Journal of Statistical Physics

, Volume 173, Issue 3–4, pp 806–844 | Cite as

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution

  • Pim van der HoornEmail author
  • Gabor Lippner
  • Dmitri Krioukov


Even though power-law or close-to-power-law degree distributions are ubiquitously observed in a great variety of large real networks, the mathematically satisfactory treatment of random power-law graphs satisfying basic statistical requirements of realism is still lacking. These requirements are: sparsity, exchangeability, projectivity, and unbiasedness. The last requirement states that entropy of the graph ensemble must be maximized under the degree distribution constraints. Here we prove that the hypersoft configuration model, belonging to the class of random graphs with latent hyperparameters, also known as inhomogeneous random graphs or W-random graphs, is an ensemble of random power-law graphs that are sparse, unbiased, and either exchangeable or projective. The proof of their unbiasedness relies on generalized graphons, and on mapping the problem of maximization of the normalized Gibbs entropy of a random graph ensemble, to the graphon entropy maximization problem, showing that the two entropies converge to each other in the large-graph limit.


Sparse random graphs Power-law degree distributions Maximum-entropy graphs 

Mathematics Subject Classification

05C80 05C82 54C70 



This work was supported by the ARO Grant No. W911NF-16-1-0391 and by the NSF Grant No. CNS-1442999.


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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of PhysicsNortheastern UniversityBostonUSA
  2. 2.Department of MathematicsNortheastern UniversityBostonUSA
  3. 3.Departments of Physics, Mathematics, and Electrical & Computer EngineeringNortheastern UniversityBostonUSA

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