Abstract—The paper studies average consensus with random topologies (intermittent links) and noisy channels. Consensus with noise in the network links leads to the bias-variance dilemma—running consensus for long reduces the bias of the final average estimate but increases its variance. We present two different compromises to this tradeoff: the algorithm modifies conventional consensus by forcing the weights to satisfy a persistence condition (slowly decaying to zero;) and the algorithm where the weights are constant but consensus is run for a fixed number of iterations, then it is restarted and rerun for a total of runs, and at the end averages the final states of the runs (Monte Carlo averaging). We use controlled Markov processes and stochastic approximation arguments to prove almost sure convergence of to a finite consensus limit and compute explicitly the mean square error (mse) (variance) of the consensus limit. We show that represents the best of both worlds—zero bias and low variance—at the cost of a slow convergence rate; rescaling the weights balances the variance versus the rate of bias reduction (convergence rate). In contrast, , because of its constant weights, converges fast but presents a different bias-variance tradeoff. For the same number of iterations, shorter runs (smaller) lead to high bias but smaller variance (larger number of runs to average over.) For a static nonrandom network with Gaussian noise, we compute the optimal gain for to reach in the shortest number of iterations, with high probability (1), ()-consensus ( residual bias). Our results hold under fairly general assumptions on the random link failures and communication noise. Index Terms—Additive noise, consensus, sensor networks, stochastic approximation, random topology. I.

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 paper presents an overview of recent work in the area. We describe convergence rate results, which are related to the number of transmittedmessages 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.

A rigid interval graph is an interval graph which has only one clique tree. In 2009, Panda and Das show that all connected unit interval graphs are rigid interval graphs. Generalizing the two classic graph search algorithms, Lexicographic Breadth-First Search (LBFS) and Maximum Cardinality Search (MCS), Corneil and Krueger propose in 2008 the so-called Maximal Neighborhood Search (MNS) and show that one sweep of MNS is enough to recognize chordal graphs. We develop the MNS properties of rigid interval graphs and characterize this graph class in several different ways. This allows us obtain several linear time multi-sweep MNS algorithms for recognizing rigid interval graphs and unit interval graphs, generalizing a corresponding 3-sweep LBFS algorithm for unit interval graph recognition designed by Corneil in 2004. For unit interval graphs, we even present a new linear time 2-sweep MNS certifying recognition algorithm. Submitted:

The interrelationship between control and communication theory is becoming of fundamental importance in many distributed control systems, such as the coordination of a team of autonomous agents. In such a problem, communication constraints impose limits on the achievable control performance. We consider as instance of coordination the consensus problem. The aim of the paper is to characterize the relationship between the amount of information exchanged by the agents and the rate of convergence to the consensus. We show that time-invariant communication networks with circulant symmetries yield slow convergence if the amount of information exchanged by the agents does not scale well with their number. On the other hand, we show that randomly time-varying communication networks allow very fast convergence rates. We also show that, by adding logarithmic quantized data links to time-invariant networks with symmetries, control performance significantly improves with little growth of the required communication effort.

Abstract — Motivated by recent interest for multi-agent systems and smart grid architectures, we discuss the synchronization problem for the network-reduced model of a power system with non-trivial transfer conductances. Our key insight is to exploit the relationship between the power network model and a first-order model of coupled oscillators. Assuming overdamped generators (possibly due to local excitation controllers), a singular perturbation analysis shows the equivalence between the classic swing equations and a non-uniform Kuramoto model characterized by multiple time constants, non-homogeneous coupling, and non-uniform phase shifts. By extending methods from synchronization theory and consensus protocols, we establish sufficient conditions for synchronization of non-uniform Kuramoto oscillators. These conditions reduce to and improve upon previously-available tests for the classic Kuramoto model. By combining our singular perturbation and Kuramoto analyses, we derive concise and purely algebraic conditions that relate synchronization and transient stability of a power network to the underlying network parameters and initial conditions. I.

