Surface effects in dense random graphs with sharp edge constraint

TL;DR

This paper investigates the asymptotic behavior of the number of triangles in dense random graphs with a fixed number of edges, revealing a surface term in the expansion analogous to physical surface effects.

Contribution

It introduces a new asymptotic expansion for triangle counts in constrained random graphs, highlighting the surface effect absent in unconstrained models.

Findings

The triangle count has an expansion with a surface term proportional to n^2.

The mean of the fluctuation term is O(n), and its standard deviation is O(n^{3/2}).

Surface effects are observed in other graph models with similar edge constraints.

Abstract

We show that the random number $T_n$ of triangles in a random graph on $n$ vertices, with a strict constraint on the total number of edges, admits an expansion $T_n = an^3 + bn^2 + F_n$, where $a$ and $b$ are numbers, with the mean $\langle F_n \rangle = O(n)$ and the standard deviation $\sigma(T_n) =\sigma(F_n)= O(n^{3/2})$. The presence of a `surface term' $bn^2$ has a significance analogous to the macroscopic surface effects of materials, and is missing in the model where the edge constraint is removed. We also find the surface effect in other graph models using similar edge constraints.