Publikationen

Artikel in Referierten Journalen

  • N. Engler, B. Jahnel, Ch. Külske, Gibbsianness of locally thinned random fields, Markov Processes and Related Fields, 28 (2022), pp. 185--214, DOI 10.48550/arXiv.2201.02651 .
    Abstract
    We consider the locally thinned Bernoulli field on ℤ d, which is the lattice version of the Type-I Matérn hardcore process in Euclidean space. It is given as the lattice field of occupation variables, obtained as image of an i.i.d. Bernoulli lattice field with occupation probability p, under the map which removes all particles with neighbors, while keeping the isolated particles. We prove that the thinned measure has a Gibbsian representation and provide control on its quasilocal dependence, both in the regime of small p, but also in the regime of large p, where the thinning transformation changes the Bernoulli measure drastically. Our methods rely on Dobrushin uniqueness criteria, disagreement percolation arguments [46], and cluster expansions

  • CH. Hirsch, B. Jahnel, S. Muirhead, Sharp phase transition for Cox percolation, Electronic Communications in Probability, 27 (2022), pp. 1--13, DOI 10.1214/22-ECP487 .
    Abstract
    We prove the sharpness of the percolation phase transition for a class of Cox percolation models, i.e., models of continuum percolation in a random environment. The key requirements are that the environment has a finite range of dependence and satisfies a local boundedness condition, however the FKG inequality need not hold. The proof combines the OSSS inequality with a coarse-graining construction.

  • S.K. Iyer, S.K. Jhawar, Phase transitions and percolation at criticality in enhanced random connection models, Mathematical Physics, Analysis and Geometry, 25 (2022), pp. 1--40, DOI 10.1007/s11040-021-09409-y .
    Abstract
    We study phase transition and percolation at criticality for three random graph models on the plane, viz., the homogeneous and inhomogeneous enhanced random connection models (RCM) and the Poisson stick model. These models are built on a homogeneous Poisson point process P?P? in R2R2 of intensity ?. In the homogeneous RCM, the vertices at x,y are connected with probability g( x ? y ), independent of everything else, where g:[0,?)?[0,1]g:[0,?)?[0,1] and ? is the Euclidean norm. In the inhomogeneous version of the model, points of P?P? are endowed with weights that are non-negative independent random variables with distribution P(W>w)=w??1[1,?)(w)P(W>w)=w??1[1,?)(w), ? >?0. Vertices located at x,y with weights Wx,Wy are connected with probability 1?exp(??WxWy x?y ?)1?exp?(??WxWy x?y ?), ?,? >?0, independent of all else. The graphs are enhanced by considering the edges of the graph as straight line segments starting and ending at points of P?P?. A path in the graph is a continuous curve that is a subset of the union of all these line segments. The Poisson stick model consists of line segments of independent random lengths and orientation with the mid point of each segment located at a distinct point of P?P?. Intersecting lines form a path in the graph. A graph is said to percolate if there is an infinite connected component or path. We derive conditions for the existence of a phase transition and show that there is no percolation at criticality.

  • B. Jahnel, Ch. Hirsch, E. Cali, Percolation and connection times in multi-scale dynamic networks, Stochastic Processes and their Applications, 151 (2022), pp. 490--518, DOI 10.1016/j.spa.2022.06.008 .
    Abstract
    We study the effects of mobility on two crucial characteristics in multi-scale dynamic networks: percolation and connection times. Our analysis provides insights into the question, to what extent long-time averages are well-approximated by the expected values of the corresponding quantities, i.e., the percolation and connection probabilities. In particular, we show that in multi-scale models, strong random effects may persist in the limit. Depending on the precise model choice, these may take the form of a spatial birth-death process or a Brownian motion. Despite the variety of structures that appear in the limit, we show that they can be tackled in a common framework with the potential to be applicable more generally in order to identify limits in dynamic spatial network models going beyond the examples considered in the present work.

