bit slow, but that's okay.  Just, okay, maybe I actually, two minutes, including questions.  Okay. Just, just to give you a flavor, quick flavor of how we, how we deal with these things.  So we start with a random process, so catholic process. Typically, if we have no statistical  knowledge, it's a good idea to start with a so-called Poisson point process. This, you should  essentially think of a uniform distribution of points in space. We can create a graph G,  by just connecting two points that are sufficiently close to each other. And there is a famous  phase transition of continuum percolation, where I see an infinite cluster emerging.  And then we can look at such a Voronoi tessellation. These are all the areas of points that are  connected to the infinite cluster. And on the second level, now, now introduce a  time process, where we associate to every edge a certain passing time. And then we can look at  the, the set of, of infected devices, when we initially have an infection at the origin,  we can look at all the devices for which there is a pass, where the passing times,  the sum of the passing times does not exceed a certain time t. So this is the set of infected  points at time t. And this is a, this is a recent result, where we, which is called a  shape theorem. So I will also move. We're interested in the limiting shape of the infected region.  This is the infected region, we scale at time t, we scale by t. And what is indeed true,  if you put some moment conditions and, and zero conditions here on the passing times. So  that we see a limiting shape, which is in fact the ball, where the parameter phi is speed.  So here's a picture. Between two time points, there is t1, you see a certain region infected and  at some later time point, you see additional infections in the, in the red area.  Yeah, some, some rescaling results, maybe I'll skip this. Thank you for your attention.  Thank you very much for this really interesting talk questions.  Let me start. You always mentioned random networks. For my little experience, I never saw in reality  a random network. So you have all some structures. Do you mean Erdisch-Renie or also other kinds of  random networks? Well, here the randomness, I mean, the simplest case, the randomness  enters, enters via the position of the devices in space, the points. And randomness here is,  is I mean, we can have a philosophical question about what is randomness, but  enters from an operator's perspective, I have no control on where, where, where you move. I mean,  I cannot control where you move in space. So I have maybe some knowledge that you have a typical  path to your office. But, but if you look at from a global perspective, I have no clue what you're  doing. Okay. Then, then this is a different perspective. I call this topology, whether it's  randomness or not. And this is a bit different in your point. I think, I think it has some  structures in topology. Yes. Yes. But okay, good, good. Then it's clear. Thank you. Further questions?  Hi. So if you have such an attack network, how do you then solve the network routing problems?  Because I guess you cannot solve them in a centralized way. Who passes messages to women on  which path? Oh, we don't solve them. We, we, we make, for instance, we evaluate expected behavior.  I see. Okay. So the, the routing takes a random path to the, it's a random path depending on the  configuration of the networks. It's a random configuration. We can say something about  the past in this configuration. And then we have a probability distribution and we can make analysis  of, for instance, the expected behavior, the unexpected behavior. Okay. Thank you.  Yeah. Okay. Maybe a quick question. You talked about  motivating people. So how can you incentivize participation in a talk peer to peer  network for mobile communication? I mean, you, you have to have your GPS on. You have to use  data all the time to relay calls or data. How do you incentivize and how do you make sure people  don't just make their own system? Like, you know, what do you call Uberization, I believe?  This, this is actually an interesting story. One of our own older projects with, with this  major telecommunication company, they're, they're a bit scared that something like Uber happens to  them in the sense that you have, for instance, only in the city, you have this company and they say,  well, we don't have infrastructure, but we all sign up for this contract and you can use your cell  phone so you can, other people can send their messages, messages via Uber. Actually, such apps  exist. For instance, Firejet played a role in the, in the protested in Hong Kong, where the  government shut down the infrastructure and then via this app, you can, you can send messages.  So for them, really, it's a bit of a threat that all of a sudden this company comes up  and they, they basically exist via an app and they, they, they, they provide good network.  So people will sign up for them and leave there. There are billions of worse of infrastructure,  it's useless. So, so they're actually interested in what is the critical number of people you have to,  you have to have in your business. So this actually works. You can have long range connectivity,  for instance. So the attraction comes from low cost, I guess, essentially.  