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.
Except for the analysis of such complex systems, there are some important results on the design of local rules that give rise to a desired global behaviour. @cite_11 designed local rules that allow a network to reach a desired steady-state degree distribution, while proposed a global-to-local design methodology to compose heterogeneous swarms for self-organized task allocation @cite_5 .
@cite_5 @cite_11
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We describe a framework for the design of collective behaviors for groups of identical mobile agents. The approach is based on decentralized simultaneous estimation and control, where each agent communicates with neighbors and estimates the global performance properties of the swarm needed to make a local control decision. Challenges of the approach include designing a control law with desired convergence properties, assuming each agent has perfect global knowledge; designing an estimator that allows each agent to make correct estimates of the global properties needed to implement the controller; and possibly modifying the controller to recover desired convergence properties when using the estimates of global performance. We apply this framework to the problem of controlling the moment statistics describing the location and shape of a swarm. We derive conditions which guarantee that the formation statistics are driven to desired values, even in the presence of a changing network topology. A key problem in the study and design of complex systems is the apparent disconnection between the microscopic and the macroscopic. It is not straightforward to identify the local interactions that give rise to an observed global phenomenon, nor is it simple to design a system that will exhibit some desired global property using only local knowledge. Here we propose a methodology that allows for the identification of local interactions that give rise to a desired global property of a network, the degree distribution. Given a set of observable processes acting on a network, we determine the conditions that must be satisfied to generate a desired steady-state degree distribution. We thereby provide a simple example for a class of tasks where a system can be designed to self-organize to a given state.