Probabilistic Methods for Dynamic Communication Networks

Publications

Articles in Refereed Journals

• M. Heida, B. Jahnel, A.D. Vu, Regularized homogenization on irregularly perforated domains, Networks and Heterogeneous Media, 20 (2025), pp. 165--212, DOI 10.3934/nhm.2025010 .
  Abstract
  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.

• CH. Hirsch, B. Jahnel, S.K. Jhawar, P. Juhász, Poisson approximation of fixed-degree nodes in weighted random connection models, Stochastic Processes and their Applications, 183 (2025), pp. 104593/1--104593/15, DOI 10.1016/j.spa.2025.104593 .
  Abstract
  We present a process-level Poisson-approximation result for the degree-$k$ vertices in a high-density weighted random connection model with preferential-attachment kernel in the unit volume. Our main focus lies on the impact of the left tails of the weight distribution for which we establish general criteria based on their small-weight quantiles. To illustrate that our conditions are broadly applicable, we verify them for weight distributions with polynomial and stretched exponential left tails. The proofs rest on truncation arguments and a recently established quantitative Poisson approximation result for functionals of Poisson point processes.

• B. Jahnel, J. Köppl, On the long-time behaviour of reversible interacting particle systems in one and two dimensions, Probability and Mathematical Physics, 6 (2025), pp. 479--503, DOI 10.2140/pmp.2025.6.479 .
  Abstract
  By refining Holley's free energy technique, we show that, under quite general assumptions on the dynamics, the attractor of a (possibly non-translation-invariant) interacting particle system in one or two spatial dimensions is contained in the set of Gibbs measures if the dynamics admits a reversible Gibbs measure. In particular, this implies that there can be no reversible interacting particle system that exhibits time-periodic behaviour and that every reversible interacting particle system is ergodic if and only if the reversible Gibbs measure is unique. In the special case of non-attractive stochastic Ising models this answers a question due to Liggett.

• L. Lüchtrath, Ch. Mönch, A very short proof of Sidorenko's inequality for counts of homomorphism between graphs, Bulletin of the Australian Mathematical Society, published online on 14.04.2025, DOI 10.1017/S000497272500019X .
  Abstract
  We provide a very elementary proof of a classical extremality result due to Sidorenko (Discrete Math. 131.1-3, 1994), which states that among all graphs G on k vertices, the k-1-edge star maximises the number of graph homomorphisms of G into any graph H.

• M. Heida, B. Jahnel, A.D. Vu, An ergodic and isotropic zero-conductance model with arbitrarily strong local connectivity, Electronic Communications in Probability, 29 (2024), pp. 66/1--66/13, DOI 10.1214/24-ECP633 .
  Abstract
  We exhibit a percolating ergodic and isotropic lattice model in all but at least two dimensions that has zero effective conductivity in all spatial directions and for all non-trivial choices of the connectivity parameter. The model is based on the so-called randomly stretched lattice where we additionally elongate layers containing few open edges.

• B. Jahnel, U. Rozikov, Gibbs measures for hardcore-solid-on-solid models on Cayley trees, Journal of Statistical Mechanics: Theory and Experiment, 2024 (2024), pp. 073202/1--073202/17, DOI 10.1088/1742-5468/ad5433 .
  Abstract
  We investigate the finite-state p-solid-on-solid model, for p=∞, on Cayley trees of order k ≥ 2 and establish a system of functional equations where each solution corresponds to a (splitting) Gibbs measure of the model. Our main result is that, for three states, k=2,3 and increasing coupling strength, the number of translation-invariant Gibbs measures behaves as 1→3 →5 →6 →7. This phase diagram is qualitatively similar to the one observed for three-state p-SOS models with p>0 and, in the case of k=2, we demonstrate that, on the level of the functional equations, the transition p → ∞ is continuous.

• B. Jahnel, U. Rozikov, Three-state $p$-SOS models on binary Cayley trees, Journal of Statistical Mechanics: Theory and Experiment, 2024 (2024), pp. 113202/1--113202/25, DOI 10.1088/1742-5468/ad8749 .
  Abstract
  We consider a version of the solid-on-solid model on the Cayley tree of order two in which vertices carry spins of value 0,1 or 2 and the pairwise interaction of neighboring vertices is given by their spin difference to the power p>0. We exhibit all translation-invariant splitting Gibbs measures (TISGMs) of the model and demonstrate the existence of up to seven such measures, depending on the parameters. We further establish general conditions for extremality and non-extremality of TISGMs in the set of all Gibbs measures and use them to examine selected TISGMs for a small and a large p. Notably, our analysis reveals that extremality properties are similar for large p compared to the case p=1, a case that has been explored already in previous work. However, for the small p, certain measures that were consistently non-extremal for p=1 do exhibit transitions between extremality and non-extremality.

