Threshold functions for small subgraphs in simple graphs and multigraphs. (August 2020)

• Record Type: Journal Article
• Authors: Collet, Gwendal; de Panafieu, Élie; Gardy, Danièle; Gittenberger, Bernhard; Ravelomanana, Vlady
• Is Part Of: European journal of combinatorics. Volume 88(2020)
• Publication Date: 2020-08
• Subjects: Combinatorial analysis -- Periodicals; Analyse combinatoire -- Périodiques; Combinatorial analysis; Periodicals; Electronic journals; 511.6
• DOI: 10.1016/j.ejc.2020.103113
• Languages: English
• ISSNs: 0195-6698

Abstract

Abstract: We revisit the problem of counting the number of copies of a fixed graph in a random graph or multigraph, for various models of random (multi)graphs. For our proofs we introduce the notion of patchworks to describe the possible overlappings of copies of subgraphs. Furthermore, the proofs are based on analytic combinatorics to carry out asymptotic computations. The flexibility of our approach allows us to tackle a wide range of problems. We obtain the asymptotic number and the limiting distribution of the number of subgraphs which are isomorphic to a graph from a given set of graphs. The results apply to multigraphs as well as to (multi)graphs with degree constraints. One application is to scale-free multigraphs, where the degree distribution follows a power law, for which we show how to obtain the asymptotic number of copies of a given subgraph and give as an illustration the expected number of small cycles.