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Growth Models for Networks

  • Reference work entry
Computational Complexity

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Article Outline

Glossary

Definition of the Subject

Introduction

Equilibrium and Nonequilibrium Random Networks

Random Recursive Graphs (Trees)

Recursive Trees Versus Equilibrium Connected Trees

The Barabási–Albert Model

General Preferential Attachment

Finite Size Effects

Hidden Variables

Condensation of Edges

Weighted Growing Networks

Connected Components in Growing Networks

Significance of Loops

Accelerated Growth of Networks

Critique of the Preferential Attachment Mechanism

Optimization-Based Models

Deterministic Graphs

Future Directions

Acknowledgments

Bibliography

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Abbreviations

Random network :

A random network is a statistical ensemble: a given set of graphs and their statistical weights – realization probabilities. Formulating a network model, one defines the full set of members of the ensemble and indicates corresponding statistical weights or rules generating these weights.

Nonequilibrium random network:

Nonequilibrium random networks are nonequilibrium statistical ensembles of graphs. A growing network is a very particular case of a nonequilibrium one, with a growing number of edges and vertices.

Recursive network:

A recursive network is a growing network, where new edges connect only new vertices to already existing ones. In recursive networks, no new edges emerge between already existing vertices.

Tree :

A tree is a graph without loops (cycles ). The absence of loops crucially simplifies the description of these networks.

Random recursive graph (tree ):

In graph theory, the random recursive graph is a tree growing in the following way: at each time step, a new vertex is attached to one of the existing vertices chosen with equal probability.

Degree :

Degree – a total number of connections of a vertex – is the simplest local property of a network. A vertex degree distribution is the probability that a vertex in a network has a given degree.

Heavy‐tailed degree distributions :

These are degree distributions with divergent higher moments in the infinite network limit. In particular, power-law distributions are heavy tailed.

Scale-free networks :

In many natural and artificial networks, a sufficiently wide region of a degree distribution may be approximately fitted by a power law. These networks are called scale-free.

Preferential attachment :

In the preferential attachment process, vertices for connection/attachment are chosen preferentially – usually, with probability proportional to a given function of their degree. The preferential attachment mechanism effectively generates networks with complex architectures including networks with heavy‐tailed degree distribution.

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Acknowledgments

The author thanks A. V. Goltsev, J. F. F. Mendes, and A. N. Samukhin for numerous discussions. This work waspartially supported by projects POCI: FAT/46241, MAT/46176, FIS/61665, and BIA-BCM/62662, and DYSONET.

Author information

Authors and Affiliations

  1. Departamento de Física, Universidade de Aveiro, Aveiro, Portugal

    Sergey N. Dorogovtsev

  2. A. F. Ioffe Physico-Technical Institute, St. Petersburg, Russia

    Sergey N. Dorogovtsev

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  1. Sergey N. Dorogovtsev
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Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Dorogovtsev, S.N. (2012). Growth Models for Networks. In: Meyers, R. (eds) Computational Complexity. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1800-9_95

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