  • B. Jahnel, A. Tóbiás, E. Cali, Phase transitions for the Boolean model of continuum percolation for Cox point processes, Brazilian Journal of Probability and Statistics, 3 (2022), pp. 20--44, DOI 10.1214/21-BJPS514 .
    Abstract
    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 non-existence 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.

  • B. Jahnel, A. Tóbiás, Absence of percolation in graphs based on stationary point processes with degrees bounded by two, Random Structures and Algorithms, 62 (2022), pp. 240--255, DOI 10.1002/rsa.21084 .
    Abstract
    We consider undirected graphs that arise as deterministic functions of stationary point processes such that each point has degree bounded by two. For a large class of point processes and edge-drawing rules, we show that the arising graph has no infinite connected component, almost surely. In particular, this extends our previous result for SINR graphs based on stabilizing Cox point processes and verifies the conjecture of Balister and Bollobás that the bidirectional $k$-nearest neighbor graph of a two-dimensional homogeneous Poisson point process does not percolate for k=2.

  • B. Jahnel, A. Tóbiás, SINR percolation for Cox point processes with random powers, Adv. Appl. Math., 54 (2022), pp. 227--253, DOI 10.1017/apr.2021.25 .
    Abstract
    Signal-to-interference plus noise ratio (SINR) percolation is an infinite-range 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.

  • S.K. Jhawar, S.K. Iyer, Poisson approximation and connectivity in a scale-free random connection model, Electronic Journal of Probability, 26 (2021), pp. 1--23, DOI 10.1214/21-EJP651 .
    Abstract
    We study an inhomogeneous random connection model in the connectivity regime. The vertex set of the graph is a homogeneous Poisson point process P s Ps of intensity s > 0 s>0 on the unit cube S = ( ? 1 2 , 1 2 ] d S=?12,12d, d ? 2 d?2 . Each vertex is endowed with an independent random weight distributed as W, where P ( W > w ) = w ? ? 1 [ 1 , ? ) ( w ) P(W>w)=w??1[1,?)(w), ? > 0 ?>0. Given the vertex set and the weights an edge exists between x , y ? P s x,y?Ps with probability ( 1 ? exp ( ? ? W x W y ( d ( x , y ) ? r ) ? ) ) , 1?exp??WxWyd(x,y)?r?, independent of everything else, where ? , ? > 0 ?,?>0, d ( ? , ? ) d(?,?) is the toroidal metric on S and r > 0 r>0 is a scaling parameter. We derive conditions on ? , ? ?,? such that under the scaling r s ( ? ) d = 1 c 0 s ( log s + ( k ? 1 ) log log s + ? + log ( ? ? k ! d ) ) , rs(?)d=1c0slogs+(k?1)loglogs+?+log??k!d, ? ? R ??R, the number of vertices of degree k converges in total variation distance to a Poisson random variable with mean e ? ? e?? as s ? ? s??, where c 0 c0 is an explicitly specified constant that depends on ? , ? , d ?,?,d and ? but not on k. In particular, for k = 0 k=0 we obtain the regime in which the number of isolated nodes stabilizes, a precursor to establishing a threshold for connectivity. We also derive a sufficient condition for the graph to be connected with high probability for large s. The Poisson approximation result is derived using the Stein?s method.

Beiträge zu Sammelwerken

  • Z. Benomar, Ch. Ghribi, E. Cali, A. Hinsen, B. Jahnel, Agent-based modeling and simulation for malware spreading in D2D networks, AAMAS '22: Proceedings of the 21st International Conference on Autonomous Agents and Multiagent Systems, Auckland, New Zealand, May 11 - 13, 2022, International Foundation for Autonomous Agents and Multiagent Systems Richland, SC, 2022, pp. 91--99.
    Abstract
    This paper presents a new multi-agent model for simulating malware propagation in device-to-device (D2D) 5G networks. This model allows to understand and analyze mobile malware-spreading dynamics in such highly dynamical networks. Additionally, we present a theoretical study to validate and benchmark our proposed approach for some basic scenarios that are less complicated to model mathematically and also to highlight the key parameters of the model. Our simulations identify critical thresholds for em no propagation and for em maximum malware propagation and make predictions on the malware-spread velocity as well as device-infection rates. To the best of our knowledge, this paper is the first study applying agent-based simulations for malware propagation in D2D.

