# Hyperbolic Geometry of Complex Networks

@article{Krioukov2010HyperbolicGO, title={Hyperbolic Geometry of Complex Networks}, author={Dmitri V. Krioukov and Fragkiskos Papadopoulos and Maksim Kitsak and Amin Vahdat and Mari{\'a}n Bogu{\~n}{\'a}}, journal={Physical review. E, Statistical, nonlinear, and soft matter physics}, year={2010}, volume={82 3 Pt 2}, pages={ 036106 } }

We develop a geometric framework to study the structure and function of complex networks. We assume that hyperbolic geometry underlies these networks, and we show that with this assumption, heterogeneous degree distributions and strong clustering in complex networks emerge naturally as simple reflections of the negative curvature and metric property of the underlying hyperbolic geometry. Conversely, we show that if a network has some metric structure, and if the network degree distribution is…

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## References

SHOWING 1-10 OF 156 REFERENCES

On curvature and temperature of complex networks

- Computer SciencePhysical review. E, Statistical, nonlinear, and soft matter physics
- 2009

It is shown that heterogeneous degree distributions in observed scale-free topologies of complex networks can emerge as a consequence of the exponential expansion of hidden hyperbolic space and a remarkable congruency between the embedding and the model is found.

Entropy of network ensembles.

- Computer SciencePhysical review. E, Statistical, nonlinear, and soft matter physics
- 2009

The structural entropy is defined and evaluated, i.e., the entropy of the ensembles of undirected uncorrelated simple networks with given degree sequence, and a solution to the paradox is proposed by proving that scale-free degree distributions are the most likely degree distribution with the corresponding value of the structural entropy.

Self-similarity of complex networks and hidden metric spaces

- Computer SciencePhysical review letters
- 2008

These findings indicate that hidden geometries underlying these real networks are a plausible explanation for their observed topologies and, in particular, for their self-similarity with respect to the degree-based renormalization.

Statistical mechanics of networks.

- Computer SciencePhysical review. E, Statistical, nonlinear, and soft matter physics
- 2004

We study the family of network models derived by requiring the expected properties of a graph ensemble to match a given set of measurements of a real-world network, while maximizing the entropy of…

Community structure in social and biological networks

- Computer ScienceProceedings of the National Academy of Sciences of the United States of America
- 2002

This article proposes a method for detecting communities, built around the idea of using centrality indices to find community boundaries, and tests it on computer-generated and real-world graphs whose community structure is already known and finds that the method detects this known structure with high sensitivity and reliability.

Elements of Asymptotic Geometry

- Mathematics
- 2007

Asymptotic geometry is the study of metric spaces from a large scale point of view, where the local geometry does not come into play. An important class of model spaces are the hyperbolic spaces (in…

Entropy measures for networks: toward an information theory of complex topologies.

- Computer SciencePhysical review. E, Statistical, nonlinear, and soft matter physics
- 2009

The quantities introduced here will play a crucial role for the formulation of null models of networks through maximum-entropy arguments and will contribute to inference problems emerging in the field of complex networks.

Tuning clustering in random networks with arbitrary degree distributions.

- Computer SciencePhysical review. E, Statistical, nonlinear, and soft matter physics
- 2005

A generator of random networks where both the degree-dependent clustering coefficient and the degree distribution are tunable and an universal relation among clustering and degree-degree correlations for all networks is unveiled.

Hierarchy measures in complex networks.

- Computer SciencePhysical review letters
- 2004

This work proposes a simple dynamical process used to construct networks which are either maximally or minimally hierarchical, and shows the extent of topological hierarchy to smoothly decline with gamma, the exponent of a degree distribution.

Hierarchical structure and the prediction of missing links in networks

- Computer ScienceNature
- 2008

This work presents a general technique for inferring hierarchical structure from network data and shows that the existence of hierarchy can simultaneously explain and quantitatively reproduce many commonly observed topological properties of networks.