  We study bond percolation on a one-parameter family of hierarchical
small-world network, and find a meta-transition between the inverted BKT
transition and the abrupt transition driven by changing the network topology.
It is found that the order parameter is continuous and fractal exponent is
discontinuous in the inverted BKT transition, and oppositely, the former is
discontinuous and the latter is continuous in the abrupt transition. The gaps
of the order parameter and fractal exponent in each transition go to vanish as
approaching the meta-transition point. This point corresponds to a marginal
power-law transition. In the renormalization group formalism, this
meta-transition corresponds to the transition between transcritical and
saddle-node bifurcations of the fixed point via a pitchfork bifurcation.