Yeah, also a bit in the direction of the two questions. If I understand correctly, you try to  avoid, of course, percolation transitions. And if you speak on good or bad messages, I mean,  percolation is good if you want to have connectivity. Okay, good. And then you can give  conditions for the companies whether to avoid or to, to reach. For instance, if it's a one parameter  model, where the only parameter is the expected number of devices per unit square kilometer,  then I can rigorously, but also by simulations, tell them there's a critical point where the system  jumps from only being local to being global. Okay, okay. Okay, that's interesting. Yeah. Yep, last question.  I'm wondering about bandwidth problems, like I'm really having the mobile phone users in the  back of my mind. So if I order a file from a central server, then I expect the server to have a  large bandwidth. But if I kind of take this route through all small devices, then maybe somebody  just doesn't have a good know them and then it will kind of stop. Yeah, that's true. Okay.  Cool. Then it's not a question. Was a statement or was it a question statement that I could  answer? Okay, but it's not a right question. Yeah, I don't I am not at the point where I can  maybe join lunch. Yeah, you should discuss. Yeah, thank you. Okay, then thank you very much.  We are really in the beginning of a discussion. What is the best situation to start the break?  Thank you very much. Okay.
07_Jahnel_0_827.png 07_Jahnel_828_3179.png 07_Jahnel_3180_7139.png 07_Jahnel_7140_8483.png 07_Jahnel_8484_15263.png 07_Jahnel_15264_19115.png 07_Jahnel_19116_21719.png 07_Jahnel_21720_21803.png 07_Jahnel_21804_27719.png 07_Jahnel_27720_30911.png 07_Jahnel_30912_33935.png 07_Jahnel_33936_35927.png 07_Jahnel_35928_37127.png 07_Jahnel_37128_37319.png 07_Jahnel_37320_37919.png 07_Jahnel_37920_38003.png 07_Jahnel_38004_38111.png 07_Jahnel_38112_38351.png 07_Jahnel_38352_39275.png 07_Jahnel_39276_39395.png 07_Jahnel_39396_39743.png 07_Jahnel_39744_39851.png 07_Jahnel_39852_39911.png 07_Jahnel_39912_40091.png 07_Jahnel_40092_40175.png 07_Jahnel_40176_40307.png 07_Jahnel_40308_40367.png 07_Jahnel_40368_40415.png 07_Jahnel_40416_40463.png 07_Jahnel_40464_40511.png 07_Jahnel_40512_40559.png 07_Jahnel_40560_40643.png 07_Jahnel_40644_40691.png 07_Jahnel_40692_40859.png 07_Jahnel_40860_40943.png 07_Jahnel_40944_41015.png 07_Jahnel_41016_41063.png 07_Jahnel_41064_41111.png 07_Jahnel_41112_41183.png 07_Jahnel_41184_41315.png 07_Jahnel_41316_47267.png 07_Jahnel_47268_48407.png 07_Jahnel_48408_48575.png 07_Jahnel_48576_48598.png
Thank you for the introduction. My name is Benedikt Gjarnel. I work at the Weierstraße Institute. It's a pleasure to be here and present some joint work within my group, DICOMnet. It's a small group at Vias in Berlin and we are all probabilists and we use stochastic geometry to model and analyze telecommunication networks. So I also realized that this is a diverse audience so I tried to essentially do my talk in three parts. Some motivation, some general ideas, but I don't want to spare you from some theorems at the end of my talk. So you have an idea of what we're actually doing every day. So the motivation, as I said, comes from telecommunication systems. I guess I don't have to emphasize this too much. There is a strong increasing demand for communications on all different levels. This is just some slides that I took from some picture that I took from a big telecommunication systems provider, Cisco, and they make predictions in their white papers and all these, they indicate that we have an explosion of data transmissions, an explosion of subscribers. There's the large field of Internet of Things where small units communicate with each other, machine to machine is a bus word in the field. So we have this just this tremendous increase in mobile communication, but also in communication of small devices like sensors. Exponential growth essentially is what we see right now and in the near future. And I want to address a very particular part of this development and how operators try to cope with these challenges. And the part that I want to speak about today are spatial networks. So not social networks in the Internet. These are spatial networks. And in particular, an aspect of spatial networks that is also already to a small extent implemented in the current 5G mobile standard, but it will be playing a more stronger role. Many people predict that it will be playing a stronger role even in the future. And this aspect is device to device communications. So as these simulated pictures should indicate, we have maybe an environment, some kind of edge system here, some kind of urban topology maybe, and we have devices that sit on in this environment. You're indicated by blue particles, blue points, and they're supposed to communicate in a peer-to-peer fashion directly with each other. And this should bring some key benefits to the systems that are listed here. It provides more robustness to the system because there's multiple paths that data can be transmitted between components of the networks. It can come with much lower costs. We can get rid of heavy infrastructure, billions worth of infrastructure, and our current days mobile phones have a lot of power themselves, for instance. Security issues come decentralized. This is a benefit, but this can also be a challenge. And we can get rid of a lot of latency if you think of peer-to-peer communications. We don't have to go to the backbone of the system. We don't have to go through base stations. The devices can communicate in a peer-to-peer fashion, makes it faster and more robust. Many more