## Abstract

We discuss how various models of scale-free complex networks approach their limiting properties when the size N of the network grows. We focus mainly on equilibrated networks and their finite-size degree distributions. Our results show that the position of the cutoff in the degree distribution, k(cutoff), scales with N in a different way than predicted for N ->infinity; that is, subleading corrections to the scaling k(cutoff)similar to N-alpha are strong even for networks of order N similar to 10(9) nodes. We observe also a logarithmic correction to the scaling for degenerated graphs with the degree distribution pi(k)similar to k(-3). On the other hand, the distribution of the maximal degree k(max) may have a different scaling than the cutoff and, moreover, it approaches the thermodynamic limit much faster. We argue that k(max)similar to N-alpha' with an exponent alpha(')=min[alpha,1/(gamma-1)], where gamma is the exponent in the power law pi(k)similar to k(-gamma). We also present some results on the cutoff function and the distribution of the maximal degree in equilibrated networks.

Original language | English |
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Article number | 061125 |

Pages (from-to) | - |

Number of pages | 9 |

Journal | Physical Review E - Statistical, Nonlinear and Soft Matter Physics |

Volume | 78 |

Issue number | 6 |

DOIs | |

Publication status | Published - Dec 2008 |

## Keywords

- complex networks
- thermodynamics
- STATISTICAL-MECHANICS
- RANDOM GRAPHS
- GROWING NETWORKS
- CONDENSATION
- MODEL