{"title":"Clustering and Cliques in Preferential Attachment Random Graphs with Edge Insertion","authors":"Caio Alves, Rodrigo Ribeiro, Rémy Sanchis","doi":"10.1007/s10955-024-03279-8","DOIUrl":null,"url":null,"abstract":"<p>In this paper, we investigate the global clustering coefficient (a.k.a transitivity) and clique number of graphs generated by a preferential attachment random graph model with an additional feature of allowing edge connections between existing vertices. Specifically, at each time step <i>t</i>, either a new vertex is added with probability <i>f</i>(<i>t</i>), or an edge is added between two existing vertices with probability <span>\\(1-f(t)\\)</span>. We establish concentration inequalities for the global clustering and clique number of the resulting graphs under the assumption that <i>f</i>(<i>t</i>) is a regularly varying function at infinity with index of regular variation <span>\\(-\\gamma \\)</span>, where <span>\\(\\gamma \\in [0,1)\\)</span>. We also demonstrate an inverse relation between these two statistics: the clique number is essentially the reciprocal of the global clustering coefficient.</p>","PeriodicalId":667,"journal":{"name":"Journal of Statistical Physics","volume":null,"pages":null},"PeriodicalIF":1.3000,"publicationDate":"2024-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Statistical Physics","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1007/s10955-024-03279-8","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper, we investigate the global clustering coefficient (a.k.a transitivity) and clique number of graphs generated by a preferential attachment random graph model with an additional feature of allowing edge connections between existing vertices. Specifically, at each time step t, either a new vertex is added with probability f(t), or an edge is added between two existing vertices with probability \(1-f(t)\). We establish concentration inequalities for the global clustering and clique number of the resulting graphs under the assumption that f(t) is a regularly varying function at infinity with index of regular variation \(-\gamma \), where \(\gamma \in [0,1)\). We also demonstrate an inverse relation between these two statistics: the clique number is essentially the reciprocal of the global clustering coefficient.
期刊介绍:
The Journal of Statistical Physics publishes original and invited review papers in all areas of statistical physics as well as in related fields concerned with collective phenomena in physical systems.