  • CH. Ghribi, E. Cali, Ch. Hirsch , B. Jahnel, Agent-based simulations for coverage extensions in 5G networks and beyond, 25th Conference on Innovation in Clouds, Internet and Networks and Workshops (ICIN) (Hybrid Event), Paris, France, March 7 - 10, 2022, 2022, pp. 1--7, DOI 10.5555/3535850.3535862 .

Preprints, Reports, Technical Reports

  • A. Hinsen, B. Jahnel, E. Cali, J.-P. Wary, Chase-escape in dynamic device-to-device networks, Preprint no. 2969, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2969 .
    Abstract, PDF (1550 kByte)
    The present paper features results on global survival and extinction of an infection in a multi-layer network of mobile agents. Expanding on a model first presented in CHJW22, we consider an urban environment, represented by line-segments in the plane, in which agents move according to a random waypoint model based on a Poisson point process. Whenever two agents are at sufficiently close proximity for a sufficiently long time the infection can be transmitted and then propagates into the system according to the same rule starting from a typical device. Inspired by wireless network architectures, the network is additionally equipped with a second class of agents that is able to transmit a patch to neighboring infected agents that in turn can further distribute the patch, leading to a chase-escape dynamics. We give conditions for parameter configurations that guarantee existence and absence of global survival as well as an in-and-out of the survival regime, depending on the speed of the devices. We also provide complementary results for the setting in which the chase-escape dynamics is defined as an independent process on the connectivity graph. The proofs mainly rest on percolation arguments via discretization and multiscale analysis.

  • A. Hinsen, B. Jahnel, E. Cali, J.-P. Wary, Connectivity in mobile device-to-device networks in urban environments, Preprint no. 2952, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2952 .
    Abstract, PDF (606 kByte)
    In this article we setup a dynamic device-to-device communication system where devices, given as a Poisson point process, move in an environment, given by a street system of random planar-tessellation type, via a random-waypoint model. Every device independently picks a target location on the street system using a general waypoint kernel, and travels to the target along the shortest path on the streets with an individual velocity. Then, any pair of devices becomes connected whenever they are on the same street in sufficiently close proximity, for a sufficiently long time. After presenting some general properties of the multi-parameter system, we focus on an analysis of the clustering behavior of the random connectivity graph. In our main results we isolate regimes for the almost-sure absence of percolation if, for example, the device intensity is too small, or the connectivity time is too large. On the other hand, we exhibit parameter regimes of sufficiently large intensities of devices, under favorable choices of the other parameters, such that percolation is possible with positive probability. Most interestingly, we also show an in-and-out of percolation as the velocity increases. The rigorous analysis of the system mainly rests on comparison arguments with simplified models via spatial coarse graining and thinning approaches. Here we also make contact to geostatistical percolation models with infinite-range dependencies.

  • B. Jahnel, S.K. Jhawar, A.D. Vu, Continuum percolation in a nonstabilizing environment, Preprint no. 2943, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2943 .
    Abstract, PDF (2463 kByte)
    We prove nontrivial phase transitions for continuum percolation in a Boolean model based on a Cox point process with nonstabilizing directing measure. The directing measure, which can be seen as a stationary random environment for the classical Poisson--Boolean model, is given by a planar rectangular Poisson line process. This Manhattan grid type construction features long-range dependencies in the environment, leading to absence of a sharp phase transition for the associated Cox--Boolean model. Our proofs rest on discretization arguments and a comparison to percolation on randomly stretched lattices established in [MR2116736].

