Weighted distances in geometric random graphs

Speaker: Júlia Komjáthy (Eindhoven University of Technology, The Netherlands)

Abstract:

In the talk we study weighted distances in scale-free spatial network models: hyperbolic random graphs (HRG), and geometric inhomogeneous random graphs (GIRG). In HRGs, n=Θ(eR/2) vertices are sampled independently from the hyperbolic disk with radius R and two vertices are connected either when they are within hyperbolic distance R, or independently with a probability depending on the hyperbolic distance. In GIRGs, each vertex is given an independent weight and location from an underlying measured metric space and Zd, respectively, and two vertices are connected independently with a probability that is a function of their distance and weights. We assign i.i.d. weights to the edges of the random graphs and study the weighted distance between two uniformly chosen vertices. In particular, we study the case when the parameters are so that the degree distribution in the graph follows a power law with exponent τ∈(2,3) (infinite variance), and the edge-weight distribution is such that it produces an explosive age-dependent branching process with power-law offspring distribution. We show that in both models, typical distances within the giant component converge in distribution as the number of vertices tends to infinity.