Preferential Attachment Random Graphs with Edge-Step Functions

Caio Alves; Rodrigo Ribeiro; Rémy Sanchis

doi:10.1007/s10959-019-00959-0

Preferential Attachment Random Graphs with Edge-Step Functions

Caio Alves, Rodrigo Ribeiro, Rémy Sanchis

Research output: Contribution to journal › Article › peer-review

7 Scopus citations

Abstract

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.

Original language	English
Pages (from-to)	438-476
Number of pages	39
Journal	Journal of Theoretical Probability
Volume	34
Issue number	1
DOIs	https://doi.org/10.1007/s10959-019-00959-0
State	Published - Mar 2021
Externally published	Yes

Funding

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

Keywords

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

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title = "Preferential Attachment Random Graphs with Edge-Step Functions",

abstract = "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.",

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Preferential Attachment Random Graphs with Edge-Step Functions

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