All Scale-Free Networks Are Sparse

Del Genio, C. I.; Gross, T.; Bassler, K. E.

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All Scale-Free Networks Are Sparse

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Del Genio, C. I.
Max Planck Institute for the Physics of Complex Systems, Max Planck Society;

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Gross, T.
Max Planck Institute for the Physics of Complex Systems, Max Planck Society;

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Bassler, K. E.
Max Planck Institute for the Physics of Complex Systems, Max Planck Society;

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Del Genio, C. I., Gross, T., & Bassler, K. E. (2011). All Scale-Free Networks Are Sparse. Physical Review Letters, 107(17): 178701.

Cite as: https://hdl.handle.net/11858/00-001M-0000-0029-8BF1-1

Abstract

We study the realizability of scale-free networks with a given degree sequence, showing that the fraction of realizable sequences undergoes two first-order transitions at the values 0 and 2 of the power-law exponent. We substantiate this finding by analytical reasoning and by a numerical method, proposed here, based on extreme value arguments, which can be applied to any given degree distribution. Our results reveal a fundamental reason why large scale-free networks without constraints on minimum and maximum degree must be sparse.