Gromov Hyperbolicity Research Articles

Gromov Hyperbolicity of Periodic Graphs

Gromov hyperbolicity grasps the essence of both negatively curved spaces and discrete spaces. The hyperbolicity of a geodesic metric space is equivalent to the hyperbolicity of a graph related to it; hence, characterizing hyperbolic graphs is a main problem in the theory of hyperbolicity. Since this is a very ambitious goal, a more achievable problem is to characterize hyperbolic graphs in particular classes of graphs. The main result in this paper is a characterization of the hyperbolicity of periodic graphs.

Metric tree‐like structures in real‐world networks: an empirical study

Based on solid theoretical foundations, we present strong evidence that a number of real‐world networks, taken from different domains (such as Internet measurements, biological data, web graphs, and social and collaboration networks) exhibit tree‐like structures from a metric point of view. We investigate a few graph parameters, namely, the tree‐distortion and the tree‐stretch, the tree‐length and the tree‐breadth, Gromov's hyperbolicity, the cluster‐diameter and the cluster‐radius in a layering partition of a graph; such parameters capture and quantify this phenomenon of being metrically close to a tree. By bringing all those parameters together, we provide efficient means for detecting such metric tree‐like structures in large‐scale networks. We also show how such structures can be used. For example, they are helpful in efficient and compact encoding of approximate distance and almost shortest path information and in quick and accurate estimation of diameters and radii of those networks. Estimating the diameter and estimating the radius of a graph (or distances between arbitrary vertices) are fundamental primitives in many network and graph mining algorithms. © 2015 Wiley Periodicals, Inc. NETWORKS, Vol. 67(1), 49–68 2016

Gromov hyperbolicity and the Kobayashi metric on convex domains of finite type

In this paper we prove necessary and sufficient conditions for the Kobayashi metric on a convex domain to be Gromov hyperbolic. In particular we show that for convex domains with $$C^\infty $$ boundary being of finite type in the sense of D’Angelo is equivalent to the Gromov hyperbolicity of the Kobayashi metric. We also show that bounded domains which are locally convexifiable and have finite type in the sense of D’Angelo have Gromov hyperbolic Kobayashi metric. The proofs use ideas from the theory of the Hilbert metric.

On Computing the Gromov Hyperbolicity

The Gromov hyperbolicity is an important parameter for analyzing complex networks which expresses how the metric structure of a network looks like a tree. It is for instance used to provide bounds on the expected stretch of greedy-routing algorithms in Internet-like graphs. However, the best-known theoretical algorithm computing this parameter runs in O ( n 3.69 ) time, which is prohibitive for large-scale graphs. In this article, we propose an algorithm for determining the hyperbolicity of graphs with tens of thousands of nodes. Its running time depends on the distribution of distances and on the actual value of the hyperbolicity. Although its worst case runtime is O ( n 4 ), it is in practice much faster than previous proposals as observed in our experimentations. Finally, we propose a heuristic algorithm that can be used on graphs with millions of nodes. Our algorithms are all evaluated on benchmark instances.

HYPERBOLIC NOTIONS ON A PLANAR GRAPH OF BOUNDED FACE DEGREE

We study the relations between strong isoperimetric inequalities and Gromov hyperbolicity on planar graphs, and give an alternative proof for the following statement: if a planar graph of bounded face degree satisfies a strong isoperimetric inequality, then it is Gromov hyperbolic. This theorem was formerly proved in the author's paper from 2014 [12] using combinatorial methods, while geometric approach is used in the present paper.

Gromov (non-)hyperbolicity of certain domains in ℂN

Abstract We prove the Gromov non-hyperbolicity with respect to the Kobayashi distance for 𝒞 1 , 1 ${\mathcal{C}^{1,1}}$ -smooth convex domains in ℂ 2 ${\mathbb{C}^{2}}$ which contain an analytic disc in the boundary or have a point of infinite type with rotation symmetry. The same is shown for “generic” product spaces, as well as for the symmetrized polydisc and the tetrablock. On the other hand, examples of smooth, non-pseudoconvex, Gromov hyperbolic domains in ℂ n ${\mathbb{C}^{n}}$ are given.