aspects that I have listed here, and of course challenges. One of the main challenges is that from an operator's perspective, you cannot control the system. It's an ATOX system. So the network components building this network, and you have to deal with the fact that this is somehow not controlled by some operator. It's actually hard to read from here. Applications. I think I've already addressed some of them. It already implemented our main sensor networks, where, for instance, in large solar fields, the different sensors communicate with each other. It makes no sense to build up some centralized base station system. The sensors communicate. They accumulate the data, and they bring it to some exit point. So you see this implemented already in a few situations in the real world, but we hope and we try to support by our research more applications in the new future. For instance, by creating networks in areas of the world where there is no infrastructure available yet. So our a priori belief is that we can only understand these systems theoretically by using probabilistic methods. As I try to convince you, there is an intrinsic randomness in the system, because we cannot control how the individual components behave in the system. So I want to use the lenses of a probabilist to model and analyze these systems. Randomness enters on various levels. There is an intrinsic randomness on the behavior of the components. From an operator's perspective, you have a lot of uncertainty how they behave. We have stochastic algorithms that are partially used, but can still be developed. And when we deal with randomness, then we can essentially look at the system from two sides. What is the expected behavior of the system? That's important from a design perspective. And something that is not yet developed so good, and we want to make contributions there. We do make contributions there is to also understand the system in its behavior, in its unexpected behavior, which is also critical. You want to understand situations where the system behaves really bad and understand why it behaves bad. This is typically also very hard to simulate if you have a model, because maybe these bad situations are very rare. So to make a statistics on the typical behavior within the rare event is difficult. We have some tools to tackle this problem. The undots we use is stochastic geometry. So that's maybe roughly 40, 50 years old field within spatial probability. And in essence, you can make a mapping between notions in stochastic geometry and the notions that come up when you speak with people from engineering in big telecommunication companies. So for instance, the device locations can be seen as point processes. So point processes would be the equivalent notion within the community of stochastic geometry for the locations of the devices or the components of the networks in space. For instance, we can deal with an environment. And in this picture, the environment is a street system. And this can be seen as a tessellation process in space, which is also a quite fundamental notion in stochastic geometry. Mobility is a big topic. How do the components move in space? And there's a large array of models within the probabilistic community that try to model this. For instance, the random waypoint model, which is it's different from a random walk, but it also has some similarities. There is medium axis. Basically, what I try to convince you that there is a, you can, this is really the right framework. We can find the appropriate notions within the stochastic geometry world for a variety of, for the models. What we want to do, and this is our domain, we want to make rigorous analysis. So by solid mathematical theorems and deductions. So this should enable us to make predictions for the future by understanding these models. With a particular interest in the critical behavior. So do we see maybe faith transitions? Do we see situations where this model dramatically changes its behavior when you just vary a certain parameter a little bit? So discontinued newities with respect to the parameter space. And many of these instances are also supported by simulation. So I present to you now three different applications and associated methods that we use. And the first application is connectivity in mobile urban networks. And just stick with the picture. Maybe there is, there is a city topology, maybe some street system. And we have mobile devices in blue. And we have some base, additional base station infrastructure. And now we want to augment the base stations or the system by the possibility to have intermediate peer to peer connections. So the green base stations, they come within certain interaction radius where they can directly be connected with devices. But there are lighter green extensions of these areas that by adding one, two, three additional peer to peer hops. And the base stations, they don't move in space, but the devices they move maybe along the streets, maybe away from the streets. And, and a characteristic of the network that we want to analyze, I put here in the display in mass symbols, don't worry too much. Maybe later I will be more precise with the notation. One, one important quantity is the connection time. So how much time does a typical device connect to an infrastructure using at most k hops. So if you build such a network, this is very important to understand the quality of the network, how much time do you actually spend connected to the infrastructure