Many phenomena in the real world can be phrased in terms of networks. Examples include the World-Wide Web, social interactions and Internet, but also the interaction patterns between proteins, food webs and citation networks. In applications, one is often interested in identifying the `important' vertices in such networks. However, being important is not uniquely defined, and as a result, a plethora of centrality measures have been, and are still being, introduced. Key examples include degree, PageRank, closeness, and betweenness centralities, each having their own benefits. The fact that there so many centrality measures calls for a mathematical investigation of their properties. In this talk, I will discuss recent work on this direction, focusing on two topics. The first topic is the recent work on large-graph limits of the PageRank distribution, and its power-law behaviour, both for directed as well as undirected graphs. The second is a method to compare centrality measures called the Centrality Comparison Curve. We assume no prior knowledge in graph theory, probability or otherwise. This joint work with George Exarchakos, Alessandro Garavaglia, Florian Henning, Nelly Litvak, Oliver Nagy, and Manish Pandey.
Remco van der Hofstad received his PhD at the University of Utrecht in 1997, under the supervision of Frank den Hollander and Richard Gill. Since 2005, he is full professor in probability at Eindhoven University of Technology. Remco was scientific director of Eurandom from 2011 until 2019, and jointly with Frank den Hollander he is responsible for the ‘Random Spatial Structures’ Program at Eurandom. He wrote some 200 papers, and three books on random graphs, complex networks, and high-dimensional percolations (the latter with Markus Heydenreich). Remco received the Rollo Davidson Prize 2007, and is a laureate of the `Innovative Research VIDI Scheme' 2003 and `Innovative Research VICI Scheme' 2008. He is also one of the 11 co-applicants of the Gravitation program NETWORKS (see https://www.thenetworkcenter.nl/ for more information). In 2018, Remco was elected in the Royal Academy of Science and Arts (KNAW), where he currently is the chair of the Mathematics Section and member of the Board Natural and Technical Sciences. 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/ for more information). Remco is contact person for the research area Grip on Complexity of the Institute for Complex Molecular Systems, and the chair of the Board of Trustees of the Applied Probability Trust.