  We study a simple model of how social behaviors, like trends and opinions,
propagate in networks where individuals adopt the trend when they are informed
by threshold $T$ neighbors who are adopters. Using a dynamic message-passing
algorithm, we develop a tractable and computationally efficient method that
provides complete time evolution of each individual's probability of adopting
the trend or of the frequency of adopters and non-adopters in any arbitrary
networks. We validate the method by comparing it with Monte Carlo based agent
simulation in real and synthetic networks and provide an exact analytic scheme
for large random networks, where simulation results match well. Our approach is
general enough to incorporate non-Markovian processes and to include
heterogeneous thresholds and thus can be applied to explore rich sets of
complex heterogeneous agent-based models.
