  We propose a family of models to study the evolution of ties in a network of
interacting agents by reinforcement and penalization of their connections
according to certain local laws of interaction. The family of stochastic
dynamical systems, on the edges of a graph, exhibits \emph{good} convergence
properties, in particular, we prove a strong-stability result: a subset of
binary matrices or graphs -- characterized by certain compatibility properties
-- is a global almost sure attractor of the family of stochastic dynamical
systems. To illustrate finer properties of the corresponding strong attractor,
we present some simulation results that capture, e.g., the conspicuous
phenomenon of emergence and downfall of leaders in social networks.
