different. For $r^{\rm tag}$ the value is $\Phi(p_{c2})=0$ but for $b$ it is $\Phi'(p_{c2})\approx -0.11$, as explained in the text.[]{data-label="fig:comp"}](fig4b.eps "fig:"){width="45.00000%"} ![(Color online) Comparisons between $r^{\rm tag}$, the average number of tagged routes from an arbitrary surface point to the midpoint, and, $b$, the number of surface points connected to the midpoint. $r^{\rm tag}$ is given by the analytical solution and $b$ is obtained from numerical simulations. (a) The number of tagged routes from the surface to the midpoint, $Br^{\rm tag}$ for various sizes $L$. All curves cross in a single point at $p_{c1}\approx
0.303$. (b) The same plot for $b$. Note the strong resemblance and the single crossing point at $p\approx 0.303(1)$. The panel (c) illustrates the close connection between $Br^{\rm tag}$ and $b$. For $p\leq0.4$ the number of routes and the number of connections are almost the same (the number of routes is slightly smaller as explained in the text). However for $p\geq
0.4$, the number of routes exceeds the number of connected surface points. The panel (d) illustrates this connection further by comparing the size-scaling exponent $1+\Phi$ obtained analytically for $r^{\rm tag}$ with the size scaling for $b$ obtained from (b). The agreement is striking for $p\leq 