One promising way to cope with the new and more complex structures that arise is to exploit probabilistic methods. Indeed, fundamental ansatzes from stochastic geometry (e.g., spatial Poisson processes, continuum percolation theory, ...) are widely used for modelling the spatial locations of the users, the relays and the base stations and their basic connectivity properties. For the description of temporal developments, standard methods from stochastic processes (stochastic interacting particle processes like bootstrap percolation or the contact process) are commonly used to model the spread of large amounts of data through a network.
Contribution of the Institute
The WIAS has performed mathematical research on connectivity and capacity problems in mobile relayaugmented probabilistic models over a period of four years within the Leibniz Group "Probabilistic methods for mobile adhoc networks" together with the LeibnizInstitute "Innovations for High Performance Microelectronics" (IHP) and in other collaborations. Its expertise includes dynamic modelling of message propagation in dense networks, bottleneck behaviour in DevicetoDevice (D2D) systems, connection times in large networks without infrastructure, wifiaugmented mobil urban communications models and interferencebased spatial analysis of message trajectories as stochastic processes.A particular focus is on the simulation and extraction of relevant data, the impact of mobility on key network characteristics, and the possible consequences of delays and buffering of data transfers.
Highlights
Highlights of recent years include several industry collaborations with a major European telecommunications company aiming to better understand large D2D networks on realistic street systems. For example, we were able to demonstrate critical behavior in relation to longrange connections, also called percolation.Another highlight was the theoretical and numerical investigation of the typical communication cell in random cellular networks based on nonrotationally invariant road systems.
Publications
Articles in Refereed Journals

