  The apparent disconnection between the microscopic and the macroscopic is a
major issue in the understanding of complex systems. To this extend, we study
the convergence of repeatedly applying local rules on a network, and touch on
the expressive power of this model. We look at network systems and study their
behavior when different types of local rules are applied on them. For a very
general class of local rules, we prove convergence and provide a certain member
of this class that, when applied on a graph, efficiently computes its k-core
and its (k-1)-crust giving hints on the expressive power of such a model.
Furthermore, we provide guarantees on the speed of convergence for an important
subclass of the aforementioned class. We also study more general rules, and
show that they do not converge. Our counterexamples resolve an open question of
(Zhang, Wang, Wang, Zhou, KDD- 2009) as well, concerning whether a certain
process converges. Finally, we show the universality of our network system, by
providing a local rule under which it is Turing-Complete.
