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arXiv · 1807.05772

Threshold functions for small subgraphs in simple graphs and multigraphs

Gwendal Collet·Élie de Panafieu·Danièle Gardy·Bernhard Gittenberger·Vlady Ravelomanana
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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 \emph{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.

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Gwendal Collet, Élie de Panafieu, Danièle Gardy, Bernhard Gittenberger, Vlady Ravelomanana. 2018-07-16. Threshold functions for small subgraphs in simple graphs and multigraphs. https://arxiv.org/abs/1807.05772

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