Dynamical networks (M. Kramar Fijavž, R. Nagel)

The term "dynamical networks" refers to the study of various dynamical processes going on on a static graph. These processes are described by a system of PDEs with boundary conditions in the vertices of the graph. In recent years we have, together with various coauthors, obtained many results regarding solvability, qualitative behavior of the solutions and controllability. The proofs (and results) intertwine continuous functional analysis with discrete mathematics.

We will present some open questions which should be of interest to specialist in linear algebra, graph theory or operator theory. Here is a sample:
Which n x n positive column stochastic matrices A satisfy that the set {v, Av, A²v, ..., A^(n-1)v} has full range if we take v = e₁, e.g. the first basis vector?

The above matrix A can be viewed as a weighted adjacent matrix of a directed graph. This condition characterizes the vertices of the graph, in which our dynamical process is controllable, but it is still unclear what are its consequences for the graph.