CH. Hirsch, B. Jahnel, Large deviations for the capacity in dynamic spatial relay networks, Markov Processes and Related Fields, 25 (2019), pp. 3373.
Abstract
We derive a large deviation principle for the spacetime evolution of users in a relay network that are unable to connect due to capacity constraints. The users are distributed according to a Poisson point process with increasing intensity in a bounded domain, whereas the relays are positioned deterministically with given limiting density. The preceding work on capacity for relay networks by the authors describes the highly simplified setting where users can only enter but not leave the system. In the present manuscript we study the more realistic situation where users leave the system after a random transmission time. For this we extend the point process techniques developed in the preceding work thereby showing that they are not limited to settings with strong monotonicity properties. 
W. König, A. Tóbiás, A Gibbsian model for message routeing in highly dense multihop networks, ALEA. Latin American Journal of Probability and Mathematical Statistics, 16 (2019), pp. 211258, DOI DOI: 10.30757/ALEA.v1608 .
Abstract
We investigate a probabilistic model for routing in relayaugmented multihop adhoc communication networks, where each user sends one message to the base station. Given the (random) user locations, we weigh the family of random, uniformly distributed message trajectories by an exponential probability weight, favouring trajectories with low interference (measured in terms of signaltointerference ratio) and trajectory families with little congestion (measured by how many pairs of hops use the same relay). Under the resulting Gibbs measure, the system targets the best compromise between entropy, interference and congestion for a common welfare, instead of a selfish optimization. We describe the joint routing strategy in terms of the empirical measure of all message trajectories. In the limit of high spatial density of users, we derive the limiting free energy and analyze the optimal strategy, given as the minimizer(s) of a characteristic variational formula. Interestingly, expressing the congestion term requires introducing an additional empirical measure. 
W. König, A. Tóbiás, Routeing properties in a Gibbsian model for highly dense multihop networks, IEEE Transactions on Information Theory, 65 (2019), pp. 68756897, DOI 10.1109/TIT.2019.2924187 .
Abstract
We investigate a probabilistic model for routeing in a multihop adhoc communication network, where each user sends a message to the base station. Messages travel in hops via the other users, used as relays. Their trajectories are chosen at random according to a Gibbs distribution that favours trajectories with low interference, measured in terms of sum of the signaltointerference ratios for all the hops, and collections of trajectories with little total congestion, measured in terms of the number of pairs of hops arriving at each relay. This model was introduced in our earlier paper [KT17], where we expressed, in the highdensity limit, the distribution of the optimal trajectories as the minimizer of a characteristic variational formula. In the present work, in the special case in which congestion is not penalized, we derive qualitative properties of this minimizer. We encounter and quantify emerging typical pictures in analytic terms in three extreme regimes. We analyze the typical number of hops and the typical length of a hop, and the deviation of the trajectory from the straight line in two regimes, (1) in the limit of a large communication area and large distances, and (2) in the limit of a strong interference weight. In both regimes, the typical trajectory turns out to quickly approach a straight line, in regime (1) with equallysized hops. Surprisingly, in regime (1), the typical length of a hop diverges logarithmically as the distance of the transmitter to the base station diverges. We further analyze the local and global repulsive effect of (3) a densely populated area on the trajectories. Our findings are illustrated by numerical examples. We also discuss a gametheoretic relation of our Gibbsian model with a joint optimization of message trajectories opposite to a selfish optimization, in case congestion is also penalized 
CH. Hirsch, B. Jahnel, E. Cali, Continuum percolation for Cox point processes, Stochastic Processes and their Applications, 366 (2019), pp. 139172 (published online on 20.11.2018), DOI 10.1016/j.spa.2018.11.002 .
Abstract
We investigate continuum percolation for Cox point processes, that is, Poisson point processes driven by random intensity measures. First, we derive sufficient conditions for the existence of nontrivial sub and supercritical percolation regimes based on the notion of stabilization. Second, we give asymptotic expressions for the percolation probability in largeradius, highdensity and coupled regimes. In some regimes, we find universality, whereas in others, a sensitive dependence on the underlying random intensity measure survives. 
CH. Hirsch, B. Jahnel, R.I.A. Patterson, Spacetime large deviations in capacityconstrained relay networks, ALEA. Latin American Journal of Probability and Mathematical Statistics, 15 (2018), pp. 587615, DOI 10.30757/ALEA.v1524 .
Abstract
We consider a singlecell network of random transmitters and fixed relays in a bounded domain of Euclidean space. The transmitters arrive over time and select one relay according to a spatially inhomogeneous preference kernel. Once a transmitter is connected to a relay, the connection remains and the relay is occupied. If an occupied relay is selected by another transmitters with later arrival time, this transmitter becomes frustrated. We derive a large deviation principle for the spacetime evolution of frustrated transmitters in the highdensity regime. 
