2008/07/31 by Bartłomiej Waclaw, B. Waclaw, L. Bogacz +2 · 1 citation
Mathematics · Physics and Astronomy · #Combinatorics #Complex Network Analysis Techniques #Complex network #Cutoff #Degree (music) #Degree distribution #Distribution (mathematics) #Exponent #Geometry #Graph theory and applications #Limit (mathematics) #Limiting #Logarithm #Logarithmic scale #Mathematical analysis #Mathematics #Physics #Position (finance) #Power law #Quantum mechanics #Scaling #Statistical physics #Statistics #Theoretical and Computational Physics #Thermodynamic limit #cond-mat.dis-nn #cond-mat.stat-mech
paper · pdf · doi:10.1103/physreve.78.061125
published as Phys. Rev. E 78, 061125 (2008) · 9 pages, 5 figures, minor changes, figures replaced
arxiv created 2008/12/09 · openalex publication_date 2008/12/23 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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, kcutoff, scales with N in a different way than predicted for N\ensuremath→\ensuremath∞; that is, subleading corrections to the scaling kcutoff\ensuremath∼N^\ensuremathα are strong even for networks of order N\ensuremath∼109 nodes. We observe also a logarithmic correction to the scaling for degenerated graphs with the degree distribution \ensuremathπ(k)\ensuremath∼k^\ensuremath-3. On the other hand, the distribution of the maximal degree kmax may have a different scaling than the cutoff and, moreover, it approaches the thermodynamic limit much faster. We argue that kmax\ensuremath∼N^\ensuremathα^\ensuremath' with an exponent \ensuremathα^\ensuremath'=min[\ensuremathα,1∕(\ensuremathγ\ensuremath-1)], where \ensuremathγ is the exponent in the power law \ensuremathπ(k)\ensuremath∼k^\ensuremath-\ensuremathγ. We also present some results on the cutoff function and the distribution of the maximal degree in equilibrated networks.