In this article, we establish orientation and connectivity based criteria for the agreement algorithm to achieve asymptotic consensus in the context of time-varying topology and communication delays. These criteria unify and extend many earlier convergence results on the agreement algorithm for deterministic and discrete-time multiagent systems.
@cite_17 proved that in the case of stochastic matrices with equal positive entries in each row (usually referred to as the equal neighbor model ,), and under assumptions A and B with @math , the agreement algorithm achieves asymptotic consensus when all the communication graphs are oriented. Their convergence result thus coincides with Theorem in the particular case of a synchronous multiagent system and with the equal neighbor model.
@cite_17
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We consider distributed plurality consensus in a complete graph of size @math with @math initial opinions. We design an efficient and simple protocol in the asynchronous communication model that ensures that all nodes eventually agree on the initially most frequent opinion. In this model, each node is equipped with a random Poisson clock with parameter @math . Whenever a node's clock ticks, it samples some neighbors, uniformly at random and with replacement, and adjusts its opinion according to the sample. A prominent example is the so-called two-choices algorithm in the synchronous model, where in each round, every node chooses two neighbors uniformly at random, and if the two sampled opinions coincide, then that opinion is adopted. This protocol is very efficient and well-studied when @math . If @math for some small @math , we show that it converges to the initial plurality opinion within @math rounds, w.h.p., as long as the initial difference between the largest and second largest opinion is @math . On the other side, we show that there are cases in which @math rounds are needed, w.h.p. One can beat this lower bound in the synchronous model by combining the two-choices protocol with randomized broadcasting. Our main contribution is a non-trivial adaptation of this approach to the asynchronous model. If the support of the most frequent opinion is at least @math times that of the second-most frequent one and @math , then our protocol achieves the best possible run time of @math , w.h.p. We relax full synchronicity by allowing @math nodes to be poorly synchronized, and the well synchronized nodes are only required to be within a certain time difference from one another. We enforce this synchronicity by introducing a novel gadget into the protocol. Consider a network of n processes each of which has a d-dimensional vector of reals as its input. Each process can communicate directly with all the processes in the system; thus the communication network is a complete graph. All the communication channels are reliable and FIFO (first-in-first-out). The problem of Byzantine vector consensus (BVC) requires agreement on a d-dimensional vector that is in the convex hull of the d-dimensional input vectors at the non-faulty processes. We obtain the following results for Byzantine vector consensus in complete graphs while tolerating up to f Byzantine failures: * We prove that in a synchronous system, n >= max(3f+1, (d+1)f+1) is necessary and sufficient for achieving Byzantine vector consensus. * In an asynchronous system, it is known that exact consensus is impossible in presence of faulty processes. For an asynchronous system, we prove that n >= (d+2)f+1 is necessary and sufficient to achieve approximate Byzantine vector consensus. Our sufficiency proofs are constructive. We show sufficiency by providing explicit algorithms that solve exact BVC in synchronous systems, and approximate BVC in asynchronous systems. We also obtain tight bounds on the number of processes for achieving BVC using algorithms that are restricted to a simpler communication pattern. In this work we present a new algorithm to solve the average-consensus problem. The main goal of this algorithm is to obtain exact convergence despite the existence of quantized communication channels between the agents. Starting from the Zoom-in Zoom-out strategy already presented in [5], we introduce the equations describing the behaviour of the algorithm and we formally prove the asymptotic agreement. We will also show that, under a reasonable hypothesis, the algorithm parameters ensuring convergence, can be chosen regardless the number of agents. This paper gives a lower bound on the convergence rate of a class of network consensus algorithms. Two different approaches using directed graphs as a main tool are introduced: one is to compute the "scrambling constants" of stochastic matrices associated with "neighbor shared graphs" and the other is to analyze random walks on a sequence of graphs. Both approaches prove that the time to reach consensus within a dynamic network is logarithmic in the relative error and is in worst case exponential in the size of the network. The paper studies asynchronous consensus problems of continuous-time multi-agent systems with discontinuous information transmission. The proposed consensus control strategy is implemented based on the state information of each agent's neighbors at some discrete times. The asynchrony means that each agent's update times, at which the agent adjusts its dynamics, are independent of others'. Furthermore, it is assumed that the communication topology among agents is time-dependent and the information transmission is with bounded time-varying delays. If the union of the communication topology across any time interval with some given length contains a spanning tree, the consensus problem is shown to be solvable. The analysis tool developed in this paper is based on nonnegative matrix theory and graph theory. The main contribution of this paper is to provide a valid distributed consensus algorithm that overcomes the difficulties caused