Summary. This paper provides a theoretical framework for analysis of consensus algorithms for multi-agent networked systems with an emphasis on the role of directed information flow, robustness to changes in network topology due to link/node failures, time-delays, and performance guarantees. An overview of basic concepts of information consensus in networks and methods of convergence and performance analysis for the algorithms are provided. Our analysis framework is based on tools from matrix theory, algebraic graph theory, and control theory. We discuss the connections between consensus problems in networked dynamic systems and diverse applications including synchronization of coupled oscillators, flocking, formation control, fast consensus in small-world networks, Markov processes and gossip-based algorithms, load balancing in networks, rendezvous in space, distributed sensor fusion in sensor networks, and belief propagation. We establish direct connections between spectral and structural properties of complex networks and the speed of information diffusion of consensus algorithms. A brief introduction is provided on networked systems with nonlocal information flow that are considerably faster than distributed systems with latticetype nearest neighbor interactions. Simulation results are presented that demonstrate the role of small-world effects on the speed of consensus algorithms and cooperative control of multi-vehicle formations.

Motivated by applications to sensor, peer-to-peer, and ad hoc networks, we study distributed algorithms, also known as gossip algorithms, for exchanging information and for computing in an arbitrarily connected network of nodes. The topology of such networks changes continuously as new nodes join and old nodes leave the network. Algorithms for such networks need to be robust against changes in topology. Additionally, nodes in sensor networks operate under limited computational, communication, and energy resources. These constraints have motivated the design of “gossip ” algorithms: schemes which distribute the computational burden and in which a node communicates with a randomly chosen neighbor. We analyze the averaging problem under the gossip constraint for an arbitrary network graph, and find that the averaging time of a gossip algorithm depends on the second largest eigenvalue of a doubly stochastic matrix characterizing the algorithm. Designing the fastest gossip algorithm corresponds to minimizing this eigenvalue, which is a semidefinite program (SDP). In general, SDPs cannot be solved in a distributed fashion; however, exploiting problem structure, we propose a distributed subgradient method that solves the optimization problem over the network. The relation of averaging time to the second largest eigenvalue naturally relates it to the mixing time of a random walk with transition probabilities derived from the gossip algorithm. We use this connection to study the performance and scaling of gossip algorithms on two popular networks: Wireless Sensor Networks, which are modeled as Geometric Random Graphs, and the Internet graph under the so-called Preferential Connectivity (PC) model.

Submitted to the IEEE Transactions on Automatic Control Technical Report CIT-CDS 2004-005 In this paper, we present a theoretical framework for design and analysis of distributed flocking algorithms. Two cases of flocking in free-space and presence of multiple obstacles are considered. We present three flocking algorithms: two for free-flocking and one for constrained flocking. A comprehensive analysis of the first two algorithms is provided. We demonstrate the first algorithm embodies all three rules of Reynolds. This is a formal approach to extraction of interaction rules that lead to the emergence of collective behavior. We show that the first algorithm generically leads to regular fragmentation, whereas the second and third algorithms both lead to flocking. A systematic method is provided for construction of cost functions (or collective potentials) for flocking. These collective potentials penalize deviation from a class of lattice-shape objects called α-lattices. We use a multi-species framework for construction of collective potentials that consist of flock-members, or α-agents, and virtual agents associated with α-agents called β- and γ-agents. We show that the tracking/migration problem for flocks can be solved using an algorithm with a peer-to-peer architecture. Each node (or macro-agent) of this peer-to-peer network is the aggregation of all three species of agents. The implication of this fact is that “flocks

We discuss an old distributed algorithm for reaching consensus that has received a fair amount of recent attention. In this algorithm, a number of agents exchange their values asynchronously and form weighted averages with (possibly outdated) values possessed by their neighbors. We overview existing convergence results, and establish some new ones, for the case of unbounded intercommunication intervals.

Abstract — We consider a network of distributed sensors, where each sensor takes a linear measurement of some unknown parameters, corrupted by independent Gaussian noises. We propose a simple distributed iterative scheme, based on distributed average consensus in the network, to compute the maximum-likelihood estimate of the parameters. This scheme doesn’t involve explicit point-to-point message passing or routing; instead, it diffuses information across the network by updating each node’s data with a weighted average of its neighbors ’ data (they maintain the same data structure). At each step, every node can compute a local weighted least-squares estimate, which converges to the global maximum-likelihood solution. This scheme is robust to unreliable communication links. We show that it works in a network with dynamically changing topology, provided that the infinitely occurring communication graphs are jointly connected. I.

We consider a stochastic model for distributed average consensus, which arises in applications such as load balancing for parallel processors, distributed coordination of mobile autonomous agents, and network synchronization. In this model, each node updates its local variable with a weighted average of its neighbors ’ values, and each new value is corrupted by an additive noise with zero mean. The quality of consensus can be measured by the total mean-square deviation of the individual variables from their average, which converges to a steady-state value. We consider the problem of finding the (symmetric) edge weights that result in the least mean-square deviation in steady state. We show that this problem can be cast as a convex optimization problem, so the global solution can be found efficiently. We describe some computational methods for solving this problem, and compare the weights and the mean-square deviations obtained by this method and several other weight design methods.

We analyze the performance of a distributed Kalman filter proposed in recent work on distributed dynamical systems. This approach to distributed estimation is novel in that it admits a systematic analysis of its performance as various network quantities such as connection density, topology, and bandwidth are varied. Our main contribution is a frequency-domain characterization of the distributed estimator’s performance; this is quantified in terms of a special matrix associated with the connection topology called the graph Laplacian, and also the rate of message exchange between immediate neighbors in the communication network. We present simulations for an array of sonar-like sensors to verify our analysis results. 1.

Developing energy efficient strategies for the extraction, transmission, and dissemination of information is a core theme in wireless sensor network research. In this paper we present a novel system for decentralized data compression and predistribution. The system simultaneously computes random projections of the sensor data and disseminates them throughout the network using a simple gossiping algorithm. These summary statistics are stored in an efficient manner and can be extracted from a small subset of nodes anywhere in the network. From these measurements one can reconstruct an accurate approximation of the data at all nodes in the network, provided the original data is compressible in a certain sense which need not be known to the nodes ahead of time. The system provides a practical and universal approach to decentralized compression and content distribution in wireless sensor networks. Two example applications, network health monitoring and field estimation, demonstrate the utility of our method.

We provide a complete analysis of the Kuramoto model of coupled nonlinear oscillators with uncertain natural frequencies and arbitrary interconnection topology. Our work extends and supersedes existing, partial results for the case of an all-to-all connected network. Using tools from spectral graph theory and control theory, we prove that for couplings above a critical value all the oscillators synchronize, resulting in convergence of all phase di#erences to a constant value, both in the case of identical natural frequencies as well as uncertain ones. We further explain the behavior of the system as the number of oscillators grows to infinity.

Abstract. This paper presents coordination algorithms for groups of mobile agents performing deployment and coverage tasks. As an important modeling constraint, we assume that each mobile agent has a limited sensing/communication radius. Based on the geometry of Voronoi partitions and proximity graphs, we analyze a class of aggregate objective functions and propose coverage algorithms in continuous and discrete time. These algorithms have convergence guarantees and are spatially distributed with respect to appropriate proximity graphs. Numerical simulations illustrate the results.