CH. Hirsch, B. Jahnel, P. Keeler, R.I.A. Patterson, Large deviations in relayaugmented wireless networks, Queueing Systems. Theory and Applications, 88 (2018), pp. 349387 (published online on 28.10.2017).
Abstract
We analyze a model of relayaugmented cellular wireless networks. The network users, who move according to a general mobility model based on a Poisson point process of continuous trajectories in a bounded domain, try to communicate with a base station located at the origin. Messages can be sent either directly or indirectly by relaying over a second user. We show that in a scenario of an increasing number of users, the probability that an atypically high number of users experiences bad quality of service over a certain amount of time, decays at an exponential speed. This speed is characterized via a constrained entropy minimization problem. Further, we provide simulation results indicating that solutions of this problem are potentially nonunique due to symmetry breaking. Also two general sources for bad quality of service can be detected, which we refer to as isolation and screening. 
CH. Hirsch, B. Jahnel, P. Keeler, R.I.A. Patterson, Traffic flow densities in large transport networks, Advances in Applied Probability, 49 (2017), pp. 10911115, DOI 10.1017/apr.2017.35 .
Abstract
We consider transport networks with nodes scattered at random in a large domain. At certain local rates, the nodes generate traffic flowing according to some navigation scheme in a given direction. In the thermodynamic limit of a growing domain, we present an asymptotic formula expressing the local traffic flow density at any given location in the domain in terms of three fundamental characteristics of the underlying network: the spatial intensity of the nodes together with their traffic generation rates, and of the links induced by the navigation. This formula holds for a general class of navigations satisfying a linkdensity and a subballisticity condition. As a specific example, we verify these conditions for navigations arising from a directed spanning tree on a Poisson point process with inhomogeneous intensity function. 
CH. Hirsch, B. Jahnel, P. Keeler, R.I.A. Patterson, Largedeviation principles for connectable receivers in wireless networks, Advances in Applied Probability, 48 (2016), pp. 10611094.
Abstract
We study largedeviation principles for a model of wireless networks consisting of Poisson point processes of transmitters and receivers, respectively. To each transmitter we associate a family of connectable receivers whose signaltointerferenceandnoise ratio is larger than a certain connectivity threshold. First, we show a largedeviation principle for the empirical measure of connectable receivers associated with transmitters in large boxes. Second, making use of the observation that the receivers connectable to the origin form a Cox point process, we derive a largedeviation principle for the rescaled process of these receivers as the connection threshold tends to zero. Finally, we show how these results can be used to develop importancesampling algorithms that substantially reduce the variance for the estimation of probabilities of certain rare events such as users being unable to connect. 
P. Keeler, N. Ross, A. Xia, B. Błaszczyszyn, Stronger wireless signals appear more Poisson, IEEE Wireless Communications Letters, 5 (2016), pp. 572575.
Abstract
Keeler, Ross and Xia [1] recently derived approximation and convergence results, which imply that the point process formed from the signal strengths received by an observer in a wireless network under a general statistical propagation model can be modelled by an inhomogeneous Poisson point process on the positive real line. The basic requirement for the results to apply is that there must be a large number of transmitters with different locations and random propagation effects. The aim of this note is to apply some of the main results of [1] in a less general but more easily applicable form to illustrate how the results can be applied in practice. New results are derived that show that it is the strongest signals, after being weakened by random propagation effects, that behave like a Poisson process, which supports recent experimental work.
[1] P. Keeler, N. Ross, and A. Xia:“When do wireless network signals appear Poisson?? ” 
H. Döring, G. Faraud, W. König, Connection times in large adhoc mobile networks, Bernoulli. Official Journal of the Bernoulli Society for Mathematical Statistics and Probability, 22 (2016), pp. 21432176.
Abstract
We study connectivity properties in a probabilistic model for a large mobile adhoc network. We consider a large number of participants of the system moving randomly, independently and identically distributed in a large domain, with a spacedependent population density of finite, positive order and with a fixed time horizon. Messages are instantly transmitted according to a relay principle, i.e., they are iteratedly forwarded from participant to participant over distances $leq 2R$, with $2R$ the communication radius, until they reach the recipient. In mathematical terms, this is a dynamic continuum percolation model. We consider the connection time of two sample participants, the amount of time over which these two are connected with each other. In the above thermodynamic limit, we find that the connectivity induced by the system can be described in terms of the counterplay of a local, random, and a global, deterministic mechanism, and we give a formula for the limiting behaviour. A prime example of the movement schemes that we consider is the wellknown random waypoint model (RWP). Here we describe the decay rate, in the limit of large time horizons, of the probability that the portion of the connection time is less than the expectation. 
P. Keeler, P.G. Taylor, Discussion on ``On the Laplace transform of the aggregate discounted claims with Markovian arrivals'' by Jiandong Ren, Volume 12 (2), North American Actuarial Journal, 19 (2015), pp. 7377.