Contributions to Collected Editions

• L. Lüchtrath, Ch. Mönch, The directed age-dependent random connection model with arc reciprocity, in: Modelling and Mining Networks, M. Dewar, B. Kamiński, D. Kaszyński, Ł. Kraiński, P. Prałat, F. Théberge, M. Wrzosek, eds., 14671 of Lecture Notes in Computer Science, Springer, 2024, pp. 97--114, DOI 10.1007/978-3-031-59205-8_7 .
  Abstract
  We introduce a directed spatial random graph model aimed at modelling certain aspects of social media networks. We provide two variants of the model: an infinite version and an increasing sequence of finite graphs that locally converge to the infinite model. Both variants have in common that each vertex is placed into Euclidean space and carries a birth time. Given locations and birth times of two vertices, an arc is formed from younger to older vertex with a probability depending on both birth times and the spatial distance of the vertices. If such an arc is formed, a reverse arc is formed with probability depending on the ratio of the endpoints' birth times. Aside from the local limit result connecting the models, we investigate degree distributions, two different clustering metrics and directed percolation.

Preprints, Reports, Technical Reports

• B. Jahnel, L. Lüchtrath, A.D. Vu, First contact percolation, Preprint no. 3164, WIAS, Berlin, 2025, DOI 10.20347/WIAS.PREPRINT.3164 .
  Abstract, PDF (506 kByte)
  We study a version of first passage percolation on Z^d where the random passage times on the edges are replaced by contact times represented by random closed sets on R. Similarly to the contact process without recovery, an infection can spread into the system along increasing sequences of contact times. In case of stationary contact times, we can identify associated first passage percolation models, which in turn establish shape theorems also for first contact percolation. In case of periodic contact times that reflect some reoccurring daily pattern, we also present shape theorems with limiting shapes that are universal with respect to the within-one-day contact distribution. In this case, we also prove a Poisson approximation for increasing numbers of within-one-day contacts. Finally, we present a comparison of the limiting speeds of three models -- all calibrated to have one expected contact per day -- that suggests that less randomness is beneficial for the speed of the infection. The proofs rest on coupling and subergodicity arguments.

• L. Lüchtrath, All spatial graphs with weak long-range effects have chemical distance comparable to Euclidean distance, Preprint no. 3154, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3154 .
  Abstract, PDF (262 kByte)
  This note provides a sufficient condition for linear lower bounds on chemical distances (compared to the Euclidean distance) in general spatial random graphs. The condition is based on the scarceness of long edges in the graph and weak correlations at large distances and is valid for all translation invariant and locally finite graphs that fulfil these conditions. The proof is based on a renormalisation scheme introduced by Berger [arXiv: 0409021 (2004)].

• E. Jacob, B. Jahnel, L. Lüchtrath, Subcritical annulus crossing in spatial random graphs, Preprint no. 3148, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3148 .
  Abstract, PDF (481 kByte)
  We consider general continuum percolation models obeying sparseness, translation invariance, and spatial decorrelation. In particular, this includes models constructed on general point sets other than the standard Poisson point process or the Bernoulli-percolated lattice. Moreover, in our setting the existence of an edge may depend not only on the two end vertices but also on a surrounding vertex set and models are included that are not monotone in some of their parameters. We study the critical emphannulus-crossing intensity λ̂_c, which is smaller or equal to the classical critical percolation intensity λ_c and derive a condition for λ̂_c > 0 by relating the crossing of annuli to the occurrence of long edges. This condition is sharp for models that have a modicum of independence. In a nutshell, our result states that annuli are either not crossed for small intensities or crossed by a single edge. Our proof rests on a multiscale argument that further allows us to directly describe the decay of the annulus-crossing probability with the decay of long edges probabilities. We apply our result to a number of examples from the literature. Most importantly, we extensively discuss the emphweight-dependent random connection model in a generalised version, for which we derive sufficient conditions for the presence or absence of long edges that are typically easy to check. These conditions are built on a decay coefficient ζ that has recently seen some attention due to its importance for various proofs of global graph properties

• B. Jahnel, J. Köppl, Y. Steenbeck, A. Zass, The variational principle for a marked Gibbs point process with infinite-range multibody interactions, Preprint no. 3126, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3126 .
  Abstract, PDF (468 kByte)
  We prove the Gibbs variational principle for the Asakura?Oosawa model in which particles of random size obey a hardcore constraint of non-overlap and are additionally subject to a temperature-dependent area interaction. The particle size is unbounded, leading to infinite-range interactions, and the potential cannot be written as a k-body interaction for fixed k. As a byproduct, we also prove the existence of infinite-volume Gibbs point processes satisfying the DLR equations. The essential control over the influence of boundary conditions can be established using the geometry of the model and the hard-core constraint.