Abstract — Given a network of interconnected nodes, each with a given initial value, we develop a distributed strategy that enables some or all of the nodes to calculate any arbitrary function of these initial values, despite the presence of some malicious (or faulty) nodes. Our scheme utilizes a linear iterative strategy where, at each time-step, each node updates its value to be a weighted average of its own previous value and those of its neighbors. We consider a node to be malicious if, instead of following the predefined linear iterative strategy, it updates its value arbitrarily at each time-step (perhaps by conspiring and coordinating with other malicious nodes). When there are up to f malicious nodes, we show that any node in the network is guaranteed to be able to calculate any arbitrary function of all initial node values if the graph of the network is at least (2f + 1)-connected. Specifically, we show that under this condition, the nodes can calculate their desired functions after running the linear iteration for a finite number of time-steps (upper bounded by the number of nodes in the network) using almost any set of weights (i.e., for all weights except for a set of measure zero). Our approach treats the problem of fault-tolerant distributed consensus, where all nodes have to calculate the same function despite the presence of faulty or malicious nodes, as a special case. I.

Abstract—We study a model of opinion dynamics introduced by Krause: each agent has an opinion represented by a real number, and updates its opinion by averaging all agent opinions that differ from its own by less than one. We give a new proof of convergence into clusters of agents, with all agents in the same cluster holding the same opinion. We then introduce a particular notion of equilibrium stability and provide lower bounds on the inter-cluster distances at a stable equilibrium. To better understand the behavior of the system when the number of agents is large, we also introduce and study a variant involving a continuum of agents, obtaining partial convergence results and lower bounds on inter-cluster distances, under some mild assumptions. Index Terms—Consensus, decentralized control, multi-agent system, opinion dynamics.

This work considers the problem of reaching consensus in an unreliable linear consensus network. A solution to this problem is relevant for several tasks in multi-agent systems including motion coordination, clock synchronization, and cooperative estimation. By modeling the unreliable nodes as unknown and unmeasurable inputs affecting the network, we recast the problem into an unknown-input system theoretic framework. Only relying on their direct measurements, the agents detect and identify the misbehaving agents using fault detection and isolation techniques. We consider both the case that misbehaviors are simply caused by faults, or that they are the product of a definite, malignant “Byzantine ” strategy. We express the solvability conditions of the two cases in a system theoretic framework, and from a graph theoretic perspective. We show that generically any node can correctly detect and identify the misbehaving agents, provided that the connectivity of the network is sufficiently high. Precisely, for a linear consensus network to be generically resilient to k concurrent faults, the connectivity of the communication graph needs to be 2k + 1, if Byzantine agents are allowed, and k + 1, if non-colluding agents are considered. We finally provide algorithms for detecting and isolating misbehaving agents. The first procedure applies standard fault detection techniques, and affords complete intrusion detection if global knowledge of the graph is available to each agent, at a high computational cost. The second method is designed to exploit the presence in a network of weakly interconnected subparts, and provides computationally efficient detection of misbehaving agents whose behavior deviates more than a threshold, which is quantified in terms of the interconnection structure.

Average consensus consists in the problem of determining the average of some quantities by means of a distributed algorithm. It is a simple instance of problems arising when designing estimation algorithms operating on data produced by sensor networks. Simple solutions based on linear estimation algorithms have already been proposed in the literature and their performance has been analyzed in detail. If the communication links which allow the data exchange between the sensors have some loss, then the estimation performance will degrade. In this contribution the performance degradation due to this data loss is evaluated.

Abstract — In this paper we propose a subgradient method for solving coupled optimization problems in a distributed way given restrictions on the communication topology. The iterative procedure maintains local variables at each node and relies on local subgradient updates in combination with a consensus process. The local subgradient steps are applied simultaneously as opposed to the standard sequential or cyclic procedure. We study convergence properties of the proposed scheme using results from consensus theory and approximate subgradient methods. The framework is illustrated on an optimal distributed finite-time rendezvous problem. I.