  Heterogeneous adoption thresholds exist widely in social contagions, but were
always neglected in previous studies. We first propose a non-Markovian
spreading threshold model with general adoption threshold distribution. In
order to understand the effects of heterogeneous adoption thresholds
quantitatively, an edge-based compartmental theory is developed for the
proposed model. We use a binary spreading threshold model as a specific
example, in which some individuals have a low adoption threshold (i.e.,
activists) while the remaining ones hold a relatively high adoption threshold
(i.e., bigots), to demonstrate that heterogeneous adoption thresholds markedly
affect the final adoption size and phase transition. Interestingly, the
first-order, second-order and hybrid phase transitions can be found in the
system. More importantly, there are two different kinds of crossover phenomena
in phase transition for distinct values of bigots' adoption threshold: a change
from first-order or hybrid phase transition to the second-order phase
transition. The theoretical predictions based on the suggested theory agree
very well with the results of numerical simulations.
