In this chapter, we have presented a new methodology to optimize the net- work topology in terms of several energy related functions. We first apply this methodology to three different settings that usually arise in industrial en- vironments where extra situation awareness is needed. We also identify the constraints that impose each of these settings and relate them with the graph models that provide the best results in general. Thus, as opposite to previous works, which introduce unrealistic parameters to reproduce the formation of the different graph models, we identify several real parameters that affect the formation of these graphs. In particular, the formation of small word graphs (SWG), scale free graphs (SFG) and random geometric graphs (RGG) have been found to characterize the topology under certain real conditions. More- over, motivated by two well known tasks in the field of signal processing, we also apply this methodology to improve their performance. In the particular case of the distributed detection problem, our optimization results on sys- tems running full detection capabilities for longer periods of time. In the case of the distributed estimation problem, our optimization results on ensuring a minimum quality of the estimations for longer periods of time. Extensive numerical results have been presented to show the efficiency of our method- ology in terms of the considered energy functions.
Topology Optimization: Discrete
Systems
5.1
Introduction
In the previous chapter, we have presented a new methodology for continu- ous systems to reach consensus in an energy efficient manner. This is based on a centralized method that performs an a priori design of the topology to minimize a set of cost functions that are based on convex combinations of the power vector defined in (2.4.1) multiplied by the convergence time in (3.1.2). However, the sensor systems in practice are usually discrete. A discrete system, as opposite to a continuous system, involves digital components work- ing with digital signals and having a countable number of states. For that rea- son, discrete systems have a direct relation to real sensor networks, since these networks are generally composed by digital devices (although inte- grated sensors may be analogic). In such systems, the moments in which the updates are performed are discrete time instants. In the particular case of a system executing consensus, these updates are performed whenever the nodes communicate with their neighbors and calculate their own new state with the data collected from them.
In addition, in the previous chapter, it is implicitly assumed the existence of a central entity in charge of obtaining the optimal topology. Once obtained, the solution is given to the deployed nodes, which communicate accordingly to reach consensus. However, since one of the main advantages of consen-
Efficient Consensus in Sensor Networks
sus algorithms is that they operate without relying in such central entities, it is also of great interest to find new methods that can be implemented in a distributed manner, so that the network topology optimization can be collabo- ratively performed by the sensor nodes as the consensus algorithm itself. For this purpose, new low complexity methods to improve both the convergence time and the power consumption of consensus algorithms are necessary.
Another important issue to deal with is that the sensor network to be opti- mized may be already executing a consensus based application over an exist- ing topology. This implies that a posteriori optimization methodologies, to be executed over the existing configuration, may be needed for such scenarios.
Although most of the results derived in Chapter 4 can be easily extended to discrete systems when a small step-size is considered, this is not always possible due to the time constraints that are present in the consensus-based applications. The main reason is that a small step-size can be far from effi- cient, since the convergence time is reduced by increasing the value of this parameter. Although the methodologies to be applied in a discrete system with a large step size are different from the ones for continuous systems, the energy parameters to be optimized are still determined by the convergence time and the power consumption of the nodes.
In this chapter, besides showing how to adapt the a priori topology de- sign presented in Chapter 4, we also consider consensus algorithms executed over already formed topologies. The network topologies to be optimized are composed by computationally constrained discrete devices that run on batter- ies. This scenario makes the results of this chapter closer to reality than the previous one, which is generally theory-driven.
For simplicity and in order to not extend unnecessarily this chapter, we only consider homogeneous networks and unicast communications. Simi- lar results can be obtained for the different network settings presented in the previous chapter. Thus, we firstly show how to adapt the topology optimiza- tion methodology proposed in Chapter 4 to discrete systems when a small enough step size is considered. We also identify the problems appeared when continuous and discrete systems are decoupled, finding, in this case, an al- ternative optimization method to improve the network topology. Finally, two distributed methodologies with different computational cost aimed to opti- mize the topology are proposed.