by at most k hops. Second, maybe how often do you have to reconnect to new base stations, all these quantities play a major role in the design of these networks. Let me also say that in many instances, our analysis will be performed in some, some kind of limiting regime. It's, it's rather hard to come up with, with vigorous results. If you fix parameters, so we want to understand essentially the edges and corners of the parameter space by, for instance, letting the number of hops become large, letting time become large. So the actual analysis is typically performed in some kind of limiting regime. Another application is data routing. So here is an, here's an, the image or the picture is as follows. One has a central base station and a number of randomly placed devices around the base station, base stations in the center. And now the idea is that every device has a certain algorithm that tells them, okay, I want to connect to the base station, the center, but I cannot go there directly. So I use I look for instance for the device that is closest to me in the direction of the base station and therefore creating a tree where messages has through via other devices towards the base station. And then for instance, the question can be how is the data, how much data does actually a point close by the, by the base station has to handle. So how much data is actually passed through nodes that are close to the base station. And also here, the system can be analyzed in its expected behavior. But what is even more interesting is to understand what are the typical configurations of the system. So for instance, a large proportion of devices cannot connect to the base station because maybe some, there's too much throughput in a certain region of space. So this is, we also do again, an analysis with respect to the expected behavior, but also the unexpected behavior. This is, this is very important. Third application is malware propagation. And here, the idea is that there is wanted data in the system, there's unwanted data. And this is a decentralized situation. So, so there is maybe a malware at the, at some typical node at the center of the network. And how does it propagate through the network? Obviously, this is, this is very much connected maybe to two models for epidemic spreading. So, so I think we make there quite a strong connection towards spatial epidemic modeling. So here you see a picture where you have susceptible devices in space, they're connected whenever they're sufficiently close to each other. And we have, we have an initial malware at the center, it's spread through the edges step by step in continuous time in this case. And, and red infected devices appear. But here comes a speciality of, of our model, which we designed together with people from, from orange, by the way, I should mention this, we have a major collaboration with, with industry. So just to make sure that this is not purely academic. So there is, there is, there is regular contracts. I think we have seven contracts already, where we analyze and simulate these systems for them. So, so there is, there is some, some grounding there. Anyways, the speciality is, is now, so how can you, how can you counteract this, this, this spreading of a malware over, of an infection through the system. And what they imagine, and already in part have, have, have prototyped is something that they call a white knight, which is, which is also a device in space. It's, it's essentially indistinguishable from, from the regular devices. And it can patch when, when it can patch an infected device whenever it is attacked. The problem here seems to be that you cannot simply patch any device because of legal reasons you have to wait until you're attacked. And only then you can patch. So this, this is, this makes the system actually quite complicated, highly non-monotonic. And what you see in the picture here is a later point in time. Initially, there is an infection at the origin. There are white knights in green in space sparsely. And then the infection grows. And whenever an infection meets a white knight, it can retaliate. And therefore, on the set of in set, it's just infected devices patch and they become green. So you see a kind of a chase escape dynamics here. The infection has a slight advantage because it can go to all the infected, the susceptible, susceptible devices, but the white knights has, they have to chase because they can only use the infected devices. So this is, this is an analysis that we do. And let me also say that, that one of the main inputs here is that we deal with a random network of positions. So many of these things have been done actually quite, quite recently on, on non-random, let's say grids or non-random topologies, geometries. In our case, we always deal with these random, random point configurations in space. So these are three applications. Now, let, let me address briefly methods that we use from stochastic geometry. One key method is called continuum percolation. This is closely linked by the way to statistical physics. And, and this is one of the, one of the key concepts that we explore and use. And it goes like this, for instance, in this picture, you have some kind of environment in green where you see, you see a certain intensity, density of points in blue, and they all come with, with a certain interaction radius that might be device dependent. So we can model this also by, by assigning individual IID areas where, where people