  In $d > 2$ dimensional, homogeneous threshold models discontinuous transition
occur, but the mean-field solution provides $1/t$ power-law activity decay and
other power-laws, thus it is called mixed-order or hybrid type. It has recently
been shown that the introduction of quenched disorder rounds the discontinuity
and second order phase transition and Griffiths phases appear. Here we provide
numerical evidence, that even in case of high graph dimensional hierarchical
modular networks the Griffiths phase of the $K=2$ threshold model is present
below the hybrid phase transition. This is due to the fragmentation of the
activity propagation by modules, which are connected via single links. This
provides a widespread mechanism in case of threshold type of heterogeneous
systems, modeling the brain or epidemics for the occurrence of dynamical
criticality in extended Griffiths phase parameter spaces. We investigate this
in synthetic modular networks with and without inhibitory links as well as in
the presence of refractory states.