Bounds on Gromov hyperbolicity constant

If X is a geodesic metric space and $$x_{1},x_{2},x_{3} \in X$$ , a geodesic triangle $$T=\{x_{1},x_{2},x_{3}\}$$ is the union of the three geodesics $$[x_{1}x_{2}]$$ , $$[x_{2}x_{3}]$$ and $$[x_{3}x_{1}]$$ in X. The space X is $$\delta $$ -hyperbolic in the Gromov sense if any side of T is contained in a $$\delta $$ -neighborhood of the union of the two other sides, for every geodesic triangle T in X. If X is hyperbolic, we denote by $$\delta (X)$$ the sharp hyperbolicity constant of X, i.e., $$\delta (X) =\inf \{ \delta \ge 0: X ~\text {is}~ \delta \text {-hyperbolic} \}.$$ To compute the hyperbolicity constant is a very hard problem. Then it is natural to try to bound the hyperbolicity constant in terms of some parameters of the graph. Denote by $$\mathcal {G}(n,m)$$ the set of (simple) graphs G with n vertices and m edges, and such that every edge has length 1. In this work we estimate $$A(n,m):=\min \{\delta (G)\mid G \in \mathcal {G}(n,m) \}$$ and $$B(n,m):=\max \{\delta (G)\mid G \in \mathcal {G}(n,m) \}$$ . In particular, we obtain good bounds for B(n, m), and we compute the precise value of A(n, m) for all values of n and m. We also study this problem for non-simple and weighted graphs.

Gromov hyperbolicity of strongly pseudoconvex almost complex manifolds

Let $D=\{\rho < 0\}$ be a smooth relatively compact domain in an almost complex manifold $(M,J)$, where $\rho$ is a smooth defining function of $D$, strictly $J$-plurisubharmonic in a neighborhood of the closure $\overline{D}$ of $D$. We prove that $D$ has a connected boundary and is Gromov hyperbolic.

Computing the Gromov hyperbolicity of a discrete metric space

We give exact and approximation algorithms for computing the Gromov hyperbolicity of an n-point discrete metric space. We observe that computing the Gromov hyperbolicity from a fixed base-point reduces to a (max,min) matrix product. Hence, using the (max,min) matrix product algorithm by Duan and Pettie, the fixed base-point hyperbolicity can be determined in O(n2.69) time. It follows that the Gromov hyperbolicity can be computed in O(n3.69) time, and a 2-approximation can be found in O(n2.69) time. We also give a (2log2⁡n)-approximation algorithm that runs in O(n2) time, based on a tree-metric embedding by Gromov. We also show that hyperbolicity at a fixed base-point cannot be computed in O(n2.05) time, unless there exists a faster algorithm for (max,min) matrix multiplication than currently known.

Duality Properties of Strong Isoperimetric Inequalities on a Planar Graph and Combinatorial Curvatures

This paper is about hyperbolic properties on planar graphs. First, we study the relations among various kinds of strong isoperimetric inequalities on planar graphs and their duals. In particular, we show that a planar graph satisfies a strong isoperimetric inequality if and only if its dual has the same property, if the graph satisfies some minor regularity conditions and we choose an appropriate notion of strong isoperimetric inequalities. Second, we consider planar graphs where negative combinatorial curvatures dominate, and use the outcomes of the first part to strengthen the results of Higuchi, ?uk, and, especially, Woess. Finally, we study the relations between Gromov hyperbolicity and strong isoperimetric inequalities on planar graphs, and give a proof that a planar graph satisfying a proper kind of a strong isoperimetric inequality must be Gromov hyperbolic if face degrees of the graph are bounded. We also provide some examples to support our results.

Computing the hyperbolicity constant of a cubic graph

In this paper we obtain information about the hyperbolicity constant of cubic graphs. They are a very interesting class of graphs with many applications; furthermore, they are also very important in the study of Gromov hyperbolicity, since for any graph G with bounded maximum degree there exists a cubic graph G* such that G is hyperbolic if and only if G* is hyperbolic. We find some characterizations for the cubic graphs which have small hyperbolicity constants, i.e. the graphs which are like trees (in the Gromov sense). Besides, we obtain bounds for the hyperbolicity constant of the complement graph of a cubic graph; our main result of this kind says that for any finite cubic graph G which is not isomorphic either to K4 or to K3, 3, the inequalities 5k/4≤δ (Ḡ)≤3k/2 hold, if k is the length of every edge in G.

Gromov hyperbolicity and quasihyperbolic geodesics

We characterize Gromov hyperbolicity of the quasihyperbolic metric space (\Omega,k) by geometric properties of the Ahlfors regular length metric measure space (\Omega,d,\mu). The characterizing properties are called the Gehring--Hayman condition and the ball--separation condition.

