Benedikt Jahnel

Christina van de Sand

Alexander Hinsen (Math+ Project: Influence of Mobility on Connectivity)

Realization of an urban street system with base
stations (green) and mobile devices (blue). Via D2D
connections the coverage zones (green shades)
of the base stations are dynamically extended.

Research topics:
  • Connectivity improvements in mobile urban D2D augmented networks
  • Data routing in D2D systems
  • Malware propagation in pure D2D systems
  • Dynamic continuum percolation
  • Large-deviations theory for space-time point processes
  • Interacting particle systems on random graphs

In an increasingly connected world with the Internet of Things (IoT), device-to-device communications (D2D), self-driving cars and interacting smart devices, an overwhelming amount of data must be rapidly transmitted through highly complex networks. The inevitable dramatic changes in the systems architecture do affect almost every aspect of the network design. For example, with the roll-out of the new 5G telecommunication standard, faster connections, higher throughput, and more capacity are envisioned via an enhanced mobile broadband. Additionally, ultra reliable low latency communications should enable the system to support time-crucial applications such as car-to-car communication, whereas massive machine-type communication systems will hold the key to a successful transition towards the industry 4.0.

Snapshot of devices (blue) transmitting data
towards a central base station (green) via a
directed routing scheme (black edges).

In light of these ever-growing opportunities, but also demands, of modern communication systems, device-to-device (D2D) communication is becoming one of the key technologies pervading a highly diverse set of use cases, see for example proximity-based messenger apps Firechat, Briar, or Bridgefy. Although sensor networks and related ad-hoc networks have been discussed at length in theory in academia, recent technological breakthroughs are only now providing the basis for the D2D technology to unfold its full potential. The envisioned benefits are manifold, ranging from coverage extensions in emerging markets to network robustness and green networking in an urban setting.

At the same time, D2D components are less controllable by operators and their performance fluctuates more wildly. A thorough and careful modeling and analysis is therefore crucial in identifying and mitigating potential threats for the quality of service in such networks. Here, probabilistic methods have proven helpful in the analysis as well as in the design of the system. Note that probability enters the picture in two ways: it is indispensable to deal with intrinsic uncertainties in the network, like the locations of equipment, but it is also extremely helpful since, e.g., stochastic algorithms can be used to manage the system.

The ubiquitous ansatz for a comprehensive modeling of mobile D2D networks is to identify the network components as points in a stochastic point process. Statistical knowledge about user behavior, environment constraints or mobility are then encoded in the distribution of the random point cloud.

Snapshot of malware (red) propagation in a D2D
network of susceptible nodes (blue) with patch-
holding white knights (green).

The way in which messages, or data packages, are transmitted from device to device is a highly complicated mechanism that incorporates coding schemes, medium access protocols, interference control, security requirements, routing schemes and many more aspects. Typically, this complexity is reflected by additional layers of random input that need to be incorporated in a meaningful comprehensive model of the dynamic system. Ultimately, the performance of the network has to be evaluated based on a variety of performance metrics, for example the throughput, that are adjusted to the specific use cases and that lead to a wide range of recommended actions for network operators and designers.

The state of the art in the field of probabilistic modeling and rigorous analysis of spatial D2D networks usually has a focus on isolated aspects of the broad picture just described. It feeds from a variety of communities ranging from engineering, physics, computer and information science to applied mathematics. The majority of the presently available work on the subject contributes to the understanding of static networks, where devices are immobile and scattered independently at random, with a focus on connectivity and capacity under the viewpoint to understand the expected behavior. Here is where the Leibniz Junior Research Group will make decisive progress. We like to understand and predict the influence of device mobility in realistic environments on the systems performance. We wish to analyze different routing and access strategies for interference reduction in multi-channel D2D communication networks and understand how wanted and unwanted data distributes through the system. For this we need to advance the theory of spatial stochastic dynamics, navigations and dependent random thinnings for point processes as well as Markovian and non-Markovian dynamics on random graphs, also with respect to their large-deviation behavior.

Simulated phase-transition line between global
survival and extinction in a random growth model
on Poisson--Gilbert graphs.

Realization of a dense set of points (green, red)
aiming to connect to the center under interfe-
rence constraints. Red points are disconnected
due to large distance and interference created
around the center by too many points.

Realization of a Poisson--Boolean model (blue)
based on a Cox point process (red) with random
environment (green) given by an independent
Poisson--Boolean model.