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. In most swarm systems, agents are either aware of the position of their direct neighbors or they possess a substrate on which they can deposit information (stigmergy). However, such resources are not always obtainable in real-world applications because of hardware and environmental constraints. In this paper we study in 2D simulation the design of a swarm system which does not make use of positioning information or stigmergy. This endeavor is motivated by an application whereby a large number of Swarming Micro Air Vehicles (SMAVs), of fixed-wing configuration, must organize autonomously to establish a wireless communication network (SMAVNET) between users located on ground. Rather than relative or absolute positioning, agents must rely only on their own heading measurements and local communication with neighbors. Designing local interactions responsible for the emergence of the SMAVNET deployment and maintenance is a challenging task. For this reason, artificial evolution is used to automatically develop neuronal controllers for the swarm of homogenous agents. This approach has the advantage of yielding original and efficient swarming strategies. A detailed behavioral analysis is then performed on the fittest swarm to gain insight as to the behavior of the individual agents. A central theme in distributed network algorithms concerns understanding and coping with the issue of locality. Yet despite considerable progress, research efforts in this direction have not yet resulted in a solid basis in the form of a fundamental computational complexity theory for locality. Inspired by sequential complexity theory, we focus on a complexity theory for distributed decision problems. In the context of locality, solving a decision problem requires the processors to independently inspect their local neighborhoods and then collectively decide whether a given global input instance belongs to some specified language. We consider the standard LOCAL model of computation and define LD(t) (for local decision) as the class of decision problems that can be solved in t communication rounds. We first study the intriguing question of whether randomization helps in local distributed computing, and to what extent. Specifically, we define the corresponding randomized class BPLD(t,p,q), containing all languages for which there exists a randomized algorithm that runs in t rounds, accepts correct instances with probability at least p, and rejects incorrect ones with probability at least q. We show that p2 p q = 1 is a threshold for the containment of LD(t) in BPLD(t,p,q). More precisely, we show that there exists a language that does not belong to LD(t) for any t=o(n) but does belong to BPLD(0,p,q) for any p,q ∈ (0,1) such that p2 p q ≤ 1. On the other hand, we show that, restricted to hereditary languages, BPLD(t,p,q)=LD(O(t)), for any function t, and any p, q ∈ (0,1) such that p2 p q > 1. In addition, we investigate the impact of nondeterminism on local decision, and establish several structural results inspired by classical computational complexity theory. Specifically, we show that nondeterminism does help, but that this help is limited, as there exist languages that cannot be decided locally nondeterministically. Perhaps surprisingly, it turns out that it is the combination of randomization with nondeterminism that enables to decide all languages in constant time. Finally, we introduce the notion of local reduction, and establish a couple of completeness results.