B. Blaszczyszyn, P. Keeler, Studying the SINR process of the typical user in Poisson networks by using its factorial moment measures, IEEE Transactions on Information Theory, 61 (2015), pp. 67746794.

B. Blaszczyszyn, M. Karray, P. Keeler, Wireless networks appear Poissonian due to strong shadowing, IEEE Transactions on Wireless Communications, 14 (2015), pp. 43794390.
Contributions to Collected Editions

B. Jahnel, W. König, Probabilistic methods for spatial multihop communication systems, in: Topics in Applied Analysis and Optimisation, M. Hintermüller, J.F. Rodrigues, eds., CIM Series in Mathematical Sciences, Springer, Springer International Publishing, 2019, pp. 239268.

P. Keeler, B. Jahnel, O. Maye, D. Aschenbach, M. Brzozowski, Disruptive events in highdensity cellular networks, in: 2018 16th International Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks (WiOpt), IEEE Xplore digital library, 2018, pp. 17789136/117789136/8, DOI 10.23919/WIOPT.2018.8362867 .
Abstract
Stochastic geometry models are used to study wireless networks, particularly cellular phone networks, but most of the research focuses on the typical user, often ignoring atypical events, which can be highly disruptive and of interest to network operators. We examine atypical events when a unexpected large proportion of users are disconnected or connected by proposing a hybrid approach based on ray launching simulation and point process theory. This work is motivated by recent results [12] using large deviations theory applied to the signaltointerference ratio. This theory provides a tool for the stochastic analysis of atypical but disruptive events, particularly when the density of transmitters is high. For a section of a European city, we introduce a new stochastic model of a single network cell that uses ray launching data generated with the open source RaLaNS package, giving deterministic path loss values. We collect statistics on the fraction of (dis)connected users in the uplink, and observe that the probability of an unexpected large proportion of disconnected users decreases exponentially when the transmitter density increases. This observation implies that denser networks become more stable in the sense that the probability of the fraction of (dis)connected users deviating from its mean, is exponentially small. We also empirically obtain and illustrate the density of users for network configurations in the disruptive event, which highlights the fact that such bottleneck behaviour not only stems from too many users at the cell boundary, but also from the nearfar effect of many users in the immediate vicinity of the base station. We discuss the implications of these findings and outline possible future research directions.
Preprints, Reports, Technical Reports

