Preferential Attachment Random Graphs with Edge-Step Functions

  • Caio Alves
  • Rodrigo RibeiroEmail author
  • Rémy Sanchis


We analyze a random graph model with preferential attachment rule and edge-step functions that govern the growth rate of the vertex set, and study the effect of these functions on the empirical degree distribution of these random graphs. More specifically, we prove that when the edge-step function f is a monotone regularly varying function at infinity, the degree sequence of graphs associated with it obeys a (generalized) power-law distribution whose exponent belongs to (1, 2] and is related to the index of regular variation of f at infinity whenever said index is greater than \(-1\). When the regular variation index is less than or equal to \(-1\), we show that the empirical degree distribution vanishes for any fixed degree.


Complex networks Preferential attachment Concentration bounds Power-law Scale-free Karamata’s theory Regularly varying functions 

Mathematics Subject Classification (2010)

Primary 05C82 Secondary 60K40 68R10 



C. A. was supported by the Deutsche Forschungsgemeinschaft (DFG). R. R. was partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq). R.S. has been partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) and by FAPEMIG (Programa Pesquisador Mineiro), Grant PPM 00600/16.


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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of LeipzigLeipzigGermany
  2. 2.IMPA, Estrada Da. CastorinaRio de JaneiroBrazil
  3. 3.Departamento de MatemáticaUniversidade Federal de Minas GeraisBelo HorizonteBrazil

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