  The so-called Granovetter-Watts model was introduced to capture a situation
in which the adoption of new ideas or technologies requires a certain
redundancy in the social environment of each agent to take effect. This model
has become a paradigm for complex contagion. Here we investigate a symmetric
version of the model: agents may be in two states that can spread equally
through the system via complex contagion. We find three possible phases: a
mixed one (dynamically active disordered state), an ordered one, and a
heterogeneous frozen phase. These phases exist for several configurations of
the contact network. Then we consider the effect of introducing aging as a
non-Markovian mechanism in the model, where agents become increasingly
resistant to change their state the longer they remain in it. We show that when
aging is present, the mixed phase is replaced, for sparse networks, by a new
phase with different dynamical properties. This new phase is characterized by
an initial disordering stage followed by a slow ordering process towards a
fully ordered absorbing state. In the ordered phase, aging modifies the
dynamical properties. For random contact networks, we develop a theoretical
description based on an Approximate Master Equation that describes with good
accuracy the results of numerical simulations for the model with and without
aging.
