  Many spreading processes in our real-life can be considered as a complex
contagion, and the linear threshold (LT) model is often applied as a very
representative model for this mechanism. Despite its intensive usage, the LT
model suffers several limitations in describing the time evolution of the
spreading. First, the discrete-time step that captures the speed of the
spreading is vaguely defined. Second, the synchronous updating rule makes the
nodes infected in batches, which can not take individual differences into
account. Finally, the LT model is incompatible with existing models for the
simple contagion. Here we consider a generalized linear threshold (GLT) model
for the continuous-time stochastic complex contagion process that can be
efficiently implemented by the Gillespie algorithm. The time in this model has
a clear mathematical definition and the updating order is rigidly defined. We
find that the traditional LT model systematically underestimates the spreading
speed and the randomness in the spreading sequence order. We also show that the
GLT model works seamlessly with the susceptible-infected (SI) or
susceptible-infected-recovered (SIR) model. One can easily combine them to
model a hybrid spreading process in which simple contagion accumulates the
critical mass for the complex contagion that leads to the global cascades.
Overall, the GLT model we proposed can be a useful tool to study complex
contagion, especially when studying the time evolution of the spreading.
