# Core congestion is inherent in hyperbolic networks

@article{Chepoi2017CoreCI, title={Core congestion is inherent in hyperbolic networks}, author={Victor Chepoi and Feodor F. Dragan and Yann Vax{\`e}s}, journal={ArXiv}, year={2017}, volume={abs/1605.03059} }

We investigate the impact the negative curvature has on the traffic congestion in large-scale networks. We prove that every Gromov hyperbolic network $G$ admits a core, thus answering in the positive a conjecture by Jonckheere, Lou, Bonahon, and Baryshnikov, Internet Mathematics, 7 (2011) which is based on the experimental observation by Narayan and Saniee, Physical Review E, 84 (2011) that real-world networks with small hyperbolicity have a core congestion. Namely, we prove that for every…

## 27 Citations

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This paper provides a simple factor 8 approximation algorithm for computing the hyperbolicity of an $n$-vertex graph $G=(V,E)$ in optimal time $O(n^2)$ (assuming that the input is the distance matrix of the graph), and leads to constant factor approximations of other graph-parameters related to hyperBolicity (thinness, slimness, and insize).

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In this paper, we characterize the hyperbolic product graphs for the Cartesian sum $$G_1\oplus G_2$$G1⊕G2: $$G_1\oplus G_2$$G1⊕G2 is always hyperbolic, unless either $$G_1$$G1 or $$G_2$$G2 is the…

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Several large scale networks, such as the backbone of the Internet, have been observed to behave like convex Riemannian manifolds of negative curvature. In particular, this paradigm explains the…

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The hyperbolicity constant of infinite circulant graphs

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Abstract If X is a geodesic metric space and x1, x2, x3 ∈ X, a geodesic triangle T = {x1, x2, x3} is the union of the three geodesics [x1x2], [x2x3] and [x3x1] in X. The space X is δ-hyperbolic (in…

## References

SHOWING 1-10 OF 51 REFERENCES

Euclidean versus Hyperbolic Congestion in Idealized versus Experimental Networks

- MathematicsInternet Math.
- 2011

A theoretical justification of the experimental exponent discrepancy observed by Narayan and Saniee between traffic loads in Gromov-hyperbolic networks from the Rocketfuel database and synthetic Euclidean lattice networks is provided.

On the Hyperbolicity of Large-Scale Networks

- Computer ScienceArXiv
- 2013

Observations indicate that -hyperbolicity is a common feature of large-scale networks, from IP-layer connectivity to citation, collaboration, co-authorship, and friendship graphs, and in conjunction with other local characteristics of networks, such as the degree distribution and clustering coecients are proposed.

On the Hyperbolicity of Small-World and Tree-Like Random Graphs

- Mathematics, Computer ScienceISAAC
- 2012

This study provides one of the first significant analytical results on the hyperbolicity of a rich class of random graphs, which shed light on the relationship between hyperBolicity and navigability of random graph models, as well as on the sensitivity of hyperbolics {\delta} to noises in random graphs.

Hyperbolic Embedding of Internet Graph for Distance Estimation and Overlay Construction

- Computer ScienceIEEE/ACM Transactions on Networking
- 2008

It is found that if the curvature, that defines the extend of the bending, is selected in the adequate range, the accuracy of Internet distance embedding can be improved, and a new efficient centralized embedding algorithm is presented that enables the accurate embedding of short distances.

Cop and Robber Game and Hyperbolicity

- MathematicsSIAM J. Discret. Math.
- 2014

It is proved that all cop-win graphs in the game in which the robber and the cop move at different speeds are established, which establishes a new---game-theoretical---characterization of Gromov hyperbolicity.

Packing and Covering delta -Hyperbolic Spaces by Balls

- Mathematics, Computer ScienceAPPROX-RANDOM
- 2007

It is shown how to construct in polynomial time a covering of S with at most Δ(S,R)balls of (slightly larger) radius R+ Δ, which is established in the general framework ofΔ-hyperbolic geodesic metric spaces and is extended to some other set families derived from balls.

Injective hulls of certain discrete metric spaces and groups

- Mathematics
- 2011

Injective metric spaces, or absolute 1-Lipschitz retracts, share a number of properties with CAT(0) spaces. In the 1960es, J. R. Isbell showed that every metric space X has an injective hull E(X).…

The Average Distance in a Random Graph with Given Expected Degrees

- Mathematics, Computer ScienceInternet Math.
- 2003

It is shown that for certain families of random graphs with given expected degrees, the average distance is almost surely of order log n/ logd̃ where d̃ is the weighted average of the sum of squares of the expected degrees.

Metric tree‐like structures in real‐world networks: an empirical study

- Computer ScienceNetworks
- 2016

This work presents strong evidence that a number of real‐world networks, taken from different domains, exhibit tree‐like structures from a metric point of view, and investigates a few graph parameters, namely, the tree‐distortion and the tree-stretch, theTree‐length and the Tree‐breadth, Gromov's hyperbolicity, the cluster‐diameter and the cluster-radius in a layering partition of a graph.

Treewidth and Hyperbolicity of the Internet

- Computer Science2011 IEEE 10th International Symposium on Network Computing and Applications
- 2011

The tree width of the Internet appears to be quite large and being far from a tree with that respect, reflecting some high degree of connectivity, which proves the existence of a well linked core in the Internet.