the tree has the maximum possible of steps, the lattice network can have at most $4$ steps. In all cases the last step corresponds to $\rho_c=0$ because for those values of $t$ a cooperator, regardless of the composition of its neighbourhood, is not able to turn a defecting neighbour into a cooperating one. Furthermore, it is also possible that, because of geometrical constraints, $\rho_c$ also vanishes for other steps. For the networks analyzed in this paper, we have confirmed that only the height of the steps depends on $C$. Furthermore, simulations show that only for the first two steps the final number of cooperators is non vanishing (see Fig.\[figsteps\]). For these reasons we have only analyzed the dependence of $rho_c$ in these first two steps, i.e. we have used only two values of $t$, $t_1$ and $t_2$, that satisfy $1<t_1<k/(k-1)$ and $k/(k-1)<t_2<k/(k-2)$.

![ Steady cooperator density $\rho_c$ as a function of the parameter $t$ for three different networks: $k=4$ (lattice), random lattice with $C=0.2$ starting from a lattice with $k=6$, and idem with $k=8$.[]{data-label="figsteps"}](steps.eps)

In Figs. \[k4\] to \[k8\] we plot the numerical results obtained from computational simulations with $1000$ to $5000$ agents. Each curve corresponds to the average fraction 