  Models of social diffusion reflect processes of how new products, ideas or
behaviors are adopted in a population. These models typically lead to a
continuous or a discontinuous phase transition of the number of adopters as a
function of a control parameter. We explore a simple model of social adoption
where the agents can be in two states, either adopters or non-adopters, and can
switch between these two states interacting with other agents through a
network. The probability of an agent to switch from non-adopter to adopter
depends on the number of adopters in her network neighborhood, the adoption
threshold $T$ and the adoption coefficient $a$, two parameters defining a Hill
function. In contrast, the transition from adopter to non-adopter is
spontaneous at a certain rate $\mu$. In a mean-field approach, we derive the
governing ordinary differential equations and show that the nature of the
transition between the global non-adoption and global adoption regimes depends
mostly on the balance between the probability to adopt with one and two
adopters. The transition changes from continuous, via a transcritical
bifurcation, to discontinuous, via a combination of a saddle-node and a
transcritical bifurcation, through a supercritical pitchfork bifurcation. We
characterize the full parameter space. Finally, we compare our analytical
results with Montecarlo simulations on annealed and quenched degree regular
networks, showing a better agreement for the annealed case. Our results show
how a simple model is able to capture two seemingly very different types of
transitions, i.e., continuous and discontinuous and thus unifies underlying
dynamics for different systems. Furthermore the form of the adoption
probability used here is based on empirical measurements.
