# Farey graphs as models for complex networks

@article{Zhang2011FareyGA, title={Farey graphs as models for complex networks}, author={Zhongzhi Zhang and Francesc Comellas}, journal={Theor. Comput. Sci.}, year={2011}, volume={412}, pages={865-875} }

Farey sequences of irreducible fractions between 0 and 1 can be related to graph constructions known as Farey graphs. These graphs were first introduced by Matula and Kornerup in 1979 and further studied by Colbourn in 1982, and they have many interesting properties: they are minimally 3-colorable, uniquely Hamiltonian, maximally outerplanar and perfect. In this paper, we introduce a simple generation method for a Farey graph family, and we study analytically relevant topological properties…

## Figures and Topics from this paper

## 36 Citations

Structure Properties of Generalized Farey graphs based on Dynamical Systems for Networks

- Medicine, MathematicsScientific Reports
- 2018

This work discusses here a category of graphs which are extension of the well-known Farey graphs and finds that the new models not only possess the properties of being small-world and highly clustered, but also possess the quality of being scale-free.

Counting spanning trees in a small-world Farey graph

- Mathematics, Physics
- 2012

The problem of spanning trees is closely related to various interesting problems in the area of statistical physics, but determining the number of spanning trees in general networks is…

Counting spanning trees of a type of generalized Farey graphs

- Mathematics
- 2020

Abstract The Farey graph F n is derived from the famous Farey sequence and it is a small-world network with a connectivity distribution decaying exponentially. By using the Matrix-Tree theorem, Zhang…

Deterministic self-similar models of complex networks based on very symmetric graphs

- Mathematics
- 2013

Using very symmetric graphs we generalize several deterministic self-similar models of complex networks and we calculate the main network parameters of our generalization. More specifically, we…

Geometric Assortative Growth Model for Small-World Networks

- Mathematics, MedicineTheScientificWorldJournal
- 2014

This paper proposes a geometrically growing model for small-world networks that displays both tunable small- world phenomenon and tunable assortativity and obtains analytical solutions of relevant topological properties.

Corona graphs as a model of small-world networks

- Mathematics, Computer Science
- 2015

The recursive corona graphs are introduced as a model of small-world networks and the critical characteristics of the model, including order and size, degree distribution, average path length, clustering coefficient, and the number of spanning trees are investigated.

Vertex Labeling and Routing for Farey-Type Symmetrically-Structured Graphs

- Computer Science, MathematicsSymmetry
- 2018

This work proposes a label-based routing protocol for Farey-type models that should help contribute toward the understanding of several physical dynamic processes.

A NETWORK MODEL GENERATED FROM THE RECURSIVE GRAPH BASED ON POLYGON

- Mathematics
- 2013

The study of network models is one of the most challenging research fields among the studies of complex networks, which have been the hot research topics in recent decades. In this paper, we…

Extended corona product as an exactly tractable model for weighted heterogeneous networks

- Computer ScienceComput. J.
- 2018

A corona product of two weighted graphs is developed, based on which and an observed updating mechanism of edge weight in real networks, a minimal generative model for inhomogeneous weighted networks is proposed, allowing to analytically treat its structural and dynamical properties.

Distinct Clusterings and Characteristic Path Lengths in Dynamic Small-World Networks with Identical Limit Degree Distribution

- Mathematics
- 2012

Many real-world networks belong to a particular class of structures, known as small-world networks, that display short distance between pair of nodes. In this paper, we introduce a simple family of…

## References

SHOWING 1-10 OF 47 REFERENCES

Pseudofractal scale-free web.

- Mathematics, MedicinePhysical review. E, Statistical, nonlinear, and soft matter physics
- 2002

It is found that scale-free random networks are excellently modeled by simple deterministic graphs and exactly and numerically with high precision all main characteristics of the graph are found.

Apollonian networks: simultaneously scale-free, small world, euclidean, space filling, and with matching graphs.

- Mathematics, MedicinePhysical review letters
- 2005

A new family of networks are introduced that are simultaneously scale-free, small-world, Euclidean, space filling, and with matching graphs, that could be applied to the geometry of fully fragmented porous media, hierarchical road systems, and area-covering electrical supply networks.

Farey Series and Maximal Outerplanar Graphs

- Mathematics
- 1982

Certain graphs representing Farey series of irreducible fractions are shown to be maximal outerplanar. For a suitable generalization of Farey series, the class of graphs obtained is exactly the class…

Random graph models of social networks

- Computer Science, MedicineProceedings of the National Academy of Sciences of the United States of America
- 2002

It is found that in some cases, the models are in remarkable agreement with the data, whereas in others the agreement is poorer, perhaps indicating the presence of additional social structure in the network that is not captured by the random graph.

Information Theory of Complex Networks: On Evolution and Architectural Constraints

- Physics
- 2004

Complex networks are characterized by highly heterogeneous distributions of links, often pervading the presence of key properties such as robustness under node removal. Several correlation measures…

Recursive graphs with small-world scale-free properties.

- Mathematics, MedicinePhysical review. E, Statistical, nonlinear, and soft matter physics
- 2004

We discuss a category of graphs, recursive clique trees, which have small-world and scale-free properties and allow a fine tuning of the clustering and the power-law exponent of their discrete degree…

Geometric fractal growth model for scale-free networks.

- Mathematics, MedicinePhysical review. E, Statistical, nonlinear, and soft matter physics
- 2002

A deterministic model for scale-free networks, whose degree distribution follows a power law with the exponent gamma, and the case that the number of offspring is the same for all vertices, finds that the degree distribution exhibits an exponential-decay behavior.

Graph Classes: A Survey

- Mathematics
- 1987

Preface 1. Basic Concepts 2. Perfection, Generalized Perfection, and Related Concepts 3. Cycles, Chords and Bridges 4. Models and Interactions 5. Vertex and Edge Orderings 6. Posets 7. Forbidden…

Emergence of scaling in random networks

- Computer Science, PhysicsScience
- 1999

A model based on these two ingredients reproduces the observed stationary scale-free distributions, which indicates that the development of large networks is governed by robust self-organizing phenomena that go beyond the particulars of the individual systems.

Hierarchical organization in complex networks.

- Computer Science, PhysicsPhysical review. E, Statistical, nonlinear, and soft matter physics
- 2003

It is found that several real networks, such as the Worldwideweb, actor network, the Internet at the domain level, and the semantic web obey this scaling law, indicating that hierarchy is a fundamental characteristic of many complex systems.