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- ItemPhase transitions for chase-escape models on Poisson–Gilbert graphs([Madralin] : EMIS ELibEMS, 2020) Hinsen, Alexander; Jahnel, Benedikt; Cali, Elie; Wary, Jean-PhilippeWe present results on phase transitions of local and global survival in a two-species model on Poisson–Gilbert graphs. Initially, there is an infection at the origin that propagates on the graph according to a continuous-time nearest-neighbor interacting particle system. The 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 well-definedness 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 finite-degree approximations of the underlying random graphs.
- ItemThe typical cell in anisotropic tessellations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Hirsch, Christian; Jahnel, Benedikt; Hinsen, Alexander; Cali, ElieThe typical cell is a key concept for stochastic-geometry 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 Manhattan-type 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.
- ItemPercolation for D2D networks on street systems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Cali, Elie; En-Najjari, Taoufik; Gafur, Nila Novita; Hirsch, Christian; Jahnel, Benedikt; Patterson, Robert I.A.We study fundamental characteristics for the connectivity of multi-hop D2D networks. Devices are randomly distributed on street systems and are able to communicate with each other whenever their separation is smaller than some connectivity threshold. We model the street systems as Poisson-Voronoi or Poisson-Delaunay tessellations with varying street lengths. We interpret the existence of adequate D2D connectivity as percolation of the underlying random graph. We derive and compare approximations for the critical device-intensity for percolation, the percolation probability and the graph distance. Our results show that for urban areas, the Poisson Boolean Model gives a very good approximation, while for rural areas, the percolation probability stays far from 1 even far above the percolation threshold.
- ItemConnection intervals in multi-scale dynamic networks(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Hirsch, Christian; Jahnel, Benedikt; Cali, ElieWe 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.