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 σ r−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-quasiconvex sets of a δ-hyperbolic graph G there exists a single ball B(c, 2 + 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)-quasiconvex sets of a δ-hyperbolic graph G, then there exists a ball B(c, r *) of radius r * := max{2 + 5δ, r + + 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 2 + 5δ intersecting all sets of a family Q of-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 κ Q in which each set is a union of at most κ-quasiconvex sets of a δ-hyperbolic graph G. Namely, we show that if r ≥ + 2δ and πr(κ Q) is the maximum number of mutually 2r-apart members of κ Q, then the minimum number of balls of radius r + 2 + 6δ intersecting all members of κ Q is at most 2κ 2 πr(κ Q) and such a hitting set and a packing can be constructed in polynomial time for every finite κ Q (this is our third main result). For set-families consisting of unions of κ balls in δ-hyperbolic graphs a similar result was obtained by Chepoi and Estellon (2007). In case of δ = 0 (trees) and = 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 κ subtrees of a tree.
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Victor Chepoi, Feodor Dragan, Yann Vaxès. Core congestion is inherent in hyperbolic networks. Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, 2017, Barcelona, Spain. ⟨10.1137/1.9781611974782.149⟩. ⟨hal-01769624⟩

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