by unreliable communication channels, such as intermittent information transmission, switching communication topology, and time-varying communication delays, and therefore has its obvious practical applications. Simulation examples are provided to demonstrate the effectiveness of the theoretical results. We consider the following distributed consensus problem: Each node in a complete communication network of size n initially holds an opinion, which is chosen arbitrarily from a finite set Σ. The system must converge toward a consensus state in which all, or almost all nodes, hold the same opinion. Moreover, this opinion should be valid, i.e., it should be one among those initially present in the system. This condition should be met even in the presence of a malicious adversary who can modify the opinions of a bounded subset of nodes, adaptively chosen in every round. We consider the 3-majority dynamics: At every round, every node pulls the opinion from three random neighbors and sets his new opinion to the majority one (ties are broken arbitrarily). Let k be the number of valid opinions. We show that, if k ≤ nα, where α is a suitable positive constant, the 3-majority dynamics converges in time polynomial in k and log n with high probability even in the presence of an adversary who can affect up to o([EQUATION]) nodes at each round. Previously, the convergence of the 3-majority protocol was known for |Σ| = 2 only, with an argument that is robust to adversarial errors. On the other hand, no anonymous, uniform-gossip protocol that is robust to adversarial errors was known for |Σ| > 2. We study the average consensus problem of multi-agent systems for general network topologies with unidirectional information flow. We propose two linear distributed algorithms, deterministic and gossip, respectively for the cases where the inter-agent communication is synchronous and asynchronous. In both cases, the developed algorithms guarantee state averaging on arbitrary strongly connected digraphs; in particular, this graphical condition does not require that the network be balanced or symmetric, thereby extending previous results in the literature. The key novelty of our approach is to augment an additional variable for each agent, called "surplus", whose function is to locally record individual state updates. For convergence analysis, we employ graph-theoretic and nonnegative matrix tools, plus the eigenvalue perturbation theory playing a crucial role. Abstract Linear consensus is a distributed algorithm which ensures that, under certain assumptions, a set of agents reach asymptotically the same opinion over a certain variable. Moreover this occurs with only a local exchange of information, namely the information exchange takes place only between agents which are neighbors in a graph representing the system communication architecture. Several performance metrics have been proposed for the evaluation of this algorithm. Particularly interesting and challenging is to relate performance to the communication topology. Different performance metrics may yield to different answers in comparing alternative communication topologies. In this paper, we present a number of possible performance metrics. Moreover, we show how these metrics are related to the communication topology. In particular, when available, we present bounds by which it is possible to relate performance and topology for general graphs, for graphs with symmetries, called Cayley graphs, and for geometric graphs. We develop an information-theoretic view of the stochastic block model, a popular statistical model for the large-scale structure of complex networks. A graph G from such a model is generated by first assigning vertex labels at random from a finite alphabet, and then connecting vertices with edge probabilities depending on the labels of the endpoints. In the case of the symmetric two-group model, we establish an explicit ‘single-letter’ characterization of the per-vertex mutual information between the vertex labels and the graph, when the graph average degree diverges. The explicit expression of the mutual information is intimately related to estimation-theoretic quantities, and -in particular- reveals a phase transition at the critical point for community detection. Below the critical point the per-vertex mutual information is asymptotically the same as if edges were independent of the vertex labels. Correspondingly, no algorithm can estimate the partition better than random guessing. Conversely, above the threshold, the per-vertex mutual information is strictly smaller than the independent-edges upper bound. In this regime there exists a procedure that estimates the vertex labels better than random guessing. We consider stationary consensus protocols for networks of dynamic agents with fixed topologies. At each time instant, each agent knows only its and its neighbors’ state, but must reach consensus on a group decision value that is function of all the agents’ initial state. We show that the agents can reach consensus if the value of such a function is time-invariant when computed over the agents’ state trajectories. We use this basic result to introduce a non-linear protocol design rule allowing consensus on a quite general set of values. Such a set includes, e.g., any generalized mean of order p of the agents’ initial states. As a second contribution we show that our protocol design is the solution of individual optimizations performed by the agents. This notion suggests a game theoretic interpretation of consensus problems as mechanism design problems. Under this perspective a supervisor entails the agents to reach a consensus by imposing individual objectives. We prove that such objectives can be chosen so that rational agents have a unique optimal protocol, and asymptotically reach consensus on a desired group decision value. We use a Lyapunov approach to prove that the asymptotical consensus can be reached when