Threshold functions for small subgraphs in simple graphs and multigraphs

TL;DR

This paper develops a flexible analytic combinatorics approach to asymptotically count and analyze the distribution of small subgraphs in various random graph and multigraph models, including scale-free networks.

Contribution

It introduces the concept of patchworks for overlapping subgraph analysis and extends asymptotic enumeration to graphs with degree constraints and power-law distributions.

Findings

Asymptotic number of subgraphs in different models

Limiting distribution of subgraph counts

Application to scale-free multigraphs and small cycles

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.