  • B. Jahnel, J. Köppl, Dynamical Gibbs variational principles for irreversible interacting particle systems with applications to attractor properties, Preprint no. 2935, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2935 .
    Abstract, PDF (355 kByte)
    We consider irreversible translation-invariant interacting particle systems on the d-dimensional cubic lattice with finite local state space, which admit at least one Gibbs measure as a time-stationary measure. Under some mild degeneracy conditions on the rates and the specification we prove, that zero relative entropy loss of a translation-invariant measure implies, that the measure is Gibbs w.r.t. the same specification as the time-stationary Gibbs measure. As an application, we obtain the attractor property for irreversible interacting particle systems, which says that any weak limit point of any trajectory of translation-invariant measures is a Gibbs measure w.r.t. the same specification as the time-stationary measure. This extends previously known results to fairly general irreversible interacting particle systems.

  • CH. Ghribi, E. Cali, Ch. Hirsch, B. Jahnel, Agent-based simulations for coverage extensions in 5G networks and beyond, Preprint no. 2920, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2920 .
    Abstract, PDF (1032 kByte)
    Device-to-device (D2D) communications is one of the key emerging technologies for the fifth generation (5G) networks and beyond. It enables direct communication between mobile users and thereby extends coverage for devices lacking direct access to the cellular infrastructure and hence enhances network capacity. D2D networks are complex, highly dynamic and will be strongly augmented by intelligence for decision making at both the edge and core of the network, which makes them particularly difficult to predict and analyze. Conventionally, D2D systems are evaluated, investigated and analyzed using analytical and probabilistic models (e.g., from stochastic geometry). However, applying classical simulation and analytical tools to such a complex system is often hard to track and inaccurate. In this paper, we present a modeling and simulation framework from the perspective of complex-systems science and exhibit an agent-based model for the simulation of D2D coverage extensions. We also present a theoretical study to benchmark our proposed approach for a basic scenario that is less complicated to model mathematically. Our simulation results show that we are indeed able to predict coverage extensions for multi-hop scenarios and quantify the effects of street-system characteristics and pedestrian mobility on the connection time of devices to the base station (BS). To our knowledge, this is the first study that applies agent-based simulations for coverage extensions in D2D.

  • Z. Benomar, Ch. Ghribi, E. Cali, A. Hinsen, B. Jahnel, Agent-based modeling and simulation for malware spreading in D2D networks, Preprint no. 2919, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2919 .
    Abstract, PDF (286 kByte)
    This paper presents a new multi-agent model for simulating malware propagation in device-to-device (D2D) 5G networks. This model allows to understand and analyze mobile malware-spreading dynamics in such highly dynamical networks. Additionally, we present a theoretical study to validate and benchmark our proposed approach for some basic scenarios that are less complicated to model mathematically and also to highlight the key parameters of the model. Our simulations identify critical thresholds for em no propagation and for em maximum malware propagation and make predictions on the malware-spread velocity as well as device-infection rates. To the best of our knowledge, this paper is the first study applying agent-based simulations for malware propagation in D2D.

  • O. Collin, B. Jahnel, W. König, The free energy of a box-version of the interacting Bose gas, Preprint no. 2914, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2914 .
    Abstract, PDF (1441 kByte)
    The interacting quantum Bose gas is a random ensemble of many Brownian bridges (cycles) of various lengths with interactions between any pair of legs of the cycles. It is one of the standard mathematical models in which a proof for the famous Bose--Einstein condensation phase transition is sought for. We introduce a simplified version of the model with an organisation of the particles in deterministic boxes instead of Brownian cycles as the marks of a reference Poisson point process (for simplicity, in Z d, instead of R d). We derive an explicit and interpretable variational formula in the thermodynamic limit for the limiting free energy of the canonical ensemble for any value of the particle density. This formula features all relevant physical quantities of the model, like the microscopic and the macroscopic particle densities, together with their mutual and self-energies and their entropies. The proof method comprises a two-step large-deviation approach for marked Poisson point processes and an explicit distinction into small and large marks. In the characteristic formula, each of the microscopic particles and the statistics of the macroscopic part of the configuration are seen explicitly; the latter receives the interpretation of the condensate. The formula enables us to prove a number of properties of the limiting free energy as a function of the particle density, like differentiability and explicit upper and lower bounds, and a qualitative picture below and above the critical threshold (if it is finite). This proves a modified saturation nature of the phase transition. However, we have not yet succeeded in proving the existence of this phase transition.