B. Jahnel, A. Tóbiás, E. Cali, Phase transitions for the Boolean model of continuum percolation for Cox point processes, Preprint no. 2704, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2704 .
Abstract, PDF (389 kByte)
We consider the Boolean model with random radii based on Cox point processes. Under a condition of stabilization for the random environment, we establish existence and nonexistence of subcritical regimes for the size of the cluster at the origin in terms of volume, diameter and number of points. Further, we prove uniqueness of the infinite cluster for sufficiently connected environments. 
A. Hinsen, B. Jahnel, E. Cali, J.P. Wary, Malware propagation in urban D2D networks, Preprint no. 2674, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2674 .
Abstract, PDF (3133 kByte)
We introduce and analyze models for the propagation of malware in pure D2D networks given via stationary CoxGilbert graphs. Here, the devices form a Poisson point process with random intensity measure λ, Λ where Λ is stationary and given, for example, by the edgelength measure of a realization of a PoissonVoronoi tessellation that represents an urban street system. We assume that, at initial time, a typical device at the center of the network carries a malware and starts to infect neighboring devices after random waiting times. Here we focus on Markovian models, where the waiting times are exponential random variables, and nonMarkovian models, where the waiting times feature strictly positive minimal and finite maximal waiting times. We present numerical results for the speed of propagation depending on the system parameters. In a second step, we introduce and analyze a counter measure for the malware propagation given by special devices called white knights, which have the ability, once attacked, to eliminate the malware from infected devices and turn them into white knights. Based on simulations, we isolate parameter regimes in which the malware survives or is eliminated, both in the Markovian and nonMarkovian setting. 
B. Jahnel, A. Tóbiás, SINR percolation for Cox point processes with random powers, Preprint no. 2659, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2659 .
Abstract, PDF (356 kByte)
Signaltointerference plus noise ratio (SINR) percolation is an infiniterange dependent variant of continuum percolation modeling connections in a telecommunication network. Unlike in earlier works, in the present paper the transmitted signal powers of the devices of the network are assumed random, i.i.d. and possibly unbounded. Additionally, we assume that the devices form a stationary Cox point process, i.e., a Poisson point process with stationary random intensity measure, in two or higher dimensions. We present the following main results. First, under suitable moment conditions on the signal powers and the intensity measure, there is percolation in the SINR graph given that the device density is high and interferences are sufficiently reduced, but not vanishing. Second, if the interference cancellation factor γ and the SINR threshold τ satisfy γ ≥ 1/(2τ), then there is no percolation for any intensity parameter. Third, in the case of a Poisson point process with constant powers, for any intensity parameter that is supercritical for the underlying Gilbert graph, the SINR graph also percolates with some small but positive interference cancellation factor. 
A. Hinsen, B. Jahnel, E. Cali, J.P. Wary, Phase transitions for chaseescape models on Gilbert graphs, Preprint no. 2642, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2642 .
Abstract, PDF (219 kByte)
We present results on phase transitions of local and global survival in a twospecies model on Gilbert graphs. At initial time there is an infection at the origin that propagates on the Gilbert graph according to a continuoustime nearestneighbor interacting particle system. The Gilbert graph consists of susceptible nodes and nodes of a second type, which we call white knights. The infection can spread on susceptible nodes without restriction. If the infection reaches a white knight, this white knight starts to spread on the set of infected nodes according to the same mechanism, with a potentially different rate, giving rise to a competition of chase and escape. We show welldefinedness of the model, isolate regimes of global survival and extinction of the infection and present estimates on local survival. The proofs rest on comparisons to the process on trees, percolation arguments and finitedegree approximations of the underlying random graphs. 
CH. Hirsch, B. Jahnel, A. Tóbiás, Lower large deviations for geometric functionals, Preprint no. 2632, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2632 .
Abstract, PDF (1268 kByte)
This work develops a methodology for analyzing largedeviation lower tails associated with geometric functionals computed on a homogeneous Poisson point process. The technique applies to characteristics expressed in terms of stabilizing score functions exhibiting suitable monotonicity properties. We apply our results to clique counts in the random geometric graph, intrinsic volumes of PoissonVoronoi cells, as well as powerweighted edge lengths in the random geometric, κnearest neighbor and relative neighborhood graph. 
A. Tóbiás, B. Jahnel, Exponential moments for planar tessellations, Preprint no. 2572, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2572 .
Abstract, PDF (276 kByte)
In this paper we show existence of all exponential moments for the total edge length in a unit disc for a family of planar tessellations based on Poisson point processes. Apart from classical such tessellations like the PoissonVoronoi, PoissonDelaunay and Poisson line tessellation, we also treat the JohnsonMehl tessellation, Manhattan grids, nested versions and Palm versions. As part of our proofs, for some planar tessellations, we also derive existence of exponential moments for the number of cells and the number of edges intersecting the unit disk. 
CH. Hirsch, B. Jahnel, A. Hinsen, E. Cali, The typical cell in anisotropic tessellations, Preprint no. 2557, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2557 .
Abstract, PDF (311 kByte)
The typical cell is a key concept for stochasticgeometry based modeling in communication networks, as it provides a rigorous framework for describing properties of a serving zone associated with a component selected at random in a large network. We consider a setting where network components are located on a large street network. While earlier investigations were restricted to street systems without preferred directions, in this paper we derive the distribution of the typical cell in Manhattantype systems characterized by a pattern of horizontal and vertical streets. We explain how the mathematical description can be turned into a simulation algorithm and provide numerical results uncovering novel effects when compared to classical isotropic networks.
Talks, Poster

A. Hinsen, Data mobility in adhoc networks: Vulnerability and security, KEIN öffentlicher Vortrag (Orange), Telecom Orange Paris, France, December 12, 2019.

