A Chemistry-Inspired Framework for Achieving Consensus in Wireless Sensor Networks

The aim of this paper is to show how simple interaction mechanisms, inspired by chemical systems, can provide the basic tools to design and analyze a mathematical model for achieving consensus in wireless sensor networks, characterized by balanced directed graphs. The convergence and stability of the model are first proven by using new mathematical tools, which are borrowed directly from chemical theory, and then validated by means of simulation results, for different network topologies and number of sensors. The underlying chemical theory is also used to derive simple interaction rules that may account for practical issues, such as the estimation of the number of neighbors and the robustness against perturbations. Finally, the proposed chemical solution is validated under real-world conditions by means of a four-node hardware implementation where the exchange of information among nodes takes place in a distributed manner (with no need for any admission control and synchronism procedure), simply relying on the transmission of a pulse whose rate is proportional to the state of each sensor.Comment: 12 pages, 10 figures, submitted to IEEE Sensors Journa

Broadcast Gossip Algorithms for Consensus on Strongly Connected Digraphs

We study a general framework for broadcast gossip algorithms which use companion variables to solve the average consensus problem. Each node maintains an initial state and a companion variable. Iterative updates are performed asynchronously whereby one random node broadcasts its current state and companion variable and all other nodes receiving the broadcast update their state and companion variable. We provide conditions under which this scheme is guaranteed to converge to a consensus solution, where all nodes have the same limiting values, on any strongly connected directed graph. Under stronger conditions, which are reasonable when the underlying communication graph is undirected, we guarantee that the consensus value is equal to the average, both in expectation and in the mean-squared sense. Our analysis uses tools from non-negative matrix theory and perturbation theory. The perturbation results rely on a parameter being sufficiently small. We characterize the allowable upper bound as well as the optimal setting for the perturbation parameter as a function of the network topology, and this allows us to characterize the worst-case rate of convergence. Simulations illustrate that, in comparison to existing broadcast gossip algorithms, the approaches proposed in this paper have the advantage that they simultaneously can be guaranteed to converge to the average consensus and they converge in a small number of broadcasts.Comment: 30 pages, submitte

Distributed consensus algorithms for wireless sensor networks: convergence analysis and optimization

Wireless sensor networks are developed to monitor areas of interest with the purpose of estimating physical parameters or/and detecting emergency events in a variety of military and civil applications. A wireless sensor network can be seen as a distributed computer, where spatially deployed sensor nodes are in charge of gathering measurements from the environment to compute a given function. The research areas for wireless sensor networks extend from the design of small, reliable hardware to low-complexity algorithms and energy saving communication protocols. Distributed consensus algorithms are low-complexity iterative schemes that have received increased attention in different fields due to a wide range of applications, where neighboring nodes communicate locally to compute the average of an initial set of measurements. Energy is a scarce resource in wireless sensor networks and therefore, the convergence of consensus algorithms, characterized by the total number of iterations until reaching a steady-state value, is an important topic of study. This PhD thesis addresses the problem of convergence and optimization of distributed consensus algorithms for the estimation of parameters in wireless sensor networks. The impact of quantization noise in the convergence is studied in networks with fixed topologies and symmetric communication links. In particular, a new scheme including quantization is proposed, whose mean square error with respect to the average consensus converges. The limit of the mean square error admits a closed-form expression and an upper bound for this limit depending on general network parameters is also derived. The convergence of consensus algorithms in networks with random topology is studied focusing particularly on convergence in expectation, mean square convergence and almost sure convergence. Closed-form expressions useful to minimize the convergence time of the algorithm are derived from the analysis. Regarding random networks with asymmetric links, closed-form expressions are provided for the mean square error of the state assuming equally probable uniform link weights, and mean square convergence to the statistical mean of the initial measurements is shown. Moreover, an upper bound for the mean square error is derived for the case of different probabilities of connection for the links, and a practical scheme with randomized transmission power exhibiting an improved performance in terms of energy consumption with respect to a fixed network with the same consumption on average is proposed. The mean square error expressions derived provide a means to characterize the deviation of the state vector with respect to the initial average when the instantaneous links are asymmetric. A useful criterion to minimize the convergence time in random networks with spatially correlated links is considered, establishing a sufficient condition for almost sure convergence to the consensus space. This criterion, valid also for topologies with spatially independent links, is based on the spectral radius of a positive semidefinite matrix for which we derive closed-form expressions assuming uniform link weights. The minimization of this spectral radius is a convex optimization problem and therefore, the optimum link weights minimizing the convergence time can be computed efficiently. The expressions derived are general and apply not only to random networks with instantaneous directed topologies but also to random networks with instantaneous undirected topologies. Furthermore, the general expressions can be particularized to obtain known protocols found in literature, showing that they can be seen as particular cases of the expressions derived in this thesis.Las redes de sensores inalámbricos se utilizan para monitorizar zonas de interés con el propósito final de estimar parámetros físicos y/o detectar situaciones de emergencia en gran variedad de aplicaciones militares y civiles. Una red de sensores inalámbricos puede ser considerada como un método de computación distribuido, donde nodos provistos de sensores toman medidas del entorno para calcular una función que depende de éstas. Las áreas de investigación comprenden desde el diseño de dispositivos hardware pequeños y fiables hasta algoritmos de baja complejidad o protocolos de comunicación de bajo consumo energético. Los algoritmos de consenso distribuidos son esquemas iterativos de baja complejidad que han suscitado mucha atención en diferentes campos debido a su gran espectro de aplicaciones, en los que nodos vecinos se comunican para calcular el promedio de un conjunto de medidas iniciales de la red. Dado que la energía es un recurso escaso en redes de sensores inalámbricos, la convergencia de dichos algoritmos de consenso, caracterizada por el número total de iteraciones hasta alcanzar un valor estacionario, es un importante tema de estudio. Esta tesis doctoral aborda problemas de convergencia y optimización de algoritmos de consenso distribuidos para la estimación de parámetros en redes de sensores inalámbricos. El impacto del ruido de cuantización en la convergencia se estudia en redes con topología fija y enlaces de comunicación simétricos. En particular, se propone un nuevo esquema que incluye el proceso de cuantización y se demuestra que el error cuadrático medio respecto del promedio inicial converge. Igualmente, se obtiene una expresión cerrada del límite del error cuadrático medio, y una cota superior para este límite que depende únicamente de parámetros generales de la red. La convergencia de los algoritmos de consenso en redes con topología aleatoria se estudia prestando especial atención a la convergencia en valor esperado, la convergencia en media cuadrática y la convergencia casi segura, y a partir del análisis se derivan expresiones cerradas útiles para minimizar el tiempo de convergencia. Para redes aleatorias con enlaces asimétricos, se obtienen expresiones cerradas del error cuadrático medio del estado suponiendo enlaces con probabilidad idéntica y con pesos uniformes, y se demuestra la convergencia en media cuadrática al promedio estadístico de las medidas iniciales. Se deduce una cota superior para el error cuadrático medio para el caso de enlaces con probabilidades de conexión diferentes y se propone, además, un esquema práctico con potencias de transmisión aleatorias, que mejora el rendimiento en términos de consumo de energía con respecto a una red fija. Las expresiones para el error cuadrático medio proporcionan una forma de caracterizar la desviación del vector de estado con respecto del promedio inicial cuando los enlaces instantáneos son asimétricos. Con el fin de minimizar el tiempo de convergencia en redes aleatorias con enlaces correlados espacialmente, se considera un criterio que establece una condición suficiente que garantiza la convergencia casi segura al espacio de consenso. Este criterio, que también es válido para topologías con enlaces espacialmente independientes, utiliza el radio espectral de una matriz semidefinida positiva para la cual se obtienen expresiones cerradas suponiendo enlaces con pesos uniformes. La minimización de dicho radio espectral es un problema de optimización convexa y, por lo tanto, el valor de los pesos óptimos puede calcularse de forma eficiente. Las expresiones obtenidas son generales y aplican no sólo para redes aleatorias con topologías dirigidas, sino también para redes aleatorias con topologías no dirigidas. Además, las expresiones generales pueden ser particularizadas para obtener protocolos conocidos en la literatura, demostrando que éstos últimos pueden ser considerados como casos particulares de las expresiones proporcionadas en esta tesis

