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Large Communities in a Scale-Free Network

Abstract

We prove the existence of a large complete subgraph w.h.p. in a preferential attachment random graph process with an edge-step. That is, we consider a dynamic stochastic process for constructing a graph in which at each step we independently decide, with probability \(p\in (0,1)\), whether the graph receives a new vertex or a new edge between existing vertices. The connections are then made according to a preferential attachment rule. We prove that the random graph \(G_{t}\) produced by this so-called generalized linear preferential (GLP) model at time t contains a complete subgraph whose vertex set cardinality is given by \(t^\alpha \), where \(\alpha = (1-\varepsilon )\frac{1-p}{2-p}\), for any small \(\varepsilon >0\) asymptotically almost surely.

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Acknowledgements

C.A. was supported by Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP), Grant No. 2013/24928-2. R.S. has been partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) and by FAPEMIG (Programa Pesquisador Mineiro), Grant No. PPM 00600/16. R.R. has been partially supported by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES).

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Correspondence to Rémy Sanchis.

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Alves, C., Ribeiro, R. & Sanchis, R. Large Communities in a Scale-Free Network. J Stat Phys 166, 137–149 (2017). https://doi.org/10.1007/s10955-016-1676-8

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  • DOI: https://doi.org/10.1007/s10955-016-1676-8

Keywords

  • Complex networks
  • Clique
  • Preferential attachment
  • Concentration bounds

Mathematics Subject Classification

  • Primary 05C82
  • Secondary 60K40
  • 68R10