111 research outputs found
Random Regular Graphs are not Asymptotically Gromov Hyperbolic
In this paper we prove that random --regular graphs with have
traffic congestion of the order where is the number
of nodes and geodesic routing is used. We also show that these graphs are not
asymptotically --hyperbolic for any non--negative almost
surely as .Comment: 6 pages, 2 figure
Traffic Analysis in Random Delaunay Tessellations and Other Graphs
In this work we study the degree distribution, the maximum vertex and edge
flow in non-uniform random Delaunay triangulations when geodesic routing is
used. We also investigate the vertex and edge flow in Erd\"os-Renyi random
graphs, geometric random graphs, expanders and random -regular graphs.
Moreover we show that adding a random matching to the original graph can
considerably reduced the maximum vertex flow.Comment: Submitted to the Journal of Discrete Computational Geometr
Symmetry in Graph Theory
This book contains the successful invited submissions to a Special Issue of Symmetry on the subject of ""Graph Theory"". Although symmetry has always played an important role in Graph Theory, in recent years, this role has increased significantly in several branches of this field, including but not limited to Gromov hyperbolic graphs, the metric dimension of graphs, domination theory, and topological indices. This Special Issue includes contributions addressing new results on these topics, both from a theoretical and an applied point of view
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 Ingeniería MatemáticaPresidente: Domingo de Guzmán Pestana Galván.- Secretario: Ana María Portilla Ferreira.- Vocal: Eva Touris Loj
Euclidean versus hyperbolic congestion in idealized versus experimental networks
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
independent of the radius R whereas, in a Euclidean ball, the same proportion
scales as 1/R^{n-1}. This discrepancy persists for the traffic load, which at
the center of the hyperbolic ball scales as the square of the volume, whereas
the same traffic load scales as the volume to the power (n+1)/n 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 data base and synthetic
Euclidean lattice networks. It is further conjectured that for networks that do
not enjoy the obvious symmetry of hyperbolic and Euclidean balls, the point of
maximum traffic is near the center of mass of the network.Comment: 23 pages, 4 figure
Report on BCTCS 2016: The 32nd British Colloquium for Theoretical Computer Science 22–24 March 2016, Queen’s University Belfast
Report on BCTCS 2016: The 32nd British Colloquium for Theoretical Computer Science 22–24 March 2016, Queen’s University Belfas
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