the communication links between nearby agents define a time-invariant undirected network. Finally we perform a simulation study concerning the vertical alignment maneuver of a team of unmanned air vehicles. Two related problems are treated in continuous time. First, the state agreement problem is studied for coupled nonlinear differential equations. The vector fields can switch within a finite family. Associated to each vector field is a directed graph based in a natural way on the interaction structure of the subsystems. Generalizing the work of Moreau, under the assumption that the vector fields satisfy a certain subtangentiality condition, it is proved that asymptotic state agreement is achieved if and only if the dynamic interaction digraph has the property of being sufficiently connected over time. The proof uses nonsmooth analysis. Second, the rendezvous problem for kinematic point-mass mobile robots is studied when the robots’ fields of view have a fixed radius. The circumcenter control law of Ando e [IEEE Trans. Robotics Automation, 15 (1999), pp. 818-828] is shown to solve the problem. The rendezvous problem is a kind of state agreement problem, but the interaction structure is state dependent. This paper is concerned with the convergence of a class of continuous-time nonlinear consensus algorithms for single integrator agents. In the consensus algorithms studied here, the control input of each agent is assumed to be a state-dependent combination of the relative positions of its neighbors in the information flow graph. Using a novel approach based on the smallest order of the nonzero derivative, it is shown that under some mild conditions the convex hull of the agents has a contracting property. A set-valued LaSalle-like approach is subsequently employed to show the convergence of the agents to a common point. The results are shown to be more general than the ones reported in the literature in some cases. An illustrative example demonstrates how the proposed convergence conditions can be verified. In the classical consensus problem, each of n processors receives a private input value and produces a decision value which is one of the original input values, with the requirement that all processors decide the same value. A central result in distributed computing is that, in several standard models including the asynchronous shared-memory model, this problem has no deterministic solution. The k-set agreement problem is a generalization of the classical consensus proposed by Chaudhuri [ Inform. and Comput., 105 (1993), pp. 132--158], where the agreement condition is weakened so that the decision values produced may be different, as long as the number of distinct values is at most k. For @math it was not known whether this problem is solvable deterministically in the asynchronous shared memory model. In this paper, we resolve this question by showing that for any k < n, there is no deterministic wait-free protocol for n processors that solves the k-set agreement problem. The proof technique is new: it is based on the development of a topological structure on the set of possible processor schedules of a protocol. This topological structure has a natural interpretation in terms of the knowledge of the processors of the state of the system. This structure reveals a close analogy between the impossibility of wait-free k-set agreement and the Brouwer fixed point theorem for the k-dimensional ball. We develop an information-theoretic view of the stochastic block model, a popular statistical model for the large-scale structure of complex networks. A graph @math from such a model is generated by first assigning vertex labels at random from a finite alphabet, and then connecting vertices with edge probabilities depending on the labels of the endpoints. In the case of the symmetric two-group model, we establish an explicit single-letter' characterization of the per-vertex mutual information between the vertex labels and the graph. The explicit expression of the mutual information is intimately related to estimation-theoretic quantities, and --in particular-- reveals a phase transition at the critical point for community detection. Below the critical point the per-vertex mutual information is asymptotically the same as if edges were independent. Correspondingly, no algorithm can estimate the partition better than random guessing. Conversely, above the threshold, the per-vertex mutual information is strictly smaller than the independent-edges upper bound. In this regime there exists a procedure that estimates the vertex labels better than random guessing. In the classical consensus problem,each of n processors receives a private input value and produces a decision value which is one of the original input values,with the requirement that all processors decide the same value. A central result in distributed computing is that,in several standard models including the asynchronous shared-memory model,this problem has no determinis- tic solution. The k-set agreement problem is a generalization of the classical consensus proposed by Chaudhuri (Inform. and Comput.,105 (1993),pp. 132-158),where the agreement condition is weak- ened so that the decision values produced may be different,as long as the number of distinct values is at most k .F or n>k ≥ 2 it was not known whether this problem is solvable deterministically in the asynchronous shared memory model. In this paper,we resolve this question by showing that for any k<n ,there is no deterministic wait-free protocol for n processors that solves the k-set agreement problem. The proof technique is new: it is based on the development of a topological structure on the set of possible processor schedules of a protocol. This topological structure has a natural interpretation in terms of the knowledge of the processors of the state of the system. This structure reveals a close analogy between the impossibility of wait-free k-set agreement and