Poisson approximation for cycles in the generalised random graph

TL;DR

This paper proves that the distribution of cycle counts in a generalized random graph converges to a Poisson process, providing rates of convergence and analyzing shortest and longest cycle lengths.

Contribution

It establishes Poisson convergence for cycle counts in generalized random graphs and extends results to related models, with explicit convergence rates.

Findings

Cycle counts converge to a Poisson point process.

Provided convergence rates under finite fourth moment assumption.

Characterized distribution of shortest and longest cycles in subcritical graphs.

Abstract

The generalised random graph contains $n$ vertices with positive i.i.d. weights. The probability of adding an edge between two vertices is increasing in their weights. We require the weight distribution to have finite second moments and study the point process $\mathcal{C}_n$ on $\{3,4,\dots\}$, which counts how many cycles of the respective length are present in the graph. We establish convergence of $\mathcal{C}_n$ to a Poisson point process. Under the stronger assumption of the weights having finite fourth moments we provide the following results. When $\mathcal{C}_n$ is evaluated on a bounded set $A$, we provide a rate of convergence. If the graph is additionally subcritical, we extend this to unbounded sets $A$ at the cost of a slower rate of convergence. From this we deduce the limiting distribution of the length of the shortest and the longest cycle when the graph is subcritical, including rates of convergence. All mentioned results also apply to the Chung-Lu model and the Norros-Reittu model.