the core $N_c$, the number of edges of the core $L_c$ and consequently the average connectivity of the core $2L_c/N_c$: there are estimators of $N i(\alpha)$, $N c(\alpha)$, $N
l(\alpha)$ and $\alpha_{eff}$, as defined before. As usual in Monte-Carlo simulations, averages have been done over many random graphs, and their confidence intervals (or error bars) are estimated by the variance of the measurements.

For each value of $\alpha$, we have generated $10\,000$ graphs of size $N=100$, $N=1000$ and $N=10\,000$, and $1000$ graphs of size $N=100\,000$ and $N=1\,000\,000$. At the transition value $\alpha=e$, $10\,000$ graphs have been generated for all the sizes. The whole computation takes a few days on a medium Sun workstation without special optimization.

Monte-Carlo results
-------------------

![ *Monte-Carlo averages (symbols) and analytical results (solid line) for the size, edges and average connectivity of the core.* []{data-label="fig:moy"}](moy.eps){width="\figwidth" height="0.9\textheight"}

On Fig. \[fig:moy\], Monte-Carlo averages of $N_c/N$, $L_c/N$ and $2L_c/N_c$ are compared with the infinite $N$ results, $c(\alpha)$, $l(\alpha)$ and $\alpha_{eff}$. Errors bars are not drawn because they are smaller than the size of symbols. This figure is a typical case of a second order transition. Far from the transition, differences between finite $N$ and thermodynamic (i.e. $N=\infty$) functions are 