• M. Gösgens, L. Lüchtrath, E. Magnanini, M. Noy, É. DE Panafieu, The Erdős--Rényi random graph conditioned on every component being a clique, Preprint no. 3111, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3111 .
  Abstract, PDF (2166 kByte)
  We consider an Erdős-Rényi random graph conditioned on the rare event that all connected components are fully connected. Such graphs can be considered as partitions of vertices into cliques. Hence, this conditional distribution defines a distribution over partitions. Using tools from analytic combinatorics, we prove limit theorems for several graph observables: the number of cliques; the number of edges; and the degree distribution. We consider several regimes of the connection probability p as the number of vertices n diverges. We prove that there is a phase transition at p=1/2 in these observables. We additionally study the near-critical regime as well as the sparse regime

• B. Jahnel, L. Lüchtrath, M. Ortgiese, Cluster sizes in subcritical soft Boolean models, Preprint no. 3106, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3106 .
  Abstract, PDF (435 kByte)
  We consider the soft Boolean model, a model that interpolates between the Boolean model and long-range percolation, where vertices are given via a stationary Poisson point process. Each vertex carries an independent Pareto-distributed radius and each pair of vertices is assigned another independent Pareto weight with a potentially different tail exponent. Two vertices are now connected if they are within distance of the larger radius multiplied by the edge weight. We determine the tail behaviour of the Euclidean diameter and the number of points of a typical maximally connected component in a subcritical percolation phase. For this, we present a sharp criterion in terms of the tail exponents of the edge-weight and radius distributions that distinguish a regime where the tail behaviour is controlled only by the edge exponent from a regime in which both exponents are relevant. Our proofs rely on fine path-counting arguments identifying the precise order of decay of the probability that far-away vertices are connected.

• J. Köppl, N. Lanchier, M. Mercer, Survival and extinction for a contact process with a density-dependent birth rate, Preprint no. 3103, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3103 .
  Abstract, PDF (860 kByte)
  To study later spatial evolutionary games based on the multitype contact process, we first focus in this paper on the conditions for survival/extinction in the presence of only one strategy, in which case our model consists of a variant of the contact process with a density-dependent birth rate. The players are located on the d-dimensional integer lattice, with natural birth rate λ and natural death rate one. The process also depends on a payoff a₁₁ = a modeling the effects of the players on each other: while players always die at rate one, the rate at which they give birth is given by λ times the exponential of a times the fraction of occupied sites in their neighborhood. In particular, the birth rate increases with the local density when a > 0, in which case the payoff a models mutual cooperation, whereas the birth rate decreases with the local density when a < 0, in which case the payoff a models intraspecific competition. Using standard coupling arguments to compare the process with the basic contact process (the particular case a = 0 ), we prove that, for all payoffs a , there is a phase transition from extinction to survival in the direction of λ. Using various block constructions, we also prove that, for all birth rates λ, there is a phase transition in the direction of a. This last result is in sharp contrast with the behavior of the nonspatial deterministic mean-field model in which the stability of the extinction state only depends on λ . This underlines the importance of space (local interactions) and stochasticity in our model.

• P.P. Ghosh, B. Jahnel, S.K. Jhawar, Large and moderate deviations in Poisson navigations, Preprint no. 3096, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3096 .
  Abstract, PDF (318 kByte)
  We derive large- and moderate-deviation results in random networks given as planar directed navigations on homogeneous Poisson point processes. In this non-Markovian routing scheme, starting from the origin, at each consecutive step a Poisson point is joined by an edge to its nearest Poisson point to the right within a cone. We establish precise exponential rates of decay for the probability that the vertical displacement of the random path is unexpectedly large. The proofs rest on controlling the dependencies of the individual steps and the randomness in the horizonal displacement as well as renewal-process arguments.

• B. Jahnel, J. Köppl, Time-periodic behaviour in one- and two-dimensional interacting particle systems, Preprint no. 3092, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3092 .
  Abstract, PDF (311 kByte)
  We provide a class of examples of interacting particle systems on $Z^d$, for $din1,2$, that admit a unique translation-invariant stationary measure, which is not the long-time limit of all translation-invariant starting measures, due to the existence of time-periodic orbits in the associated measure-valued dynamics. This is the first such example and shows that even in low dimensions, not every limit point of the measure-valued dynamics needs to be a time-stationary measure.

Talks, Poster

• B. Jahnel, Dynamical Gibbs variational principles for irreversible interacting particle systems with applications to attractor properties, Colloquium Mathematics, University of Groningen, Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence, Groningen, Netherlands, April 23, 2025.

• B. Jahnel, First contact percolation, Rhein--Main-Kolloquium Stochastik, Technische Universität Darmstadt, Fachbereich Mathematik, April 25, 2025.

