3,037 research outputs found
Graphs with small hyperbolicity constant and hyperbolic minor graphs
Hyperbolic spaces, defined by Gromov in, play an important role in geometric group theory and in the geometry of negatively curved spaces. The concept
of Gromov hyperbolicity grasps the essence of negatively curved spaces like the classical hyperbolic space, Riemannian manifolds of negative sectional curvature bounded away from 0, and of discrete spaces like trees and the Cayley graphs of many finitely generated groups. It is remarkable that a simple concept leads to such a rich general theory.
The first works on Gromov hyperbolic spaces deal with finitely generated groups. Initially, Gromov spaces were applied to the study of automatic groups in the science of
computation; indeed, hyperbolic groups are strongly geodesically automatic,
i.e., there is an automatic structure on the group.
The concept of hyperbolicity appears also in discrete mathematics, algorithms and networking.
For example, it has been shown empirically in that the internet topology
embeds with better accuracy into a hyperbolic space than into an Euclidean space of comparable
dimension; furthermore, it is evidenced that many real networks are hyperbolic. A few algorithmic problems in hyperbolic spaces and hyperbolic graphs
have been considered in recent papers. Another important application
of these spaces is the study of the spread of viruses through the internet. Furthermore,
hyperbolic spaces are useful in secure transmission of information on the network. The hyperbolicity has also been used extensively in the context of
random graphs. For example, it was shown that several types of small-world networks and networks with given expected degrees are not hyperbolic
in some sense.
The study of Gromov hyperbolic graphs is a subject of increasing interest in graph theory; and the references therein.
In our study on the hyperbolicity in graphs we use the notations (...)Programa Oficial de Doctorado en IngenierÃa MatemáticaPresidente: Domingo de Guzmán Pestana Galván.- Secretario: Ana MarÃa Portilla Ferreira.- Vocal: Eva Touris Loj
Hyperbolicity Measures "Democracy" in Real-World Networks
We analyze the hyperbolicity of real-world networks, a geometric quantity
that measures if a space is negatively curved. In our interpretation, a network
with small hyperbolicity is "aristocratic", because it contains a small set of
vertices involved in many shortest paths, so that few elements "connect" the
systems, while a network with large hyperbolicity has a more "democratic"
structure with a larger number of crucial elements.
We prove mathematically the soundness of this interpretation, and we derive
its consequences by analyzing a large dataset of real-world networks. We
confirm and improve previous results on hyperbolicity, and we analyze them in
the light of our interpretation.
Moreover, we study (for the first time in our knowledge) the hyperbolicity of
the neighborhood of a given vertex. This allows to define an "influence area"
for the vertices in the graph. We show that the influence area of the highest
degree vertex is small in what we define "local" networks, like most social or
peer-to-peer networks. On the other hand, if the network is built in order to
reach a "global" goal, as in metabolic networks or autonomous system networks,
the influence area is much larger, and it can contain up to half the vertices
in the graph. In conclusion, our newly introduced approach allows to
distinguish the topology and the structure of various complex networks
On the hyperbolicity of random graphs
Let be a connected graph with the usual (graph) distance metric
. Introduced by Gromov, is
-hyperbolic if for every four vertices , the two largest
values of the three sums differ
by at most . In this paper, we determinate the value of this
hyperbolicity for most binomial random graphs.Comment: 20 page
Lack of Hyperbolicity in Asymptotic Erd\"os--Renyi Sparse Random Graphs
In this work we prove that the giant component of the Erd\"os--Renyi random
graph for c a constant greater than 1 (sparse regime), is not Gromov
-hyperbolic for any positive with probability tending to one
as . As a corollary we provide an alternative proof that the giant
component of when c>1 has zero spectral gap almost surely as
.Comment: Updated version with improved results and narrativ
Hierarchical hyperbolicity of graphs of multicurves
We show that many graphs naturally associated to a connected, compact,
orientable surface are hierarchically hyperbolic spaces in the sense of
Behrstock, Hagen and Sisto. They also automatically have the coarse median
property defined by Bowditch. Consequences for such graphs include a distance
formula analogous to Masur and Minsky's distance formula for the mapping class
group, an upper bound on the maximal dimension of quasiflats, and the existence
of a quadratic isoperimetric inequality. The hierarchically hyperbolic
structure also gives rise to a simple criterion for when such graphs are Gromov
hyperbolic.Comment: 27 pages, 4 figures. Minor changes from previous version. Addition of
appendix describing a hierarchically hyperbolic structure on the arc grap
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