• Stability of directed networks: Random graphs are used to model real-world complex networks that are either too large to be analyzed directly or can be constantly changing. They are also very useful for determining whether a real graph is likely to have certain properties, such as short distances between vertices or large connected components. For directed graphs, the connectivity properties are a bit more subtle than in the undirected case, however, it is known that for several popular random graph models there is a threshold that determines whether there will exist a strongly connected component containing a positive fraction of all the vertices in the graph. The project I have in mind consists in analyzing a couple of models to try to determine how stable the size of the strongly connected component is with respect to the addition/removal of arcs.  Closely related to this question, is whether typical distances between vertices remain stable under the same type of perturbations.
• Inversion methods for the computation of solutions to distributional equations: Distributional fixed-point equations appear in a wide range of problems in operations research, computer science, mathematics, statistical physics, etc. In particular, many interesting problems lead to fixed-point equations that live on weighted trees, and the goal is to compute the distribution of one solution in particular. However, the exploding nature of trees makes the implementation of standard numerical methods very difficult, creating the need more efficient approaches. Some of my prior work includes the analysis of a Monte Carlo algorithm, known as “population dynamics”, that in many cases provides an efficient and accurate method. Unfortunately, it does not work under some ill-behaved but rather important settings, and for those cases we still need to find an efficient method. The project involves looking into Laplace inversion techniques to provide an alternative that works when the population dynamics algorithm fails.

I am currently looking for PhD students to work on problems related to these two themes. If you are interested, please send me an email to molvera@unc.edu.