• B. Jahnel, tba, Stat-Mech in Créteil 2025, June 10 - 11, 2025, Université Paris-Est Créteil, Créteil, France.

• B. Jahnel, Dynamical Gibbs variational principles for irreversible interacting particle systems with applications to attractor properties, Probability Colloquium Augsburg--Munich, Universität Augsburg, Institut für Mathematik, January 24, 2025.

• L. Lüchtrath, Existence of an extinction phase for the contact process on one-dimensional networks with heavy-tailed degree distribution, Workshop on Stochastic Processes on Random Geometries, February 17 - 21, 2025, Technische Universiät Braunschweig, Institut für Mathematische Stochastik, February 20, 2025.

• J. Hörmann, Geometrische Dichten für nicht-isotrope Boolesche Modelle, Hochschule Pforzheim, Fakultät für Technik, April 17, 2024.

• J. Köppl, Dynamical Gibbs variational principles and applications, 4th Italian Meeting on Probability and Mathematical Statistics, June 10 - 14, 2024, University of Rome, Tor Vergata, Sapienza University of Rome, The University of Roma Tre, LUISS, Rome, Italy, June 10, 2024.

• J. Köppl, Dynamical Gibbs variational principles and applications to attractor properties (online talk), Postgraduate Online Probability Seminar (POPS), February 28, 2024.

• J. Köppl, Dynamical Gibbs variational principles and applications to attractor properties (online talk), Oberseminar Stochastik, Universität Paderborn, Institut für Mathematik, May 15, 2024.

• J. Köppl, Syncronisation and time-periodic behaviour in interacting particle systems, Interacting particles in the continuum, September 9 - 13, 2024, EURANDOM, Eindhoven, Netherlands, September 12, 2024.

• J. Köppl, The long-time behaviour of interacting particle systems: A Lyapunov functional approach (online talk), Probability Seminar, University of California Los Angeles (UCLA), Department of Mathematics, Los Angeles, USA, February 15, 2024.

• A.D. Vu, A contact process in random environment, 19. Doktorandentreffen der Stochastik, August 27 - 30, 2024, Brandenburgische Technische Universität Cottbus--Senftenberg, August 28, 2024.

• A.D. Vu, First contact percolation, Probability-Autumn-School on Point Processes and their dynamics, November 19 - 22, 2024, Universität Münster, Institut für Mathematische Stochastik, November 20, 2024.

• B. Jahnel, Gibbs point processes in random environment, Random Geometric Systems, Third Annual Conference of SPP 2265, October 28 - 30, 2024, Harnack-Haus, Tagungsstätte der Max-Planck-Gesellschaft, October 29, 2024.

• B. Jahnel, Cluster sizes in subcritical soft Boolean models, Long-range Phenomena in Percolation, September 23 - 27, 2024, Universität zu Köln, Mathematisches Institut, September 26, 2024.

• B. Jahnel, Cluster sizes in subcritical soft Boolean models, GrHyDy2024: Random Spatial Models, October 23 - 25, 2024, Université de Lille, Institut Mines-Télécom, Lille, France, October 24, 2024.

• B. Jahnel, Poisson approximation of fixed-degree nodes in weighted random connection models, Bernoulli-IMS 11th World Congress in Probability and Statistics, Session OCPS 07 ``Spatial Random Graphs'', August 12 - 16, 2024, Ruhr-Universität Bochum, August 16, 2024.

• B. Jahnel, Time-periodic behavior in one- and two-dimensional interacting particle systems (online talk), International Scientific Conference on Gibbs Measures and the Theory of Dynamical Systems (online event), May 20 - 21, 2024, Ministry of Higher Education, Science and Innovations of the Republic of Uzbekistan, Romanovskiy Institute of Mathematics and University of Exact and Social Sciences, Tashkent, Uzbekistan, May 20, 2024.

• B. Jahnel, Subcritical annulus crossing in spatial random graphs, Seminar, Institut für Mathematische Stochastik, Georg-August-Universität Göttingen, December 4, 2024.

• L. Lüchtrath, A random cluster graph, Oberseminar Stochastik, Universität zu Köln, Mathematisches Institut, October 23, 2024.

• L. Lüchtrath, Cluster sizes in soft Boolean models, Probability and Analysis 2024, April 22 - 26, 2024, Wroclaw University of Science and Technology, Będlewo, Poland, April 22, 2024.

• L. Lüchtrath, Cluster sizes in subcritical soft Boolean models, Geneva Mathematical Physics Seminar, Université de Genève, Section de Mathématiques, Switzerland, December 2, 2024.

• L. Lüchtrath, The random cluster graph, Workshop Frauenchiemsee 2024, January 14 - 17, 2024, Universität Augsburg, Mathematisch-Naturwissenschaftlich-Technische Fakultät, January 16, 2024.