Gossip Algorithms for Distributed Signal Processing

Gossip algorithms are attractive for in-network processing in sensor networks because they do not require any specialized routing, there is no bottleneck or single point of failure, and they are robust to unreliable wireless network conditions. Recently, there has been a surge of activity in the computer science, control, signal processing, and information theory communities, developing faster and more robust gossip algorithms and deriving theoretical performance guarantees. This article presents an overview of recent work in the area. We describe convergence rate results, which are related to the number of transmitted messages and thus the amount of energy consumed in the network for gossiping. We discuss issues related to gossiping over wireless links, including the effects of quantization and noise, and we illustrate the use of gossip algorithms for canonical signal processing tasks including distributed estimation, source localization, and compression.Comment: Submitted to Proceedings of the IEEE, 29 page

Average Consensus in the Presence of Delays and Dynamically Changing Directed Graph Topologies

Classical approaches for asymptotic convergence to the global average in a distributed fashion typically assume timely and reliable exchange of information between neighboring components of a given multi-component system. These assumptions are not necessarily valid in practical settings due to varying delays that might affect transmissions at different times, as well as possible changes in the underlying interconnection topology (e.g., due to component mobility). In this work, we propose protocols to overcome these limitations. We first consider a fixed interconnection topology (captured by a - possibly directed - graph) and propose a discrete-time protocol that can reach asymptotic average consensus in a distributed fashion, despite the presence of arbitrary (but bounded) delays in the communication links. The protocol requires that each component has knowledge of the number of its outgoing links (i.e., the number of components to which it sends information). We subsequently extend the protocol to also handle changes in the underlying interconnection topology and describe a variety of rather loose conditions under which the modified protocol allows the components to reach asymptotic average consensus. The proposed algorithms are illustrated via examples.Comment: 37 page

When gossip meets consensus : convergence in correlated random WSNs

Peer ReviewedPostprint (author’s final draft

On the mean square error of randomized averaging algorithms

This paper regards randomized discrete-time consensus systems that preserve the average "on average". As a main result, we provide an upper bound on the mean square deviation of the consensus value from the initial average. Then, we apply our result to systems where few or weakly correlated interactions take place: these assumptions cover several algorithms proposed in the literature. For such systems we show that, when the network size grows, the deviation tends to zero, and the speed of this decay is not slower than the inverse of the size. Our results are based on a new approach, which is unrelated to the convergence properties of the system.Comment: 11 pages. to appear as a journal publicatio