Subgraph statistics in subcritical graph classes

TL;DR

This paper proves that in subcritical graph classes, the count of a fixed subgraph in a random graph follows a normal distribution with predictable mean and variance, using advanced analytic methods.

Contribution

It establishes a general normal limit law for subgraph counts in subcritical graph classes, extending previous probabilistic results with a new analytic framework.

Findings

Number of subgraph occurrences follows a normal distribution

Explicit formulas for triangles and 4-cycles in series-parallel graphs

Analytic framework applicable to infinite systems of functional equations

Abstract

Let $H$ be a fixed graph and $\mathcal{G}$ a subcritical graph class. In this paper we show that the number of occurrences of $H$ (as a subgraph) in a uniformly at random graph of size $n$ in $\mathcal{G}$ follows a normal limiting distribution with linear expectation and variance. The main ingredient in our proof is the analytic framework developed by Drmota, Gittenberger and Morgenbesser to deal with infinite systems of functional equations. As a case study, we get explicit expressions for the number of triangles and cycles of length four for the family of series-parallel graphs.