  • CH. Hirsch, B. Jahnel, E. Cali, Connection intervals in multi-scale dynamic networks, Preprint no. 2895, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2895 .
    Abstract, PDF (1634 kByte)
    We consider a hybrid spatial communication system in which mobile nodes can connect to static sinks in a bounded number of intermediate relaying hops. We describe the distribution of the connection intervals of a typical mobile node, i.e., the intervals of uninterrupted connection to the family of sinks. This is achieved in the limit of many hops, sparse sinks and growing time horizons. We identify three regimes illustrating that the limiting distribution depends sensitively on the scaling of the time horizon.

  • M. Heida, B. Jahnel, A.D. Vu, Stochastic homogenization on irregularly perforated domains, Preprint no. 2880, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2880 .
    Abstract, PDF (668 kByte)
    We study stochastic homogenization of a quasilinear parabolic PDE with nonlinear microscopic Robin conditions on a perforated domain. The focus of our work lies on the underlying geometry that does not allow standard homogenization techniques to be applied directly. Instead we prove homogenization on a regularized geometry and demonstrate afterwards that the form of the homogenized equation is independent from the regularization. Then we pass to the regularization limit to obtain the anticipated limit equation. Furthermore, we show that Boolean models of Poisson point processes are covered by our approach.

  • B. Jahnel, Ch. Külske, Gibbsianness and non-Gibbsianness for Bernoulli lattice fields under removal of isolated sites, Preprint no. 2878, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2878 .
    Abstract, PDF (426 kByte)
    We consider the i.i.d. Bernoulli field μ p on Z d with occupation density p ∈ [0,1]. To each realization of the set of occupied sites we apply a thinning map that removes all occupied sites that are isolated in graph distance. We show that, while this map seems non-invasive for large p, as it changes only a small fraction p(1-p)2d of sites, there is p(d) <1 such that for all p ∈ (p(d), 1) the resulting measure is a non-Gibbsian measure, i.e., it does not possess a continuous version of its finite-volume conditional probabilities. On the other hand, for small p, the Gibbs property is preserved.

  • C.F. Coletti, L.R. DE Lima, A. Hinsen, B. Jahnel, D.R. Valesin, Limiting shape for first-passage percolation models on random geometric graphs, Preprint no. 2877, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2877 .
    Abstract, PDF (2340 kByte)
    Let a random geometric graph be defined in the supercritical regime for the existence of a unique infinite connected component in Euclidean space. Consider the first-passage percolation model with independent and identically distributed random variables on the random infinite connected component. We provide sufficient conditions for the existence of the asymptotic shape and we show that the shape is an Euclidean ball. We give some examples exhibiting the result for Bernoulli percolation and the Richardson model. For the Richardson model we further show that it converges weakly to a branching process in the joint limit of large intensities and slow passing times.

Vorträge, Poster

  • J. Köppl, Dynamical Gibbs variational principles for irreversible interacting particle systems and applications to attractor properties, DMV Jahrestagung 2022, September 12 - 16, 2022, Freie Universität Berlin Fachbereich Mathematik & Informatik, September 15, 2022.

