  We perform a thorough and complete analysis of the Anderson localization
transition on several models of random graphs with regular and random
connectivity. The unprecedented precision and abundance of our exact
diagonalization data (both spectra and eigenstates), together with new finite
size scaling and statistical analysis of the graph ensembles, unveils a
universal behavior which is described by two simple, integer, scaling
exponents. A by-product of such analysis is a reconciliation of the tension
between the results of perturbation theory coming from strong disorder and
earlier numerical works, which seemed to suggest that there should be a
non-ergodic region above a given value of disorder $W_{E}$ which is strictly
less than the Anderson localization critical disorder $W_C$, and that of other
works which suggest that there is no such region. We find that, although no
separate $W_{E}$ exists from $W_C$, the length scale at which fully developed
ergodicity is found diverges like $|W-W_C|^{-1}$, while the critical length
over which delocalization develops is $\sim |W-W_C|^{-1/2}$. The separation of
these two scales at the critical point allows for a true non-ergodic,
delocalized region. In addition, by looking at eigenstates and studying leading
and sub-leading terms in system size-dependence of participation entropies, we
show that the former contain information about the non-ergodicity volume which
becomes non-trivial already deep in the delocalized regime. We also discuss the
quantitative similarities between the Anderson transition on random graphs and
many-body localization transition.
