Variance Analysis of Randomized Consensus in Switching Directed Networks

In this paper, we study the asymptotic properties of distributed consensus algorithms over switching directed random networks. More specifically, we focus on consensus algorithms over independent and identically distributed, directed Erdos-Renyi random graphs, where each agent can communicate with any other agent with some exogenously specified probability $p$. While it is well-known that consensus algorithms over Erdos-Renyi random networks result in an asymptotic agreement over the network, an analytical characterization of the distribution of the asymptotic consensus value is still an open question. In this paper, we provide closed-form expressions for the mean and variance of the asymptotic random consensus value, in terms of the size of the network and the probability of communication $p$. We also provide numerical simulations that illustrate our results.Comment: 6 pages, 3 figures, submitted to American Control Conference 201

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

Multi-hop Diffusion LMS for Energy-constrained Distributed Estimation

We propose a multi-hop diffusion strategy for a sensor network to perform distributed least mean-squares (LMS) estimation under local and network-wide energy constraints. At each iteration of the strategy, each node can combine intermediate parameter estimates from nodes other than its physical neighbors via a multi-hop relay path. We propose a rule to select combination weights for the multi-hop neighbors, which can balance between the transient and the steady-state network mean-square deviations (MSDs). We study two classes of networks: simple networks with a unique transmission path from one node to another, and arbitrary networks utilizing diffusion consultations over at most two hops. We propose a method to optimize each node's information neighborhood subject to local energy budgets and a network-wide energy budget for each diffusion iteration. This optimization requires the network topology, and the noise and data variance profiles of each node, and is performed offline before the diffusion process. In addition, we develop a fully distributed and adaptive algorithm that approximately optimizes the information neighborhood of each node with only local energy budget constraints in the case where diffusion consultations are performed over at most a predefined number of hops. Numerical results suggest that our proposed multi-hop diffusion strategy achieves the same steady-state MSD as the existing one-hop adapt-then-combine diffusion algorithm but with a lower energy budget.Comment: 14 pages, 12 figures. Submitted for publicatio

Collective Relaxation Dynamics of Small-World Networks

Complex networks exhibit a wide range of collective dynamic phenomena, including synchronization, diffusion, relaxation, and coordination processes. Their asymptotic dynamics is generically characterized by the local Jacobian, graph Laplacian or a similar linear operator. The structure of networks with regular, small-world and random connectivities are reasonably well understood, but their collective dynamical properties remain largely unknown. Here we present a two-stage mean-field theory to derive analytic expressions for network spectra. A single formula covers the spectrum from regular via small-world to strongly randomized topologies in Watts-Strogatz networks, explaining the simultaneous dependencies on network size N, average degree k and topological randomness q. We present simplified analytic predictions for the second largest and smallest eigenvalue, and numerical checks confirm our theoretical predictions for zero, small and moderate topological randomness q, including the entire small-world regime. For large q of the order of one, we apply standard random matrix theory thereby overarching the full range from regular to randomized network topologies. These results may contribute to our analytic and mechanistic understanding of collective relaxation phenomena of network dynamical systems.Comment: 12 pages, 10 figures, published in PR

When gossip meets consensus : convergence in correlated random WSNs

Peer ReviewedPostprint (author’s final draft

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

Weight Optimization for Consensus Algorithms with Correlated Switching Topology

We design the weights in consensus algorithms with spatially correlated random topologies. These arise with: 1) networks with spatially correlated random link failures and 2) networks with randomized averaging protocols. We show that the weight optimization problem is convex for both symmetric and asymmetric random graphs. With symmetric random networks, we choose the consensus mean squared error (MSE) convergence rate as optimization criterion and explicitly express this rate as a function of the link formation probabilities, the link formation spatial correlations, and the consensus weights. We prove that the MSE convergence rate is a convex, nonsmooth function of the weights, enabling global optimization of the weights for arbitrary link formation probabilities and link correlation structures. We extend our results to the case of asymmetric random links. We adopt as optimization criterion the mean squared deviation (MSdev) of the nodes states from the current average state. We prove that MSdev is a convex function of the weights. Simulations show that significant performance gain is achieved with our weight design method when compared with methods available in the literature.Comment: 30 pages, 5 figures, submitted to IEEE Transactions On Signal Processin

On Endogenous Random Consensus and Averaging Dynamics

Motivated by various random variations of Hegselmann-Krause model for opinion dynamics and gossip algorithm in an endogenously changing environment, we propose a general framework for the study of endogenously varying random averaging dynamics, i.e.\ an averaging dynamics whose evolution suffers from history dependent sources of randomness. We show that under general assumptions on the averaging dynamics, such dynamics is convergent almost surely. We also determine the limiting behavior of such dynamics and show such dynamics admit infinitely many time-varying Lyapunov functions