54 research outputs found
Embedding of hyperbolic spaces in the product of trees
We show that for each n\ge 2 there is a quasi-isometric embedding of the hyperbolic space H^n in the product T^n=Tx...xT of n copies of a (simplicial) metric tree T. On the other hand, we prove that there is no quasi-isometric embedding H^2 --> TxR^m for any metric tree T and any m\ge
Connecting geodesics and security of configurations in compact locally symmetric spaces
A pair of points in a riemannian manifold makes a secure configuration if the
totality of geodesics connecting them can be blocked by a finite set. The
manifold is secure if every configuration is secure. We investigate the
security of compact, locally symmetric spaces.Comment: 27 pages, 2 figure
Hyperbolic Geometry of Complex Networks
We develop a geometric framework to study the structure and function of
complex networks. We assume that hyperbolic geometry underlies these networks,
and we show that with this assumption, heterogeneous degree distributions and
strong clustering in complex networks emerge naturally as simple reflections of
the negative curvature and metric property of the underlying hyperbolic
geometry. Conversely, we show that if a network has some metric structure, and
if the network degree distribution is heterogeneous, then the network has an
effective hyperbolic geometry underneath. We then establish a mapping between
our geometric framework and statistical mechanics of complex networks. This
mapping interprets edges in a network as non-interacting fermions whose
energies are hyperbolic distances between nodes, while the auxiliary fields
coupled to edges are linear functions of these energies or distances. The
geometric network ensemble subsumes the standard configuration model and
classical random graphs as two limiting cases with degenerate geometric
structures. Finally, we show that targeted transport processes without global
topology knowledge, made possible by our geometric framework, are maximally
efficient, according to all efficiency measures, in networks with strongest
heterogeneity and clustering, and that this efficiency is remarkably robust
with respect to even catastrophic disturbances and damages to the network
structure
The hyperbolic dimension of metric spaces
We introduce a new quasi-isometry invariant of metric spaces called the hyperbolic dimension, hypdim, which is a version of the Gromov's asymptotic dimension, asdim. The hyperbolic dimension is at most the asymptotic dimension, however, unlike the asymptotic dimension, the hyperbolic dimension of any Euclidean space R^n is zero (while asdim R^n=n.) This invariant possesses usual properties of dimension like monotonicity and product theorems. Our main result says that the hyperbolic dimension of any Gromov hyperbolic space X (with mild restrictions) is at least the topological dimension of the boundary at infinity plus 1. As an application we obtain that there is no quasi-isometric embedding of the real hyperbolic space H^n into the (n-1)-fold metric product of metric trees stabilized by any Euclidean factor
Geodesics avoiding open subsets in surfaces of negative curvature
We prove existence and non-existence results for geodesics avoiding -separated sets on a surface of negative curvature
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