Network formation,temporal evolution, stochastic process, financial networks, critical transitions

D85, C62, G21

We present a new approach to the study of networks where the formation of links is driven by unilateral initiative of nodes. First, we propose a mathematical description of the extreme introvert and extrovert model (XIE), a dynamic network model in which the number of links fluctuates over time according to the degree-preferences of nodes. Second, we introduce a generalization of the model in which intermittent states can make the evolution of connectivity slower. In this class of networks, system evolution has nodes which can either create or destroy links according to their target attitudes. This model belongs to a class of networks not based on the rationale of linking probability between pair of nodes, but it is based on the initiative of the single units when they act. We select the minimal case of a bipartite network where we have two groups of nodes, each group has nodes with a given capability to bear links. One group (high-target) is composed by nodes that create as many links as possible, the extroverts. The other group (low-target) is composed by nodes that delete as many links as possible, i.e. the introverts. We here provide a novel analytical formulation of the system evolution through coupled Master Equations for the two interacting populations, recovering the steady state degree distributions and a new analytical description of the transient dynamics to the equilibrium. Moreover, we provide numerical evidence supporting the existence of an extreme Thouless effect in this model, i.e. a mixed-phase transition at which the order parameter, here given by the network connectivity, displays a discontinuous jump and large dynamic fluctuations. Fluctuations are also shown to be connected to a peak in degree correlation at the transition, corresponding to same size populations of extroverts and introverts.