 uniformly random choices correspond to setting $a=0$ in the expression for $p_k$, and then we can expect the graphs that pass the correlation threshold to be clustered around the points of $\rho\sim \rho_1$ and $S(X,Y)\sim 0$.

We also note a sharp variation in how the fixation probabilities of the graphs relate to the asymptotic fixation probabilities of the $K$-funnel as a mutant’s fitness is increased. For $r=1.1$, the graphs exhibiting the highest fixation probabilities, and also the highest slopes, are such that $\rho$ is somewhere between $\rho_2$ and $\rho_3$. For $r=2.0$, though, this happens between $\rho_1$ and $\rho_2$ ($=0.75$, not shown), therefore providing considerably less amplification. Part of the reason why this happens may be simply that the more potent amplifiers are harder to generate by our layer-selection mechanism as $r$ is increased. But it is also important to realize that, even for the $K$-funnel, achieving a fixation probability near $\rho_K$ requires progressively larger graphs as $r$ is increased. This is illustrated in Fig. \[fig:funnel\] for $K=3$ and the same two values of $r$.

![(Color online) Simulation results for the $3$-funnel. Dashed lines mark the values of $\rho_3$.[]{data-label="fig:funnel"}](funnel.eps)

Additional simulation results, for the much larger case of $K=10$ and $n=10\,000$, are presented in Fig. \[fig:10layers\] for $r=1.1$ and $a=1,2,3,4$. Computationally, this case is much more demanding than those of Fig. \[fig:5layers\], owing mainly to the number of distinct networks that can occur, as discussed earlier (in fact, for $K=10$ and $n=10\,000$, this number is at least of the order of $10^{33}$). Consequently, many fewer graphs surpassing the $0.9$ correlation threshold were obtained. Even so, one possible reading is that results similar to those reported in Fig. \[fig:5layers\] can be expected, but this remains to be seen.

In summary, we have demonstrated that strongly connected layered networks can be grown for which the fixation probability signi