BioND — Dynamics of Biological Networks

All scale-free networks are sparse

Charo I. del Genio, Thilo Gross, and Kevin E. Bassler
Physical Review Letters 107, 178701, 2011.
arXiv:1106.5150

Abstract

Phase Diagram

We study the graphicality of power-law distributed degree sequences, showing that the fraction of graphical sequences undergoes two sharp transitions at the values 0 and 2 of the power-law exponent. We characterize these transitions as first-order, and provide an analytic explanation of their nature. Further numerical calculations, based on extreme value arguments, verify this treatment, and introduce a method to determine transition points for any given degree distribution. Our results reveal a fundamental reason why scale-free networks with no constraints on minimum and maximum degree must be sparse for positive power-law exponents, and dense otherwise.

Media Coverage

Sitabhra Sinha, Physics, 2011-10-17
How nodes connect to each other may explain why we don’t see certain classes of networks.
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