  The structure and dynamic of social network are largely determined by the
heterogeneous interaction activity and social capital allocation of
individuals. These features interplay in a non-trivial way in the formation of
network and challenge a rigorous dynamical system theory of network evolution.
Here we study seven real networks describing temporal human interactions in
three different settings: scientific collaborations, Twitter mentions, and
mobile phone calls. We find that the node's activity and social capital
allocation can be described by two general functional forms that can be used to
define a simple stochastic model for social network dynamic. This model allows
the explicit asymptotic solution of the Master Equation describing the system
dynamic, and provides the scaling laws characterizing the time evolution of the
social network degree distribution and individual node's ego network. The
analytical predictions reproduce with accuracy the empirical observations
validating the theoretical approach. Our results provide a rigorous dynamical
system framework that can be extended to include other features of networks'
formation and to generate data driven predictions for the asymptotic behavior
of large-scale social networks.
