GRAphs At Luxembourg and Singapore (GRAALS)

The project at a glance

Start date:

01 Mar 2023
Duration in months:

24
Funding:

FNR
Principal Investigator(s):

Ivan NOURDIN

Giovanni PECCATI

Nicolas Privault (external)

Adrian Röllin (external)

About

Large random graphs are increasingly used for modeling real networks, and there is thus an important need for mathematical tools to study them. In this project, we are mostly interested in second order results, that is, in fluctuations around the limit. We work around three main themes: – Approximations of functionals of random graphs, by combining the Malliavin-Stein approach based on the use of second-order Poincaré inequalities and related estimates, with some recent improvements of the classical (firstorder) Poincaré inequality on the Poisson space by Nourdin, Peccati and Yang, in order to prove Berry-Esseen bounds that are commensurate to the inverse of the standard deviation; – connectivity of random weblike complex systems embedded in spatial regions, by designing an algorithm for the recursive computation of the joint moments of all orders of k-hop counts using decompositions into multiple Poisson stochastic integrals, and by using Stein’s method to derive Berry-Esseen bounds for the asymptotic convergence of renormalized k-hop path counts to the normal distribution, based on expressions obtained by Privault in a recent series of papers; – Fluctuation theory for graph limits, by extending the recent theory of Gaur and Röllin for dense graphs to sparse graphs, and by developing the analog of Donsker’s theorem at graph-process level. It is a widely recognised fact that Luxembourg and Singapore represent two research hubs that have a role of world leaders in the domain of probabilistic approximations and their applications — with competences that perfectly dovetail and complement each other. This collaboration consolidates existing links and creates new ones, securing the position of Luxembourg and Singapore at the forefront of the Stein’s method methodology.