can, where, where the devices can, can form connections. And then the question in, in continuum percolation is if you now tune the intensity of the devices, the expected number of devices you see in a unit volume, does when, if you tune this up, so you see more and more devices in expectation, is there a point where the system dramatically change from a locally, a local behavior where you see only local clusters of connected devices to a global situation where all of a sudden with a positive probability, a typical device can actually connect to infinity. For this is, for us, this is a, this is a rough estimate of a, of a breaking point of the system where it jumps from a purely local behavior to an global behavior. So, so for us, this, this is, is, is some kind of a, a first indication, how many people do you actually have to attract to such a system, have to be part of the system. So you see long range, the possibility of long range connectivity. The second method that is quite underdeveloped in the field is the, is the method of large deviations. So as I mentioned, briefly large deviations is a, is a, is a highly developed machinery methodology within Socastics in general to deal with rare events. And the rare event is, for instance, this, this random variable Xn, it wants to be, it has an expected value where, where, where it wants to be, there's always a running parameter and I will maybe later explain more. And we, we are interested in what is actually the probability that the system behaves away from what it wants to be, where, from, from its expected behavior. And in many instances, if you have enough independence in the system, this goes exponentially. And there is a certain speed, rate function I, and this rate function can be used to understand the typical behavior within the atypical event. And, and this, this can be, for instance, employed to, to, to build an important sampling algorithm. For instance, in this picture, we were interested, okay, here, here, all these, these many, many devices, they want to connect to the base session, there's interference. So the problem is, there is a cocktail party problem. If, if a person wants to speak to another person, but there are many, many people around, it makes it very hard to form this connection. And we're interested in what is the typical behavior of the system in a situation where you see a certain atypical number of this disconnected devices. And in this case, you even see a face, a symmetry breaking phase transition. So it's also from a purely mathematical point of view, an interesting theory and interesting applications for, for large deviations within the realm of space-time point processes. The third method is interacting particle systems on random graphs. This is a quite old theory where you look at Markov processes, or maybe beyond Markov processes on fixed configurations. You have a configuration of, of, of, of particles, let's say on a grid, they come in different local states. And now we, we, we devise exponential rates, interacting, depending on the behavior of a device. And one point depends on the neighbors, for instance. And then we look at the, at the long-time behavior of these Markov processes. And for instance, in the, in the case that I gave you with the white nights and the, and the infection spreading, we can, we can, we can model this in the, in the framework of interacting particle systems, but now on random graphs. And in some cases, for instance, derive face diagrams of global extinction and global survival rigorously in the framework of interacting particle systems. How much time do I have? When did I start? Oh, two minutes. Okay. So I was a bit slow, but that's okay. Just, okay, maybe I actually, two minutes, including questions. Okay. Just, just to give you a flavor, quick flavor of how we, how we deal with these things. So we start with a random process, so catholic process. Typically, if we have no statistical knowledge, it's a good idea to start with a so-called Poisson point process. This, you should essentially think of a uniform distribution of points in space. We can create a graph G, by just connecting two points that are sufficiently close to each other. And there is a famous phase transition of continuum percolation, where I see an infinite cluster emerging. And then we can look at such a Voronoi tessellation. These are all the areas of points that are connected to the infinite cluster. And on the second level, now, now introduce a time process, where we associate to every edge a certain passing time. And then we can look at the, the set of, of infected devices, when we initially have an infection at the origin, we can look at all the devices for which there is a pass, where the passing times, the sum of the passing times does not exceed a certain time t. So this is the set of infected points at time t. And this is a, this is a recent result, where we, which is called a shape theorem. So I will also move. We're interested in the limiting shape of the infected region. This is the infected region, we scale at time t, we scale by t. And what is indeed true, if you put some moment conditions and, and zero conditions here on the passing times. So that we see a limiting shape, which is in fact the ball, where the parameter phi is speed. So here's a picture. Between two time points, there is t1, you see a certain region infected and at some later time point, you see additional infections in the, in the red area. Yeah, some, some