Mutual selection in time-varying networks.

Time-varying networks play an important role in the investigation of the stochastic processes that occur on complex networks. The ability to formulate the development of the network topology on the same time scale as the evolution of the random process is important for a variety of applications, including the spreading of diseases. Past contributions have investigated random processes on time-varying networks with a purely random attachment mechanism. The possibility of extending these findings towards a time-varying network that is driven by mutual attractiveness is explored in this paper. Mutual attractiveness models are characterized by a linking function that describes the probability of the existence of an edge, which depends mutually on the attractiveness of the nodes on both ends of that edge. This class of attachment mechanisms has been considered before in the fitness-based complex networks literature but not on time-varying networks. Also, the impact of mutual selection is investigated alongside opinion formation and epidemic outbreaks. We find closed-form solutions for the quantities of interest using a factorizable linking function. The voter model exhibits an unanticipated behavior as the network never reaches consensus in the case of mutual selection but stays forever in its initial macroscopic configuration, which is a further piece of evidence that time-varying networks differ markedly from their static counterpart with respect to random processes that take place on them. We also find that epidemic outbreaks are accelerated by uncorrelated mutual selection compared to previously considered random attachment.