Preferential Attachment Random Graphs with Edge-Step Functions
Abstract
We propose a random graph model with preferential attachment rule and \emph{edge-step functions} that govern the growth rate of the vertex set. We 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 is a \emph{monotone regularly varying function} at infinity, the sequence of graphs associated to it obeys a power-law degree distribution whose exponent is related to the index of regular variation of at infinity whenever said index is greater than . When the regularly variation index is less than or equal to , we show that the proportion of vertices with degree smaller than any given constant goes to a. s..
Cite
@article{arxiv.1704.08276,
title = {Preferential Attachment Random Graphs with Edge-Step Functions},
author = {Caio Alves and Rodrigo Ribeiro and Remy Sanchis},
journal= {arXiv preprint arXiv:1704.08276},
year = {2019}
}
Comments
We greatly expanded the scope and generality of all the results, and, as a consequence, we needed to devote many more pages for the study of the Power-law empirical distribution. For readability issues and for the sake of better exposition, we decided to focus only on the results regarding the characterization of the empirical degree distribution in this paper