  The Anderson transition in random graphs has raised great interest, partly
because of its analogy with the many-body localization (MBL) transition. Unlike
the latter, many results for random graphs are now well established, in
particular the existence and precise value of a critical disorder separating a
localized from an ergodic delocalized phase. However, the renormalization group
flow and the nature of the transition are not well understood. In turn, recent
works on the MBL transition have made the remarkable prediction that the flow
is of Kosterlitz-Thouless type. Here we show that the Anderson transition on
graphs displays the same type of flow. Our work attests to the importance of
rare branches along which wave functions have a much larger localization length
$\xi_\parallel$ than the one in the transverse direction, $\xi_\perp$.
Importantly, these two lengths have different critical behaviors:
$\xi_\parallel$ diverges with a critical exponent $\nu_\parallel=1$, while
$\xi_\perp$ reaches a finite universal value ${\xi_\perp^c}$ at the transition
point $W_c$. Indeed, $\xi_\perp^{-1} \approx {\xi_\perp^c}^{-1} + \xi^{-1}$,
with $\xi \sim (W-W_c)^{-\nu_\perp}$ associated with a new critical exponent
$\nu_\perp = 1/2$, where $\exp( \xi)$ controls finite-size effects. The
delocalized phase inherits the strongly non-ergodic properties of the critical
regime at short scales, but is ergodic at large scales, with a unique critical
exponent $\nu=1/2$. This shows a very strong analogy with the MBL transition:
the behavior of $\xi_\perp$ is identical to that recently predicted for the
typical localization length of MBL in a phenomenological renormalization group
flow. We demonstrate these important properties for a smallworld complex
network model and show the universality of our results by considering different
network parameters and different key observables of Anderson localization.