A. Hinsen, IPS in telecommunication I, Workshop on Probability, Analysis and Applications (PAA), September 23  October 4, 2019, African Institute for Mathematical Sciences  Ghana (AIMS Ghana), Accra, Ghana, October 4, 2019.

A. Hinsen, IPS in telecommunication II, Workshop on Probability, Analysis and Applications (PAA), September 23  October 4, 2019, African Institute for Mathematical Sciences  Ghana (AIMS Ghana), Accra, Ghana, October 4, 2019.

A. Hinsen, Introduction to interacting particles systems (IPS), Workshop on Probability, Analysis and Applications (PAA), September 23  October 4, 2019, African Institute for Mathematical Sciences  Ghana (AIMS Ghana), Accra, Ghana, October 2, 2019.

A. Hinsen, The White Knight model  An epidemic on a spatial random network, Bocconi Summer School in Advanced Statistics and Probability, Lake Como School of Advanced Studies, Lake Como, Italy, July 8  19, 2019.

B. Jahnel, Continuum percolation in random environment I  IV, Workshop on Probability, Analysis and Applications (PAA), September 23  October 4, 2019, African Institute for Mathematical Sciences  Ghana (AIMS Ghana), Accra, Ghana.

A. Hinsen, Random Malware Propagation, MATH+ Center Days 2018, October 31  November 2, 2018, ZuseInstitut Berlin (ZIB), Berlin, October 31, 2018.

B. Jahnel, Telecommunication models in random environments, BIMoS Day : The Mathematics of Quantum Information, May 23, 2018, Technische Universität Berlin, Berlin, May 23, 2018.

W. König, Probabilistic Methods in Telecommunication, MATH+ Center Days 2018, October 31  November 2, 2018, ZuseInstitut Berlin (ZIB), Berlin, October 31, 2018.

A. Wapenhans, Data mobility in adhoc networks: Vulnerability & security, Telecom Orange Paris, France, November 17, 2017.

B. Jahnel, Continuum percolation for Cox processes, Seminar, Ruhr Universität Bochum, Fakultät für Mathematik, October 27, 2017.

B. Jahnel, Continuum percolation theory applied to Device to Device, Telecom Orange Paris, France, November 17, 2017.

B. Jahnel, Stochastic geometry in telecommunications, Summer School 2017: Probabilistic and Statistical Methods for Networks, August 21  September 1, 2017, Technische Universität Berlin, Berlin Mathematical School.

CH. Hirsch, Large deviations in relayaugmented wireless networks, Workshop on Dynamical Networks and Network Dynamics, January 17  22, 2016, International Centre for Mathematical Science, Edinburgh, UK, January 18, 2016.

P. Keeler, Signaltointerference ratio in wireless communication networks, Workshop on Dynamical Networks and Network Dynamics, January 17  24, 2016, International Centre for Mathematical Science, Edinburgh, UK, January 18, 2016.

W. König, Connection times in large adhoc mobile networks, Workshop on Dynamical Networks and Network Dynamics, January 18  21, 2016, International Centre for Mathematical Science, Edinburgh, UK, January 18, 2016.

P. Keeler, Largedeviation theory and coverage in mobile phone networks, Seminar ``Applied Probability'', The University of Melbourne, Department of Mathematics and Statistics, Australia, August 17, 2015.

P. Keeler, The PoissonDirichlet process and coverage in mobile phone networks, Stochastic Processes and Special Functions Workshop, August 13  14, 2015, The University of Melbourne, Melbourne, Australia, August 14, 2015.

P. Keeler, When do wireless network signals appear Poisson?, Simons Conference on Networks and Stochastic Geometry, May 18  21, 2015, University of Texas, Austin, USA, May 20, 2015.

G. Faraud, Connection times in large adhoc networks, Ecole de Printemps ``Marches Aléatoires, Milieux Aléatoires, Renforcements'' (MEMEMO2), June 10  14, 2013, Aussois, France, June 13, 2013.