Core congestion is inherent in hyperbolic networks

Abstract

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 graph with δ-thin geodesic triangles (in particular, of a δ-hyperbolic graph) G there exists a vertex m of G such that the ball B(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. Moreover, we prove a primal-dual result showing that, for any commodity graph R on X and any r ≥ 8δ, the size σr(R) of the least r-multi-core (i.e., the number of balls of radius r) intercepting all pairs of R is upper bounded by the maximum number of pairwise (2r − 5δ)-apart pairs of R and that an r-multi-core of size σr−5δ (R) can be computed in polynomial time for every finite set X.Our result about total r-multi-cores is based on a Helly-type theorem for quasiconvex sets in δ-hyperbolic graphs (this is our second main result). Namely, we show that for any finite collection 𝒬 of pairwise intersecting ϵ-quasiconvex sets of a δ-hyperbolic graph G there exists a single ball B(c, 2ϵ + 5δ) intersecting all sets of 𝒬. More generally, we prove that if 𝒬 is a collection of 2r-close (i.e., any two sets of 𝒬 are at distance ≤ 2r) ϵ-quasiconvex sets of a δ-hyperbolic graph G, then there exists a ball B(c, r*) of radius r* := max{2ϵ + 5δ, r + ϵ + 3δ} intersecting all sets of 𝒬. These kind of Helly-type results are also useful in geometric group theory.Using the Helly theorem for quasiconvex sets and a primal-dual approach, we show algorithmically that the minimum number of balls of radius 2ϵ + 5δ intersecting all sets of a family 𝒬 of ϵ-quasiconvex sets does not exceed the packing number of 𝒬 (maximum number of pairwise disjoint sets of 𝒬). We extend the covering and packing result to set-families k𝒬 in which each set is a union of at most kϵ-quasiconvex sets of a δ-hyperbolic graph G. Namely, we show that if r ≥ ϵ + 2δ and πr(k𝒬) is the maximum number of mutually 2r-apart members of k𝒬, then the minimum number of balls of radius r + 2ϵ + 6δ intersecting all members of k𝒬 is at most 2k2πr(k𝒬) and such a hitting set and a packing can be constructed in polynomial time for every finite k𝒬 (this is our third main result). For set-families consisting of unions of k balls in δ-hyperbolic graphs a similar result was obtained by Chepoi and Estellon (2007). In case of δ = 0 (trees) and ϵ = r = 0, (subtrees of a tree) we recover the result of Alon (2002) about the transversal and packing numbers of a set-family in which each set is a union of at most k subtrees of a tree.