  We study the quantum critical behavior of networks consisting of
Lipkin-Meshkov-Glick models with an anisotropic ferromagnetic coupling. We
focus on the low-energy properties of the system within a mean-field approach
and the quantum corrections around the mean-field solution. Our results show
that the weak-coupling regime corresponds to the paramagnetic phase when the
local field dominates the dynamics, but the local anisotropy leads to the
existence of an exponentially-degenerate ground state. In the strong-coupling
regime, the ground state is twofold degenerate and possesses long-range
magnetic ordering. Analytical results for a network with the ring topology are
obtained.
