Remco van der Hofstad is professor in probability at Eindhoven University of Technology (TU/e). Remco was scientific director of Eurandom from 2011–2019. He received the Rollo Davidson Prize in 2007. He is a laureate of prestigious VIDI 2003 and VICI 2008 grants, and is a PI in the 10-year Gravitation program NETWORKS. In 2018, he was elected in the Netherlands Royal Academy of Science and Arts, where he currently is the chair of the Mathematics Section. He is the author of three monographs on random graphs and percolation.

Remco is editor-in-chief of the Network Pages, an interactive website by the networks community for everyone interested in networks (see https://www.networkpages.nl/), contact person for Grip on Complexity at the TU/e Institute for Complex Molecular Systems, and the chair of the Board of Trustees of the Applied Probability Trust.

This Medallion Lecture will be given at the 11th World Congress in Probability and Statistics.

Critical percolation on scale-free random graphs

Empirical findings have shown that many real-world networks are scale free, in the sense that there is a high variability in the number of connections of the elements of the networks. Spurred by these empirical findings, models have been proposed for such networks. Mathematically, scale-free networks have degree sequences whose second moments diverge.

Percolation on networks is one of the simplest models for network functionality. It can be interpreted as describing the effect of random attacks on the network, where edges are removed independently with a fixed probability called the percolation threshold, or the result of a simple epidemic on the network. On finite networks, it is unclear how to define the percolation critical value, and definitions are often ad hoc. Remarkably, percolation on scale-free networks is robust, in the sense that a giant component containing a positive proportion of the vertices exists for every positive percolation threshold.

We investigate the percolation critical behavior for a popular model of complex networks, the Poisson random graph, which can be interpreted either as a model with multiple edges between vertex pairs, or as a model with single edges by collapsing the multiple edges between vertex pairs. We identify what the percolation critical values in both settings are, and how they scale with the graph size. Interestingly, this scaling turns out to be rather different for the multi-edge case compared to the single-edge case in the scale-free regime. This clears up some confusion in the physics literature. Furthermore, the single-edge case has a surprising phase transition at the appropriate scale of the percolation parameter. Here the size of the largest component jumps from a random value to a much larger almost deterministic value that is proportional to the square root of the graph size. This phase transition is virtually impossible to detect using simulations.

This is joint work with Shankar Bhamidi and Souvik Dhara.