Research Remco van der Hofstad

Random graphs and complex networks

In the past two decades, it has become clear that many real networks share fascinating features in being small worlds and scale-free. Such networks are typically modeled using random graphs. Random graphs are closely related to percolation, the difference being that random graphs tend to have finite size, while percolation systems tend to be infinite. The empirical findings on real networks have ignited research on various models for complex networks. The focus of the research of the group was the study of distances in models of complex networks where power-law degrees are observed, as well as the behavior of stochastic processes on random graphs.

My research has primarily focussed on studying distances in random graphs and stochastic processes on them. These models include
  1. the configuration model,
  2. various versions of generalized random graphs,
  3. and preferential attachment models.
Other aspects that draw my attention is the size of the connected components and the related phase transitions, as well as the behavior of stochastic processes on random graphs, and its relation to the topology of the underlying graph.

A key question in random graph theory related to universality, that is, to which extent do models with similar properties show similar behavior.

Percolation and high-dimensional statistical physics

Percolation is one of the paradigm models in statistical physics, displaying extremely rich critical behavior. It is a model of a porous medium, where the materials consist of substance and holes. My main focus of research in the past period has been on the study of percolation models close to criticality for high-dimensional systems. Related problems include lattice trees and lattice animals, and the Ising model, in high-dimensions.


In this research, we mainly make use of four key methods:
  1. Coupling methods and comparisons with branching processes.
  2. Weak convergence methods for stochastic processes.
  3. Large deviations.
  4. Combinatorial expansion techniques for high-dimensional systems, using the lace expansion.

Applications of Probability

I have always been interested in applications of probability, particularly in electrical engineering, computer science and theoretical physics. So if you have a cool problem, please knock on my door!