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.
More generally, we see two approaches connecting hyperbolicity with efficient routing in graphs. One approach study efficient computation of graph properties, such as diameters, centers, approximating trees, and packings and coverings for low hyperbolic- @math graphs and metric spaces @cite_16 @cite_0 @cite_34 @cite_32 @cite_30 . In large part, the reason for this interest is that there are often direct consequences for navigation and routing in these graphs @cite_34 @cite_32 @cite_38 @cite_33 . While these results are of interest for general low hyperbolic- @math graphs, they can be less interesting when applied to small-world and other low-diameter random models of complex networks. To take one example, @cite_16 provides a simple construction of a distance approximating tree for @math -hyperbolic graphs on @math vertices; but the @math additive-error guarantee is clearly less interesting for models in which the diameter of the graph is @math . Unfortunately, this @math arises for a very natural reason in the analysis, and it is nontrivial to improve it for popular tree-like complex network models.
@cite_30 @cite_38 @cite_33 @cite_32 @cite_34 @cite_0 @cite_16
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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. We present the H3 layout technique for drawing large directed graphs as node-link diagrams in 3D hyperbolic space. We can lay out much larger structures than can be handled using traditional techniques for drawing general graphs because we assume a hierarchical nature of the data. We impose a hierarchy on the graph by using domain-specific knowledge to find an appropriate spanning tree. Links which are not part of the spanning tree do not influence the layout but can be selectively drawn by user request. The volume of hyperbolic 3-space increases exponentially, as opposed to the familiar geometric increase of euclidean 3-space. We exploit this exponential amount of room by computing the layout according to the hyperbolic metric. We optimize the cone tree layout algorithm for 3D hyperbolic space by placing children on a hemisphere around the cone mouth instead of on its perimeter. Hyperbolic navigation affords a Focus+Context view of the structure with minimal visual clutter. We have successfully laid out hierarchies of over 20,000 nodes. Our implementation accommodates navigation through graphs too large to be rendered interactively by allowing the user to explicitly prune or expand subtrees. Abstract This paper proposes a mathematical justification of the phenomenon of extreme congestion at a very limited number of nodes in very large networks. It is argued that this phenomenon occurs as a combination of the negative curvature property of the network together with minimum-length routing. More specifically, it is shown that in a large n-dimensional hyperbolic ball B of radius R viewed as a roughly similar model of a Gromov hyperbolic network, the proportion of traffic paths transiting through a small ball near the center is Θ(1), whereas in a Euclidean ball, the same proportion scales as Θ(1 R n−1). This discrepancy persists for the traffic load, which at the center of the hyperbolic ball scales as volume2(B), whereas the same traffic load scales as volume1+1 n (B) in the Euclidean ball. This provides a theoretical justification of the experimental exponent discrepancy observed by Narayan and Saniee between traffic loads in Gromov-hyperbolic networks from the Rocketfuel database and synthetic ... A prominent tool in many problems involving metric spaces is a notion of randomized low-diameter decomposition. Loosely speaking, ( )-decomposition refers to a probability distribution over partitions of the metric into sets of low diameter, such that nearby points (parameterized by ( >0 )) are likely to be “clustered” together. Applying this notion to the shortest-path metric in edge-weighted graphs, it is known that n-vertex graphs admit an (O( n) )-padded decomposition (Bartal, 37th annual symposium on foundations of computer science. IEEE, pp 184–193, 1996), and that excluded-minor graphs admit O(1)-padded decomposition (, 25th annual ACM symposium on theory of computing, pp 682–690, 1993; Fakcharoenphol and Talwar, J Comput Syst Sci 69(3), 485–497, 2004; , Proceedings of the 46th annual ACM symposium on theory of computing. STOC ’14, pp 79–88. ACM, New York, NY, USA, 2014). We design decompositions to the family of p-path-separable graphs, which was defined by Abraham and Gavoille (Proceedings of the twenty-fifth annual acm symposium on principles of distributed computing, PODC ’06, pp 188–197, 2006) and refers to graphs that admit vertex-separators consisting of at most p shortest paths in the graph. Our main result is that every p-path-separable n-vertex graph admits an (O( (p n)) )-decomposition, which refines the (O( n) ) bound for general graphs, and provides new bounds for families like bounded-treewidth graphs. Technically, our clustering process differs from previous ones by working in (the shortest-path metric of) carefully chosen subgraphs.