The Poisson lilypond model is a model for a hard-core germ-grain system introduced by Olle Häggström and Ronald Meester in the paper Nearest neighbor and hard sphere models in continuum percolation. From each grain from a homogeneous Poisson point process a grain starts growing and the growth stops once grains get into contact. This model has been generalized by Sven Ebert and Günter Last to random birth times and by Daryl Daley, Sven Ebert and Günter Last to line-segment based grains. If the line segments are directed along the coordinate axis, I showed absence of percolation. Together with Gary Delaney and Volker Schmidt, I also considered a model with random birth times that gives rise to a fractal germ-grain model.

Watch the grains grow! Also don't miss out to get the tikz code of your favorite realization!

Consider a homogeneous Poisson point process $X$ in a $d$-dimensional box of side length $n\ge1$. Attach independent radii $\{R_x\}_{x\in X}$ to the Poisson points, where we assume that the radius distribution is heavy-tailed. That is, $\mathbb{P}(R_x>r)\approx r^{-\alpha}$ for some $\alpha>0$. Now, define the scale-free Gilbert graph by drawing an edge $x\to y$ if $y\in B_{R_x}(x)$.

Starting from this directed graph, there are two natural possibilities to derive an undirected graph. In the first option, an edge is drawn between $x$ and $y$ if there is a directed edge from $x$ to $y$ *and* a directed edge from $y$ to $x$. In the second option, an edge is drawn between $x$ and $y$ if there is a directed edge from $x$ to $y$ *or* a directed edge from $y$ to $x$. Surprisingly, the two models are radically different when considering the asymptotic behavior of chemical distances, i.e., minimum number of hops needed to get from one point to another. A lattice version of the first model has been investigated by Joe Yukich in his paper on ultra-small scale-free geometric networks. You can find some results for the second model in my recent preprint From heavy-tailed Boolean models to scale-free Gilbert graphs. Generate some Gilbert graphs!

In classical cellular wireless networks, users communicate with a base station via direct message exchange. In order to offload some traffic from the base station or to extend the coverage of a cell, network operators have used the possibility to install certain fixed relays whose purpose is to forward messages from the base station to the user (and vice versa). In the setting of the Leibniz group on Probabilistic methods for mobile ad-hoc networks my colleagues and I are investigating new technological possibilities, where the mobile users can serve as relays themselves. In our preprint Large deviations in relay-augmented wireless networks we derive a large deviation principle for the family of frustrated transmitters in a high-density regime.

Depending on the precise formulation of the frustration event, the minimizer of the rate function can exhibit a rich geometric structure. For network operators, this rate function provides information about possible bottle necks in the system, as it represents the most likely cause for a frustration events.

This observation is already apparant in a substantially simplified scenario, of static users uniformly distributed in a square-shaped sampling window receiving data from the base station at the window center in a single hop. Consider the requirement that the signal-to-interference ratio (SIR) is below the threshold required for the desired quality of service at two specific points. The two points together with the quotient of the SIR in a typical situation in comparison to the frustration event is shown in the left-hand figure. The right-hand figure illustrates the most likely configuration leading to this frustration event. Observe that close to the two distinguished points the user density is substantially higher than the a priori distribution. Moreover, the highest density of users is observed midway between the two distinguished points, as transmitters at this location cause substantial interference at both points.