small. Finite size effects are at least of order $1/N$, because the analytical calculations do not take into account the loops of finite size: their number is $O(1)$, so their contributions are $O(1/N)$. The simplest example is the “triangle” subgraph made of three vertices and three edges, not connected to the rest of the graph. Obviously, the triangles have no leaf and belong to the core: their contribution to $N_c$ is $(\alpha
e^{-\alpha})^3/2$.

We have verified that finite size effects are of order $O(1/N)$ (but not larger) for $c(\alpha)$ and $l(\alpha)$ far from the transition. For $\alpha_{eff}$, this is probably true but less clear because fluctuations are stronger. On the other hand, in the critical region $\alpha \sim e$, finite size effects are larger and some critical exponents can be defined. They are discussed later.

![ *Monte-Carlo estimations of the variance of the size of the core $\chi_c(\alpha)$ and the variance of the number of edges of the core $\chi_l(\alpha)$.* []{data-label="fig:var"}](var.eps){width="\figwidth" height="0.9\textheight"}

We have also examined the variances of the size, number of edges and average connectivity of the core. For $\alpha$ not too close to $e$ and large $N$, we expect that the fluctuations (square root of the variance) 