rescaling results, maybe I'll skip this. Thank you for your attention. Thank you very much for this really interesting talk questions. Let me start. You always mentioned random networks. For my little experience, I never saw in reality a random network. So you have all some structures. Do you mean Erdisch-Renie or also other kinds of random networks? Well, here the randomness, I mean, the simplest case, the randomness enters, enters via the position of the devices in space, the points. And randomness here is, is I mean, we can have a philosophical question about what is randomness, but enters from an operator's perspective, I have no control on where, where, where you move. I mean, I cannot control where you move in space. So I have maybe some knowledge that you have a typical path to your office. But, but if you look at from a global perspective, I have no clue what you're doing. Okay. Then, then this is a different perspective. I call this topology, whether it's randomness or not. And this is a bit different in your point. I think, I think it has some structures in topology. Yes. Yes. But okay, good, good. Then it's clear. Thank you. Further questions? Hi. So if you have such an attack network, how do you then solve the network routing problems? Because I guess you cannot solve them in a centralized way. Who passes messages to women on which path? Oh, we don't solve them. We, we, we make, for instance, we evaluate expected behavior. I see. Okay. So the, the routing takes a random path to the, it's a random path depending on the configuration of the networks. It's a random configuration. We can say something about the past in this configuration. And then we have a probability distribution and we can make analysis of, for instance, the expected behavior, the unexpected behavior. Okay. Thank you. Yeah. Okay. Maybe a quick question. You talked about motivating people. So how can you incentivize participation in a talk peer to peer network for mobile communication? I mean, you, you have to have your GPS on. You have to use data all the time to relay calls or data. How do you incentivize and how do you make sure people don't just make their own system? Like, you know, what do you call Uberization, I believe? This, this is actually an interesting story. One of our own older projects with, with this major telecommunication company, they're, they're a bit scared that something like Uber happens to them in the sense that you have, for instance, only in the city, you have this company and they say, well, we don't have infrastructure, but we all sign up for this contract and you can use your cell phone so you can, other people can send their messages, messages via Uber. Actually, such apps exist. For instance, Firejet played a role in the, in the protested in Hong Kong, where the government shut down the infrastructure and then via this app, you can, you can send messages. So for them, really, it's a bit of a threat that all of a sudden this company comes up and they, they basically exist via an app and they, they, they, they provide good network. So people will sign up for them and leave there. There are billions of worse of infrastructure, it's useless. So, so they're actually interested in what is the critical number of people you have to, you have to have in your business. So this actually works. You can have long range connectivity, for instance. So the attraction comes from low cost, I guess, essentially. Yeah, also a bit in the direction of the two questions. If I understand correctly, you try to avoid, of course, percolation transitions. And if you speak on good or bad messages, I mean, percolation is good if you want to have connectivity. Okay, good. And then you can give conditions for the companies whether to avoid or to, to reach. For instance, if it's a one parameter model, where the only parameter is the expected number of devices per unit square kilometer, then I can rigorously, but also by simulations, tell them there's a critical point where the system jumps from only being local to being global. Okay, okay. Okay, that's interesting. Yeah. Yep, last question. I'm wondering about bandwidth problems, like I'm really having the mobile phone users in the back of my mind. So if I order a file from a central server, then I expect the server to have a large bandwidth. But if I kind of take this route through all small devices, then maybe somebody just doesn't have a good know them and then it will kind of stop. Yeah, that's true. Okay. Cool. Then it's not a question. Was a statement or was it a question statement that I could answer? Okay, but it's not a right question. Yeah, I don't I am not at the point where I can maybe join lunch. Yeah, you should discuss. Yeah, thank you. Okay, then thank you very much. We are really in the beginning of a discussion. What is the best situation to start the break? Thank you very much. Okay.
probabilists and we use stochastic geometry to an ATOX system. So the network components addressed some of them. It already implemented What is the expected behavior of the system? stochastic geometry for the locations of the devices or the components of the networks in And the first application is connectivity in quantities play a major role in the design of is very important. Third application is malware you, how can you counteract this, this, this such a system, have to be part of the system. So with a so-called Poisson point process. This, you no control on where, where, where you move. I