  In this paper we consider an extension of the Rosenzweig-Porter (RP) model,
the L\'evy-RP (L-RP) model, in which the off-diagonal matrix elements are
broadly distributed, providing a more realistic benchmark to develop an
effective description of non-ergodic extended (NEE) states in interacting
many-body disordered systems. We put forward a simple, general, and intuitive
argument that allows one to unveil the multifractal structure of the mini-bands
in the local spectrum when hybridization is due to anomalously large transition
amplitudes in the tails of the distribution. The idea is that the energy
spreading of the mini-bands can be determined self-consistently by requiring
that the maximum of the matrix elements between a site $i$ and the other
$N^{D_1}$ sites of the support set is of the same order of the Thouless energy
itself $N^{D_1 - 1}$. This argument yields the fractal dimensions that
characterize the statistics of the multifractal wave-functions in the NEE
phase, as well as the whole phase diagram of the L-RP ensemble. Its predictions
are confirmed both analytically, by a thorough investigation of the
self-consistent equation for the local density of states obtained using the
cavity approach, and numerically, via extensive exact diagonalizations.
