Hyperbolicity is a property of a graph that may be viewed as being a "soft" version of a tree, and recent empirical and theoretical work has suggested that many graphs arising in Internet and related data applications have hyperbolic properties. We consider Gromov's notion of -hyperbolicity, and establish several results for small-world and tree-like random graph models. First, we study the hyperbolicity of Kleinberg small-world random graphs and show that the hyperbolicity of these random graphs is not significantly improved comparing to graph diameter even when it greatly improves decentralized navigation. Next we study a class of tree-like graphs called ringed trees that have constant hyperbolicity. We show that adding random links among the leaves similar to the small-world graph constructions may easily destroy the hyperbolicity of the graphs, except for a class of random edges added using an exponentially decaying probability function based on the ring distance among the leaves. Our study provides one of the first significant analytical results on the hyperbolicity of a rich class of random graphs, which shed light on the relationship between hyperbolicity and navigability of random graphs, as well as on the sensitivity of hyperbolic to noises in random graphs.
 We propose a scalable and reliable point-to-point routing algorithm for ad hoc wireless networks and sensor-nets. Our algorithm assigns to each node of the network a virtual coordinate in the hyperbolic plane, and performs greedy geographic routing with respect to these virtual coordinates. Unlike other proposed greedy routing algorithms based on virtual coordinates, our embedding guarantees that the greedy algorithm is always successful in finding a route to the destination, if such a route exists. We describe a distributed algorithm for computing each node's virtual coordinates in the hyperbolic plane, and for greedily routing packets to a destination point in the hyperbolic plane. (This destination may be the address of another node of the network, or it may be an address associated to a piece of content in a Distributed Hash Table. In the latter case we prove that the greedy routing strategy makes a consistent choice of the node responsible for the address, irrespective of the source address of the request.) We evaluate the resulting algorithm in terms of both path stretch and node congestion. We introduce a novel measure called e-four-pointscondition (e-4PC), which assigns a value e ∈ [0,1] to every metric space quantifying how close the metric is to a tree metric. Data-sets taken from real Internet measurements indicate remarkable closeness of Internet latencies to tree metrics based on this condition. We study embeddings of e-4PC metric spaces into trees and prove tight upper and lower bounds. Specifically, we show that there are constants c1 and c2 such that, (1) every metric (X,d) which satisfies the e-4PC can be embedded into a tree with distortion (1+e)c1log|X|, and (2) for every e ∈: [0,1] and any number of nodes, there is a metric space (X,d) satisfying the e-4PC that does not embed into a tree with distortion less than (1+e)c2log|X|. In addition, we prove a lower bound on approximate distance labelings of e-4PC metrics, and give tight bounds for tree embeddings with additive error guarantees. δ-Hyperbolic metric spaces have been defined by M. Gromov in 1987 via a simple 4-point condition: for any four points u,v,w,x, the two larger of the distance sums d(u,v)+d(w,x),d(u,w)+d(v,x),d(u,x)+d(v,w) differ by at most 2δ. They play an important role in geometric group theory, geometry of negatively curved spaces, and have recently become of interest in several domains of computer science, including algorithms and networking. In this paper, we study unweighted δ-hyperbolic graphs. Using the Layering Partition technique, we show that every n-vertex δ-hyperbolic graph with δ≥1 2 has an additive O(δlog n)-spanner with at most O(δn) edges and provide a simpler, in our opinion, and faster construction of distance approximating trees of δ-hyperbolic graphs with an additive error O(δlog n). The construction of our tree takes only linear time in the size of the input graph. As a consequence, we show that the family of n-vertex δ-hyperbolic graphs with δ≥1 2 admits a routing labeling scheme with O(δlog 2 n) bit labels, O(δlog n) additive stretch and O(log 2(4δ)) time routing protocol, and a distance labeling scheme with O(log 2 n) bit labels, O(δlog n) additive error and constant time distance decoder. A graph G is δ-hyperbolic if for any four vertices u,v,x,y of G the two larger of the three distance sums dG(u,v) + dG(x,y), dG(u,x) + dG(v,y), dG(u,y) + dG(v,x) differ by at most δ, and the smallest δ ≥ 0 for which G is δ-hyperbolic is called the hyperbolicity of G. In this paper, we construct a distance labeling scheme for bounded hyperbolicity graphs, that is a vertex labeling such that the distance between any two vertices of G can be estimated from their labels, without any other source of information. More precisely, our scheme assigns labels of O(log2n) bits for bounded hyperbolicity graphs with n vertices such that distances can be approximated within an additive error of O(log n). The label length is optimal for every additive error up to ne. We also show a lower bound of Ω(log log n) on the approximation factor, namely every s-multiplicative approximate distance labeling scheme on bounded hyperbolicity graphs with polylogarithmic labels requires s = Ω(log log n). Let G= (V, E) be a connected graph endowed with the standard graph-metric dGand in which longest induced simple cycle has length? (G). We prove that there exists a tree T= (V,F ) such that| dG(u, v) ?dT(u, v)| ? ??(G)2? +?for all vertices u, v?V, where?= 1 if ?(G) ?= 4, 5 and ?= 2 otherwise. The case ?(G) = 3 (i.e., G is a chordal graph) has been considered in Brandstadt, Chepoi, and Dragan, (1999) J.Algorithms 30. The proof contains an efficient algorithm for determining such a treeT . δ-Hyperbolic metric spaces have been defined by M. Gromov via a simple 4-point condition: for any four points u,v,w,x, the two larger of the sums d(u,v)+d(w,x), d(u,w)+d(v,x), d(u,x)+d(v,w) differ by at most 2δ. Given a finite set S of points of a δ-hyperbolic space, we present simple and fast methods for approximating the diameter of S with an additive error 2δ and computing an approximate radius and center of a smallest enclosing ball for S with an additive error 3δ. These algorithms run in linear time for classical hyperbolic spaces and for δ-hyperbolic graphs and networks. Furthermore, we show that for δ-hyperbolic graphs G=(V,E) with uniformly bounded degrees of vertices, the exact center of S can be computed in linear time O(|E|). We also provide a simple construction of distance approximating trees of δ-hyperbolic graphs G on n vertices with an additive error O(δlog2 n). This construction has an additive error comparable with that given by Gromov for n-point δ-hyperbolic spaces, but can be implemented in O(|E|) time (instead of O(n2)). Finally, we establish that several geometrical classes of graphs have bounded hyperbolicity.