# Random Hyperbolic Graphs: Degree Sequence and Clustering

@article{Gugelmann2012RandomHG, title={Random Hyperbolic Graphs: Degree Sequence and Clustering}, author={Luca Gugelmann and K. Panagiotou and Ueli Peter}, journal={ArXiv}, year={2012}, volume={abs/1205.1470} }

In the last decades, the study of models for large real-world networks has been a very popular and active area of research. A reasonable model should not only replicate all the structural properties that are observed in real world networks (for example, heavy tailed degree distributions, high clustering and small diameter), but it should also be amenable to mathematical analysis. There are plenty of models that succeed in the first task but are hard to analyze rigorously. On the other hand, a… Expand

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#### References

SHOWING 1-10 OF 31 REFERENCES

Mathematical results on scale‐free random graphs

- Computer Science
- 2005

There has been much interest in studying large-scale real-world networks and attempting to model their properties using random graphs, and the work in this field falls very roughly into the following categories. Expand

Popularity based random graph models leading to a scale-free degree sequence

- Computer Science, Mathematics
- Discret. Math.
- 2004

Confirming non-rigorous arguments of Dorogovtsev et al. and Drinea et al., this shows that for such a, the proportion P(d) of vertices of degree d almost surely obeys a power law, where P( d) is of the form d-^2^-^a for large d. Expand

The degree sequence of a scale-free random graph process

- Mathematics, Computer Science
- Random Struct. Algorithms
- 2001

Here the authors obtain P(d) asymptotically for all d≤n1/15, where n is the number of vertices, proving as a consequence that γ=3.9±0.1 is obtained. Expand

Affiliation networks

- Computer Science
- STOC '09
- 2009

This paper presents the first model that provides a simple, realistic, and mathematically tractable generative model that intrinsically explains all the well-known properties of the social networks, as well as densification and shrinking diameter. Expand

Graphs over time: densification laws, shrinking diameters and possible explanations

- Mathematics, Computer Science
- KDD '05
- 2005

A new graph generator is provided, based on a "forest fire" spreading process, that has a simple, intuitive justification, requires very few parameters (like the "flammability" of nodes), and produces graphs exhibiting the full range of properties observed both in prior work and in the present study. Expand

Random evolution in massive graphs

- Computer Science
- 2001

This paper gives three increasingly general directed graph models and one general undirected graph model for generating power law graphs by adding at most one node and possibly one or more edges at a time and describes a method for scaling the time in the evolution model such that the power law of the degree sequences remains invariant. Expand

Hyperbolic Geometry of Complex Networks

- Mathematics, Medicine
- Physical review. E, Statistical, nonlinear, and soft matter physics
- 2010

It is shown that targeted transport processes without global topology knowledge are maximally efficient, according to all efficiency measures, in networks with strongest heterogeneity and clustering, and that this efficiency is remarkably robust with respect to even catastrophic disturbances and damages to the network structure. Expand

Clustering in complex networks. I. General formalism.

- Mathematics, Physics
- Physical review. E, Statistical, nonlinear, and soft matter physics
- 2006

A key concept is introduced, the edge multiplicity, that measures the number of triangles passing through an edge that extends the clustering coefficient in that it involves the properties of two- and not just one-vertices. Expand

First to market is not everything: an analysis of preferential attachment with fitness

- Mathematics, Computer Science
- STOC '07
- 2007

A rigorous analysis of preferential attachment with fitness, suggested by Bianconi and Barabási and studied by Motwani and Xu, in which the degree of a vertex is scaled by its quality to determine its attractiveness. Expand

Collective dynamics of ‘small-world’ networks

- Computer Science, Medicine
- Nature
- 1998

Simple models of networks that can be tuned through this middle ground: regular networks ‘rewired’ to introduce increasing amounts of disorder are explored, finding that these systems can be highly clustered, like regular lattices, yet have small characteristic path lengths, like random graphs. Expand