Asymptotic behaviours of a class of threshold models for collective action in social networks

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 neighbours 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 neighbours. 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 characterisation of the asymptotic behaviours 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. Numerical validation on a real ego network shows that the theoretical results obtained for simple graph structures provide useful insights on the network behaviour in more complex settings.