Adaptive social networks: Strong attractors and emergence and downfall of leaders

We formulate a set of time-varying stochastic networked dynamical systems to model the evolution of tie strength among interacting agents. The dynamics of the strength of connections abide by local laws of reinforcement and penalization due to interactions among the agents. The proposed stochastic dynamical systems exhibit a strong-attractor as a certain subset of the set of binary matrices. Moreover, the family of models adapts well to capture the phenomenon of emergence and downfall of leaders in social networks as it will be illustrated via numerical simulations.