  A class of dynamic threshold models is proposed, for describing the upset of
collective actions in social networks. The agents of the network have to decide
whether to undertake a certain action or not. They make their decision by
comparing the activity level of their neighbors with a time-varying threshold,
evolving according to a time-invariant opinion dynamic model. Key features of
the model are a parameter representing the degree of self-confidence of the
agents, and the mechanism adopted by the agents to evaluate the activity level
of their neighbors. The case in which a radical agent, initially eager to
undertake the action, interacts with a group of ordinary agents, is considered.
The main contribution of the paper is the complete analytic characterization of
the asymptotic behaviors of the network, for three different graph topologies.
The asymptotic activity patterns are determined as a function of the
self-confidence parameter and of the initial threshold of the ordinary agents.
