  We analyze the ergodic properties of a metallic wavefunction for the Anderson
model in a disordered random-regular graph with branching number $k=2.$ A few
q-moments $I_q$ associated with the zero energy eigenvector are numerically
computed up to sizes $N=4\times 10^6.$ We extract their corresponding fractal
dimensions $D_q$ in the thermodynamic limit together with correlated volumes
$N_q$ that control finite-size effects. At intermediate values of disorder $W,$
we obtain ergodicity $D_q=1$ for $q=1,2$ and correlation volumes that increase
fast upon approaching the Anderson transition $\log(\log(N_q))\sim W.$ We then
focus on the extraction of the volume $N_0$ associated with the typical value
of the wavefunction $e^{<\log|\psi|^2>},$ which follows a similar tendency as
the ones for $N_1$ or $N_2.$ Its value at intermediate disorders is close, but
smaller, to the so-called ergodic volume previously found via the
super-symmetric formalism and belief propagator algorithms. None of the
computed correlated volumes shows a tendency to diverge up to disorders
$W\approx 15$, specifically none with exponent $\nu=1/2$. Deeper in the metal,
we characterize the crossover to system sizes much smaller than the first
correlated volume $N_1\gg N.$ Once this crossover has taken place, we obtain
evidence of a scaling in which the derivative of the first fractal dimension
$D_1$ behaves critically with an exponent $\nu=1.$
