EF4 – Particles and Agents



Influence of mobility on connectivity

Project Heads

Benedikt Jahnel, Wolfgang König

Project Members

Alexander Hinsen (né Wapenhans) (WIAS) 

Project Duration

01.01.2019 – 31.12.2021

Located at



We consider a spatial system of devices (smartphones and Internet-of-Things devices) on a random street system in Requipped with a realistic transmission mechanism respecting interference and delays. The devices form an ad-hoc communication system where smartphones move independently according to the random waypoint model on the streets. We study two separate models (A) propagation of malware and efficiency of countermeasures, and (B) throughput of messages, if sparse infrastructure is added. Our main interest lies on the influence of mobility on the propagation, respectively on the throughput.

In the first of our two models, we consider a spatial Poisson point process of locations of users, and an interacting particle process on it, presenting the spread of the malware over all the users as a function of time.  The infection takes place after independent exponential times between neighboring users. First we identify regimes of the parameters in which the malware spreads over unboundedly many, and we give bounds on the velocity. Then we add countermeasures, so-called white knights, to the system, which are able to neutralize and erase a device that is infected by the malware, as soon as it attempts to infect the white knight. The set-up of this model has been designed in close collaboration with our industry partner. In this model, we identify parameter regimes in which the counter measure is successful on the long run, and it gives bounds on its efficiency. The novelty in this part or our research lies in the randomness of the underlying location process (instead of just a deterministic grid) and in the details of the functionality of the counter measure. So far, mobility of the users is not present in the model.

In the second of our models, we assume that users are initially located at a home, and the homes form a spatial Poisson point process. This process may be so sparse that the malware spread would be only local on a long run. Additionally, each user performs independently a random trajectory of a time interval, starting from the home. In this way, the possibility of creating contacts with other users is introduced in the model. The goal is to identify conditions under which the mobility of the users implies, on a long run, a global connectivity in large parts of the communication system with high probability, and to roughly explain some details of this global connectivity.

In both model types, the main mathematical tool box stems from spatial probability, more precisely, the theory of (static and dynamic) random point processes in the Euclidean space, in  particular the theory of continuum percolation and the Boolean model. The designs of our models  and the questions that we aim at are developed in discussions with our industry partner.

Selected Publications

  • E. Cali, A. Hinsen, B. Jahnel, J.-P. Wary: Malware propagation in urban D2D networks,
    Preprint available at arXiv:2002.06248
  • B. Jahnel, A. Tóbiás: SINR percolation for Cox point processes with random powers,
    Preprint available at arXiv:1912.07895
  • C. Hirsch, B. Jahnel, A. Tóbiás: Lower large deviations for geometric functionals,
    Preprint available at arXiv:1910.05993
  • E. Cali, B. Jahnel, A. Tóbiás: Phase transitions for the Boolean model of continuum percolation for Cox point processes,
    Preprint available at arXiv:2003.06206

Selected Pictures

Malware propagation in urban D2D networks

Realization of randomly placed devices on a street system of Poisson-Voronoi tessellation type. Upper row: The Markovian SIG-model (Susceptible-Infected-Goodware) stopped at the time the malware has reached the radius u=2.5km (left) and u=5km (right), indicated in black. Lower row: The non-Markovian SIG-model with uniform waiting times on [40sec,120sec] for both infected and immune devices, stopped at the time the malware has reached the radius u=2.5km (left) and u=5km (right), indicated in black.

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