Conference Agenda

Preferential attachment models are a well-known class of random graphs that model the evolution of real-world complex networks over time. We study a more general model that incorporates vertex death and thus, more realistically, models evolving networks that not only increase but also decrease in size. An important result in the study of preferential attachment models is the occurence of persistence of the maximum degree, where a fixed vertex attains the maximum degree for all but finitely many steps. A clear phase transition is known to exist for the occurence of persistence. We present recent findings for persistence in preferential attachment trees with vertex death, in which we uncover regimes in which a similar phase transition exists, and regimes where persistence never occurs. This is joint work with Markus Heydenreich.

We consider the soft Boolean model, a model that interpolates between the Boolean model and long-range percolation, where vertices are given via a stationary Poisson point process. Each vertex carries an independent heavy-tailed radius and each pair of vertices is assigned another independent heavy-tailed edge-weight with a potentially different tail exponent. Two vertices are now connected if they are within distance of the larger radius multiplied by the edge weight. We determine the tail behaviour of the Euclidean diameter and the number of points of a typical maximally connected component in a subcritical percolation phase. For this, we present a sharp criterion in terms of the tail exponents of the edge-weight and radius distributions that distinguish a regime where the tail behaviour is controlled only by the edge exponent from a regime in which both exponents are relevant. We explain the principle mechansims in both regimes and explain how they lead to the observed behaviour. If time allows, we sketch the most steps of the proof.

We consider the preferential attachment graph with vertices $\{1,\dots,n\}$ where we connect two vertices $i$ and $j$ independently with probability $\beta\, (i \vee j)^{\gamma-1} \, (i \wedge j)^{-\gamma}$. In the regime of $\gamma\in(\tfrac{1}{2},1)$ it can be shown that the largest connected component always makes up a large fraction of the vertices, that is this model is robust under percolation with critical parameter $\beta_c =0$. Moreover the degrees of the vertices have infinite variance, which makes it mathematically interesting. We show the exact size of the largest component is $\exp\left(-\tfrac{c}{\beta}\right)$ if $\beta\to0$, by a constructive proof which consists of path counting techniques. There we rely heavily on the fact that these networks are ultra-small, i.e. typical distances between vertices are only of order $\log\log n$.

In recent years, many models in mathematical physics have been encoded into graphical models, which are more accessible through the lens of probability theory. These graphical models often exhibit a natural percolation structure which is easier to investigate in terms of criticallity of the model. One of the main questions is whether there exists a difference in the critical value for loops and the percolation, i.e. $\beta_c^{\text{link}}<\beta_c^{\text{loop}} $. In my talk I want to give an introduction to the topic and an overview of the results which are known so far.