Core congestion is inherent in hyperbolic networks

We investigate the impact the negative curvature has on the traffic congestion in large-scale networks. We prove that every Gromov hyperbolic network G admits a core, thus answering in the positive a conjecture by Jonckheere, Lou, Bonahon, and Baryshnikov, Internet Mathematics, 7 (2011) which is based on the experimental observation by Narayan and Saniee, Physical Review E, 84 (2011) that real-world networks with small hyperbolicity have a core congestion. Namely, we prove that for every subset X of vertices of a δ-hyperbolic graph G there exists a vertex m of G such that the disk D(m, 4δ) of radius 4δ centered at m intercepts at least one half of the total flow between all pairs of vertices of X, where the flow between two vertices x, y ∈ X is carried by geodesic (or quasi-geodesic) (x, y)-paths. A set S intercepts the flow between two nodes x and y if S intersect every shortest path between x and y. Differently from what was conjectured by Jonckheere et al., we show that m is not (and cannot be) the center of mass of X but is a node close to the median of X in the so-called injective hull of X. In case of non-uniform traffic between nodes of X (in this case, the unit flow exists only between certain pairs of nodes of X defined by a commodity graph R), we prove a primal-dual result showing that for any ρ > 5δ the size of a ρ-multi-core (i.e., the number of disks of radius ρ) intercepting all pairs of R is upper bounded by the maximum number of pairwise (ρ− 3δ)-apart