Modelling and analysis of large-scale networks for mobile agents

Percolation theory has been widely studied for about fifty years in discrete and continuous settings [3,5], both motivated by its numerous applications in physics, materials science, geology etc. as well as by its sheer mathematical beauty. Percolation theory describes a static picture, even if the underlying framework may evolve in time. However, recent applications like long-range connection in wireless communication networks with mobile users or information flow/rumor spreading in evolving social networks, in which percolation makes sense, manifest a dynamic aspect that cannot be neglected. This is exactly the goal of the current proposal: to study random networks which are both geometric and dynamic, and to investigate their connectivity properties which (from a mathematical point of view) can be expressed in a percolative setup. Indeed, models for wireless communication networks as well as social and collaborative networks can be interpreted as large scale (random) graphs in which vertices may represent either cellphone users, base stations, social network accounts, Wikipedia pages etc. In order to build more reliable, efficient and resilient networks, it becomes essential to gain a deeper understanding of how their connectivity properties depend on their geometrical structure and how they evolve in time. 

Due to the development of the Internet of Things (IoT) and the appearance of autonomous vehicles, there is an increasing demand for fast and reliable data exchange in communication systems involving mobile devices. In this context, Device-to-Device (D2D) communication is considered as one of the key concepts pervading a highly diverse set of use cases. The study of connectivity of telecommunication networks through the lens of percolation has received quite a bit of attention in recent years (see for example [1,2,4]. However, most of them fail to take into account the mobility of the users of these networks (in a realistic fashion). This project aims to fill this gap by proposing realistic models and new mathematical tools to investigate their connectivity properties. 

In a D2D network, the signal propagation through chains of (close) users can be naturally interpreted in terms of percolation. Hence, long-range connection in a D2D network corresponds to the occurrence of an unbounded cluster of connected users. A precise picture of different percolation regimes has been given in [2] for a D2D network with static users in an urban media (modelled by a Delaunay triangulation). Understanding how long-range connection in such network behaves when users are now allowed to move in the urban media is a very natural and challenging question. In [6], the authors study a Poisson Boolean model on Rd where centers move according to independent trajectories and state that if percolation occurs at time t=0 then this remains true over time. We aim to extend this dynamical percolation result to cases where the centers (i.e. the users) are no longer independent, because either they are conditioned to evolve in an (random) urban media or some interaction between them has been introduced in the dynamics. Another relevant question that is not treated in [6] concerns the uniqueness of the unbounded cluster over time. 

Finally on the numerical side, simulation of the newly developed dynamical models as well as validation against experimental data will of course be of great interest