  We consider the dynamics of an extension of the influential Granovetter model
of social behavior, where individuals are affected by their personal
preferences and observation of the neighbors' behavior. Individuals are
arranged in a network (usually, the square lattice) and each has a state and a
fixed threshold for behavior changes. We simulate the system asynchronously
either by picking a random individual and either update its state or exchange
it with another randomly chosen individual (mixing). We describe the dynamics
analytically in the fast-mixing limit by using the mean-field approximation and
investigate it mainly numerically in case of a finite mixing. We show that the
dynamics converge to a manifold in state space, which determines the possible
equilibria, and show how to estimate the projection of manifold by using
simulated trajectories, emitted from different initial points.
  We show that the effects of considering the network can be decomposed into
finite-neighborhood effects, and finite-mixing-rate effects, which have
qualitatively similar effects. Both of these effects increase the tendency of
the system to move from a less-desired equilibrium to the "ground state". Our
findings can be used to probe shifts in behavioral norms and have implications
for the role of information flow in determining when social norms that have
become unpopular (such as foot binding or female genital cutting) persist or
vanish.
