# On the Convergence of Network Systems

## TL;DR

This paper investigates the convergence behavior of local rules applied to network systems, demonstrating convergence for certain classes, providing efficiency guarantees, and establishing the model's expressive power including Turing-completeness.

## Contribution

It proves convergence for a broad class of local rules, offers efficiency guarantees, resolves an open question on non-converging rules, and shows the model's universality and Turing-completeness.

## Key findings

- Proves convergence for a general class of local rules.
- Provides efficient computation of k-core and (k-1)-crust.
- Shows non-convergence for more general rules.

## Abstract

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.