Cop and Robber Game and Hyperbolicity

In this note, we prove that all cop-win graphs G in the game in which the robber and the cop move at different speeds s and s' with s'<s, are \delta-hyperbolic with \delta=O(s^2). We also show that the dependency between \delta and s is linear if s-s'=\Omega(s) and G obeys a slightly stronger condition. This solves an open question from the paper (J. Chalopin et al., Cop and robber games when the robber can hide and ride, SIAM J. Discr. Math. 25 (2011) 333-359). Since any \delta-hyperbolic graph is cop-win for s=2r and s'=r+2\delta for any r>0, this establishes a new - game-theoretical - characterization of Gromov hyperbolicity. We also show that for weakly modular graphs the dependency between \delta and s is linear for any s'<s. Using these results, we describe a simple constant-factor approximation of the hyperbolicity \delta of a graph on n vertices in O(n^2) time when the graph is given by its distance-matrix.

Gromov hyperbolicity of planar graphs

AbstractWe prove that under appropriate assumptions adding or removing an infinite amount of edges to a given planar graph preserves its non-hyperbolicity, a result which is shown to be false in general. In particular, we make a conjecture that every tessellation graph of ℝ2 with convex tiles is non-hyperbolic; it is shown that in order to prove this conjecture it suffices to consider tessellation graphs of ℝ2 such that every tile is a triangle and a partial answer to this question is given. A weaker version of this conjecture stating that every tessellation graph of ℝ2 with rectangular tiles is non-hyperbolic is given and partially answered. If this conjecture were true, many tessellation graphs of ℝ2 with tiles which are parallelograms would be non-hyperbolic.

Gromov hyperbolicity of Denjoy domains through fundamental domains

Gromov hyperbolicity of Denjoy domains through fundamental domains

Bounds on Gromov hyperbolicity constant in graphs

If X is a geodesic metric space and x1,x2,x3 ∈ X, a geodesic triangleT = {x1,x2,x3} is the union of the three geodesics [x1x2], [x2x3] and [x3x1] in X. The space X is δ-hyperbolic (in the Gromov sense) if any side of T is contained in a δ-neighborhood of the union of two other sides, for every geodesic triangle T in X. If X is hyperbolic, we denote by δ(X) the sharp hyperbolicity constant of X, i.e. \(\delta(X)=\inf\{\delta\ge 0: \, X \, \text{ is $\delta$-hyperbolic}\}\,. \) In this paper we relate the hyperbolicity constant of a graph with some known parameters of the graph, as its independence number, its maximum and minimum degree and its domination number. Furthermore, we compute explicitly the hyperbolicity constant of some class of product graphs.

Gromov hyperbolicity of Denjoy domains with hyperbolic and quasihyperbolic metrics

We derive explicit and simple conditions which in many cases allow one to decide, whether or not a Denjoy domain endowed with the Poincare or quasihyperbolic metric is Gromov hyperbolic. The criteria are based on the Euclidean size of the complement. As a corollary, the main theorem allows us to deduce the non-hyperbolicity of any periodic Denjoy domain.

Gromov hyperbolicity of negatively curved Finsler manifolds

Let (M, F) be a closed C∞ Finsler manifold. The lift of the Finsler metric F to the universal covering space defines an asymmetric distance \({\widetilde d}\) on \({\widetilde M}\). It is well-known that the classical comparison theorem of Aleksandrov does not exist in the Finsler setting. Therefore, it is necessary to introduce new Finsler tools for the study of the asymmetric metric space \({(\widetilde M, \widetilde d)}\). In this paper, by using the geometric flip map and the unstable-stable angle introduced in [2], we prove that if (M, F) is a closed Finsler manifold of negative flag curvature, then \({(\widetilde M, \widetilde d)}\) is an asymmetric δ-hyperbolic space in the sense of Gromov.

A VERY SIMPLE CHARACTERIZATION OF GROMOV HYPERBOLICITY FOR A SPECIAL KIND OF DENJOY DOMAINS

In this paper we provide characterizations for the Gromov hyperbolicity of some particular Denjoy domains and besides some sufficient conditions to guarantee or discard the hyperbolicity of some others. The conditions obtained involve just the lengths of some special simple closed geodesics in the domain. These results, on the one hand, show the extraordinary complexity of determining the hyperbolicity of a domain and, on the other hand, allow us to construct easily a large variety of both hyperbolic and non-hyperbolic domains.

Hyperbolizing metric spaces

It was proved by M. Bonk, J. Heinonen and P. Koskela that the quasihyperbolic metric hyperbolizes (in the sense of Gromov) uniform metric spaces. In this paper we introduce a new metric that hyperbolizes all locally compact noncomplete metric spaces. The metric is generic in the sense that (1) it can be defined on any metric space; (2) it preserves the quasiconformal geometry of the space; (3) it generalizes the j j -metric, the hyperbolic cone metric and the hyperbolic metric of hyperspaces; and (4) it is quasi-isometric to the quasihyperbolic metric of uniform metric spaces. In particular, the Gromov hyperbolicity of these metrics also follows from that of our metric.