06.06.2023, 15 Uhr
Phenomena in High Dimensions
Pierre Youssef (New York University, Abu Dhabi)
Dieter Mitsche (University of Lyon 1), David Coupier (University of Valenciennes)
14:00 Dieter Mitsche (University of Lyon 1): Introduction to random hyperbolic graphs
15:15 David Coupier (Université Polytechnique Hauts-de-France, Valenciennes): Infinite branches of two geometric random graphs in Euclidean and hyperbolic spaces
Dieter Mitsche (University of Lyon 1): Introduction to random hyperbolic graphs
Random hyperbolic graphs (RHG) were proposed in 2010 as a model of real-world networks by Krioukov et al. Informally speaking, they are like random geometric graphs where the underlying metric space has negative curvature (i.e., is hyperbolic). In contrast to other models of complex networks, RHG simultaneously and naturally exhibit characteristics such as sparseness, small diameter, non-negligible clustering coefficient and power law degree distribution.
We will give a slow pace introduction to RHG, explain why they have attracted a fair amount of attention and then survey most of what is known about this promising model of real-world networks, in particular component sizes, diameter, clustering coefficient, spectral gap.
David Coupier (Université Polytechnique Hauts-de-France, Valenciennes): Infinite branches of two geometric random graphs in Euclidean and hyperbolic spaces
We consider continuum percolation models, that extend discrete percolation models to continuous space, in both Euclidean and hyperbolic spaces. After recalling basic facts on hyperbolic geometry, we will compare the radically different behaviors of the continuum percolation model in Euclidean and hyperbolic spaces.
This serves as our motivation to study the infinite branches (i.e. the topological ends) of two geometric random graphs, namely the Radial Spanning Tree (RST) and the Directed Spanning Forest (DSF). There again, we will exhibit different behaviors in Euclidean and hyperbolic spaces.