Contrôle du trafic sur les réseaux à géométrie hyperbolique. Vers une théorie géométrique de la sécurité de l'acheminement de l'information

Jonckheere, Edmond; Ray, Priyadip

doi:10.3166/jesa.37.145-159

Cited by 37 publications

(54 citation statements)

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“…The concept of hyperbolicity appears also in discrete mathematics, in particular, a number of algorithmic problems in hyperbolic spaces and hyperbolic graphs has been considered in recent papers [12,13,15,28]. Another application of these spaces is secure transmission of information on the internet [21][22][23], playing a significant role in the spread of viruses through the network [21,23]. It has been shown empirically in [42] that the internet topology embeds with better accuracy into a hyperbolic space than into an Euclidean space of comparable dimension.…”

Section: Introductionmentioning

confidence: 99%

Gromov hyperbolicity of planar graphs

et al. 2013

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We prove that under appropriate assumptions adding or removing an infinite amount of edges to a given planar graph preserves its non-hyperbolicity, a result which is shown to be false in general. In particular, we make a conjecture that every tessellation graph of R 2 with convex tiles is non-hyperbolic; it is shown that in order to prove this conjecture it suffices to consider tessellation graphs of R 2 such that every tile is a triangle and a partial answer to this question is given. A weaker version of this conjecture stating that every tessellation graph of R 2 with rectangular tiles is non-hyperbolic is given and partially answered. If this conjecture were true, many tessellation graphs of R 2 with tiles which are parallelograms would be non-hyperbolic. MSC:05C10, 05C63, 05C75, 05A20

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Section: Introductionmentioning

confidence: 99%

Gromov hyperbolicity of planar graphs

et al. 2013

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“…For example, it has been shown empirically in [13] that the Internet topology embeds with better accuracy into a hyperbolic space than into a Euclidean space of comparable dimension (formal proofs that the distortion is related to the hyperbolicity can be found in [14]); furthermore, it is evidenced that many real networks are hyperbolic (see, e.g., [15][16][17][18][19]). Recently, among the practical network applications, hyperbolic spaces were used to study secure transmission of information on the Internet or the way viruses are spread through the network (see [20,21]); also to traffic flow and effective resistance of networks [22][23][24].…”

Section: Introductionmentioning

confidence: 99%

“…(see, for example, [6][7][8][9]14,[18][19][20][21][25][26][27][28][29][30][31][32][33][34][35]). x, y ∈ J(G) \ V(G), z ∈ J(G), …”

Section: Introductionmentioning

confidence: 99%

Gromov Hyperbolicity in Mycielskian Graphs

et al. 2017

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Since the characterization of Gromov hyperbolic graphs seems a too ambitious task, there are many papers studying the hyperbolicity of several classes of graphs. In this paper, it is proven that every Mycielskian graph G M is hyperbolic and that δ(G M ) is comparable to diam(G M ). Furthermore, we study the extremal problems of finding the smallest and largest hyperbolicity constants of such graphs; in fact, it is shown that 5/4 ≤ δ(G M ) ≤ 5/2. Graphs G whose Mycielskian have hyperbolicity constant 5/4 or 5/2 are characterized. The hyperbolicity constants of the Mycielskian of path, cycle, complete and complete bipartite graphs are calculated explicitly. Finally, information on δ(G) just in terms of δ(G M ) is obtained.

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“…The study of mathematical properties of Gromov hyperbolic spaces and its applications is a topic of recent and increasing interest in graph theory; see, for instance [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18].…”

Section: Introductionmentioning

confidence: 99%

“…A few algorithmic problems in hyperbolic spaces and hyperbolic graphs have been considered in recent papers (see [21,22,23,24]). Another interesting application of these spaces is secure transmission of information on the internet (see [9,10,11]). In particular, the hyperbolicity plays a key role in the spread of viruses through the network (see [10,11]).…”

Section: Introductionmentioning

confidence: 99%