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Clustering in preferential attachment random graphs with edge-step

Part of: Special processes Discrete mathematics in relation to computer science Graph theory

Published online by Cambridge University Press:  22 November 2021

Caio Alves ,
Rodrigo Ribeiro Open the ORCID record for Rodrigo Ribeiro [Opens in a new window]  and
Rémy Sanchis
Caio Alves*
Affiliation:
University of Leipzig
Rodrigo Ribeiro*
Affiliation:
Pontificia Universidad Católica de Chile
Rémy Sanchis*
Affiliation:
Universidade Federal de Minas Gerais
*
*Postal address: Faculty of Mathematics and Computer Science, University of Leipzig, Germany. Email: caio.alves@math.uni-leipzig.de
**Postal address: Pontificia Universidad Católica de Chile, Mathematics, Santiago, Chile. Email: rbotelho@mat.uc.cl
***Postal address: Universidade Federal de Minas Gerais, Belo Horizonte, MG Brazil. Email: rsanchis@mat.ufmg.br
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Abstract

We prove concentration inequality results for geometric graph properties of an instance of the Cooper–Frieze [5] preferential attachment model with edge-steps. More precisely, we investigate a random graph model that at each time $t\in \mathbb{N}$ , with probability p adds a new vertex to the graph (a vertex-step occurs) or with probability $1-p$ an edge connecting two existent vertices is added (an edge-step occurs). We prove concentration results for the global clustering coefficient as well as the clique number. More formally, we prove that the global clustering, with high probability, decays as $t^{-\gamma(p)}$ for a positive function $\gamma$ of p, whereas the clique number of these graphs is, up to subpolynomially small factors, of order $t^{(1-p)/(2-p)}$ .

Keywords

Complex networksclustering coefficientsconcentration boundstransitivityclique number

MSC classification

Primary: 05C82: Small world graphs, complex networks
Secondary: 60K40: Other physical applications of random processes 68R10: Graph theory (including graph drawing)
Type
Original Article
Information
Journal of Applied Probability , Volume 58 , Issue 4 , December 2021 , pp. 890 - 908
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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