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 information 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 and wifiaugmented mobil urban communications models.Highlights
One of the highlights of the past years is an industry collaboration with a large European telecommunications company with the goal to better understand large scale D2D networks build on realistic street models.Publications
Articles in Refereed Journals

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, Institute of Electrical and Electronics Engineers. 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.
Preprints, Reports, Technical Reports

CH. Hirsch, B. Jahnel, R.I.A. Patterson, Spacetime large deviations in capacityconstrained relay networks, Preprint no. 2308, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2308 .
Abstract, PDF (311 kByte)
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. 
E. Bolthausen, A. Cipriani, N. Kurt, Exponential decay of covariances for the supercritical membrane model, Preprint no. 2301, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2301 .
Abstract, PDF (291 kByte)
We consider the membrane model, that is the centered Gaussian field on Z^{d} whose covariance matrix is given by the inverse of the discrete Bilaplacian. We impose a δpinning condition, giving a reward of strength ε for the field to be 0 at any site of the lattice. In this paper we prove that in dimensions d≥5 covariances of the pinned field decay at least exponentially, as opposed to the field without pinning, where the decay is polynomial. The proof is based on estimates for certain discrete weighted norms, a percolation argument and on a Bernoulli domination result. 
CH. Hirsch, B. Jahnel, P. Keeler, R.I.A. Patterson, Traffic flow densities in large transport networks, Preprint no. 2221, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2221 .
Abstract, PDF (476 kByte)
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, Large deviations in relayaugmented wireless networks, Preprint no. 2173, WIAS, Berlin, 2015.
Abstract, PDF (2647 kByte)
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.
Talks, Poster

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.