18.06.2018 16:30 Prof. Dr. Claudia Klüppelberg (TUM) :
Max-linear models on infinite graphs generated by Bernoulli bond percolation2.01.10 (Parkring 11, 85748 Garching-Hochbrück)

We extend finite-dimensional max-linear models to models on infinite graphs, and investigate their relations to classical percolation theory, more precisely to nearest neighbour bond percolation. We focus on the square lattice $\mathbb{Z}^2$ with edges to the nearest neighbours, where we direct all edges in a natural way (north-east) resulting in a directed acyclic graph (DAG) on $\mathbb{Z}^2$. On this infinite DAG a random sub-DAG may be constructed by choosing vertices and edges between them at random. In a Bernoulli bond percolation DAG edges are independently declared open with probability $p\in [0,1]$ and closed otherwise. The random DAG consists then of the vertices and the open directed edges. We find for the subcritical case where $p\le 1/2$ that two random variables of the max-linear model become independent with probability 1, whenever their distance tends to infinity. In contrast, for the supercritical case where $p>1/2$ two random variables are dependent with positive probability, even when their node distance tends to infinity. We also consider changes in the dependence properties of random variables on a sub-DAG $H$ of a finite or infinite graph in $\mathbb{Z}^2$, when enlarging this subgraph. The method of enlargement consists of adding nodes and edges of Bernoulli percolation clusters. Here we start with $X_i$ and $X_j$ independent in $H$, and answer the question, whether they can become dependent in the enlarged graph. As a possible application we discuss extreme opinions in social networks The talk is based on joint work with Ercan Sönmez and the following paper. [1] Klüppelberg, C. and Sönmez, E. (2018) Max-linear models on infinite graphs generated by Bernoulli bond percolation. In preparation.