  • J. Köppl, Dynamical Gibbs variational principles for irreversible interacting particle systems and applications to attractor properties, BMS --- BGSMath JUNIOR MEETING 2022, September 5 - 7, 2022, Centre de Recerca Matematica, Campus de Ballaterra, Bacelona, Spain, October 6, 2022.

  • A.D. Vu, An Application for Percolation Theory in Analysis, Spring School on Random geometric graphs, March 28 - April 1, 2022, Technische Universität Darmstadt, Fachbereich Mathematik, March 31, 2022.

  • A.D. Vu, Percolation theory and the effective conductivity, 21st Workshop on Stochastic Geometry, Stereology and Image Analysis, June 5 - 10, 2022, Nesuchyne, Czech Republic, June 6, 2022.

  • B. Jahnel, Continuum Percolation in Random Environments (Part 1), Stochastic modelling in the life science: From evolution to medicine, August 1 - 3, 2022, Hausdorff Center for Mathematics, Universität Bonn, August 1, 2022.

  • B. Jahnel, Continuum Percolation in Random Environments (Part 2), Stochastic modelling in the life science: From evolution to medicine, August 1 - 3, 2022, Hausdorff Center for Mathematics, Universität Bonn, August 2, 2022.

  • B. Jahnel, First-passage percolation and chase-escape dynamics on random geometric graphs, Spring School on Random geometric graphs, March 28 - April 1, 2022, Technische Universität Darmstadt, Fachbereich Mathematik, March 30, 2022.

  • B. Jahnel, First-passage percolation and chase-escape dynamics on random geometric graphs, Spring School: Random geometric graphs, Technische Universität Darmstadt, Fachbereich Mathematik, March 30, 2022.

  • B. Jahnel, Malware propagation in mobile device-to-device networks (online talk), IEEE CloudNet '2; Workshop: AI@EDGE -- INSPIRE-5G (Online Event), November 8, 2022.

  • B. Jahnel, Phase transitions and large deviations for the Boolean model of continuum percolation for Cox point processes, Probability Seminar University Padua, Università di Padova, Dipartimento di Matematica, Italy, March 25, 2022.

  • B. Jahnel , Stochastic geometry for telecommunications, Leibniz MMS Days 2022, April 25 - 27, 2022, WIAS, Potsdam, April 26, 2022.

  • A. Hinsen, Limiting shape for first-passage percolation models on random geometric graphs (online talk), German Probability & Statistics Days Mannheim (Online Event), September 27 - October 1, 2021, Universität Mannheim, September 27, 2021.

  • B. Jahnel, Connectivity improvements in mobile device-to-device networks (online talk), Telecom Orange Paris, France, July 6, 2021.

  • B. Jahnel, First-passage percolation and chase-escape dynamics on random geometric graphs, Stochastic Geometry Days, November 15 - 19, 2021, Dunkerque, France, November 17, 2021.

  • B. Jahnel, Gibbsian representation for point processes via hyperedge potentials (online talk), Thematic Einstein Semester on Geometric and Topological Structure of Materials, Summer Semester 2021, Technische Universität Berlin, May 20, 2021.

  • B. Jahnel, Phase transitions for the Boolean model for Cox point processes, Workshop on Randomness Unleashed Geometry, Topology, and Data, September 22 - 24, 2021, University of Groningen, Faculty of Science and Engineering, Groningen, Netherlands, September 23, 2021.

  • B. Jahnel, Phase transitions for the Boolean model for Cox point processes (online talk), DYOGENE Seminar (Online Event), INRIA Paris, France, January 11, 2021.

  • B. Jahnel, Phase transitions for the Boolean model for Cox point processes (online talk), Probability Seminar Bath (Online Event), University of Bath, Department of Mathematical Sciences, UK, October 18, 2021.

  • B. Jahnel, Stochastic geometry for epidemiology (online talk), Monday Biostatistics Roundtable, Institute of Biometry and Clinical Epidemiology (Online Event), Campus Charité, June 14, 2021.