On the hyperbolicity of random graphs

Let $G=(V,E)$ be a connected graph with the usual (graph) distance metric $d:V \times V \to N \cup \{0 \}$. Introduced by Gromov, $G$ is $\delta$-hyperbolic if for every four vertices $u,v,x,y \in V$, the two largest values of the three sums $d(u,v)+d(x,y), d(u,x)+d(v,y), d(u,y)+d(v,x)$ differ by at most $2\delta$. In this paper, we determinate the value of this hyperbolicity for most binomial random graphs.Comment: 20 page

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

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

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 $G(n,c/n)$ for c a constant greater than 1 (sparse regime), is not Gromov $\delta$-hyperbolic for any positive $\delta$ with probability tending to one as $n\to\infty$. As a corollary we provide an alternative proof that the giant component of $G(n,c/n)$ when c>1 has zero spectral gap almost surely as $n\to\infty$.Comment: Updated version with improved results and narrativ

Random Regular Graphs are not Asymptotically Gromov Hyperbolic

In this paper we prove that random $d$--regular graphs with $d\geq 3$ have traffic congestion of the order $O(n\log_{d-1}^{3}(n))$ where $n$ is the number of nodes and geodesic routing is used. We also show that these graphs are not asymptotically $\delta$--hyperbolic for any non--negative $\delta$ almost surely as $n\to\infty$.Comment: 6 pages, 2 figure

Random groups arising as graph products

In this paper we study the hyperbolicity properties of a class of random groups arising as graph products associated to random graphs. Recall, that the construction of a graph product is a generalization of the constructions of right-angled Artin and Coxeter groups. We adopt the Erdos - Renyi model of a random graph and find precise threshold functions for the hyperbolicity (or relative hyperbolicity). We aslo study automorphism groups of right-angled Artin groups associated to random graphs. We show that with probability tending to one as $n\to \infty$, random right-angled Artin groups have finite outer automorphism groups, assuming that the probability parameter $p$ is constant and satisfies $0.2929 <p<1$