All Scale-free networks are sparse

CI del Genio, T Gross, KE Bassler

Research output: Contribution to journalArticle (Academic Journal)peer-review

123 Citations (Scopus)


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.
Translated title of the contributionAll Scale-free networks are sparse
Original languageEnglish
Article number178701
Number of pages4
JournalPhysical Review Letters
Publication statusPublished - 2011

Structured keywords

  • Engineering Mathematics Research Group


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