the Brouwer fixed point theorem for the k-dimensional ball. The convergence of products of stochastic matrices has proven to be critical in establishing the effectiveness of distributed coordination algorithms for multi-agent systems. After reviewing some classic and recent results on infinite backward products of stochastic matrices, we provide a new necessary and sufficient condition for the convergence in terms of matrices from the Sarymsakov class of stochastic matrices, which complements the known other necessary and sufficient conditions. To gain insight into the somewhat obscure definition of the Sarymsakov class, we generalize some conditions in the definition and prove that the resulted set of matrices is exactly the set of indecomposable, aperiodic, stochastic matrices that has been extensively studied in the past. To apply the gained knowledge about the Sarymsakov class to the coordination of multi-agent systems, we investigate a specific coordination task with asynchronous update events. Then the set of scrambling stochastic matrices, a subclass of the Sarymsakov class, is utilized to establish the convergence of the system's state even when there is no common clock for the agents to synchronize their update actions. In this paper, we formulate and solve a randomized optimal consensus problem for multi-agent systems with stochastically time-varying interconnection topology. The considered multi-agent system with a simple randomized iterating rule achieves an almost sure consensus meanwhile solving the optimization problem min"z"@?"R"^"[email protected]?"i"="1^nf"i(z), in which the optimal solution set of objective function f"i can only be observed by agent i itself. At each time step, simply determined by a Bernoulli trial, each agent independently and randomly chooses either taking an average among its neighbor set, or projecting onto the optimal solution set of its own optimization component. Both directed and bidirectional communication graphs are studied. Connectivity conditions are proposed to guarantee an optimal consensus almost surely with proper convexity and intersection assumptions. The convergence analysis is carried out using convex analysis. We compare the randomized algorithm with the deterministic one via a numerical example. The results illustrate that a group of autonomous agents can reach an optimal opinion by each node simply making a randomized trade-off between following its neighbors or sticking to its own opinion at each time step. When solving consensus optimization problems over a graph, there is often an explicit characterization of the convergence rate of Gradient Descent (GD) using the spectrum of the graph Laplacian. The same type of problems under the Alternating Direction Method of Multipliers (ADMM) are, however, poorly understood. For instance, simple but important non-strongly-convex consensus problems have not yet being analyzed, especially concerning the dependency of the convergence rate on the graph topology. Recently, for a non-strongly-convex consensus problem, a connection between distributed ADMM and lifted Markov chains was proposed, followed by a conjecture that ADMM is faster than GD by a square root factor in its convergence time, in close analogy to the mixing speedup achieved by lifting several Markov chains. Nevertheless, a proof of such a claim is is still lacking. Here we provide a full characterization of the convergence of distributed over-relaxed ADMM for the same type of consensus problem in terms of the topology of the underlying graph. Our results provide explicit formulas for optimal parameter selection in terms of the second largest eigenvalue of the transition matrix of the graph's random walk. Another consequence of our results is a proof of the aforementioned conjecture, which interestingly, we show it is valid for any graph, even the ones whose random walks cannot be accelerated via Markov chain lifting. In decentralized consensus optimization, a connected network of agents collaboratively minimize the sum of their local objective functions over a common decision variable, where their information exchange is restricted between the neighbors. To this end, one can first obtain a problem reformulation and then apply the alternating direction method of multipliers (ADMM). The method applies iterative computation at the individual agents and information exchange between the neighbors. This approach has been observed to converge quickly and deemed powerful. This paper establishes its linear convergence rate for the decentralized consensus optimization problem with strongly convex local objective functions. The theoretical convergence rate is explicitly given in terms of the network topology, the properties of local objective functions, and the algorithm parameter. This result is not only a performance guarantee but also a guideline toward accelerating the ADMM convergence. This paper proves a necessary and sufficient condition for the existence of iterative, algorithms that achieve approximate Byzantine consensus in arbitrary directed graphs, where each directed edge represents a communication channel between a pair of nodes. The class of iterative algorithms considered in this paper ensures that, after each iteration of the algorithm, the state of each fault-free node remains in the convex hull of the states of the fault-free nodes at the end of the previous iteration. The following convergence requirement is imposed: for any e > 0, after a sufficiently large number of iterations, the states of the fault-free nodes are guaranteed to be within e of each other. To the best of our knowledge, tight necessary and sufficient conditions for the existence of such iterative consensus algorithms in synchronous arbitrary point-to-point networks in presence of Byzantine faults, have not been developed previously. The methodology and results presented in this paper can also be extended to asynchronous systems.