Consensus Seeking Over Random Weighted Directed Graphs

ISR develops, applies and teaches advanced methodologies of design and analysis to solve complex, hierarchical, heterogeneous and dynamic problems of engineering technology and systems for industry and government.ISR is a permanent institute of the University of Maryland, within the A. James Clark School of Engineering. It is a graduated National Science Foundation Engineering Research Center. AbstractA consensus problem consists of a group of dynamic agents who seek to agree upon certain quantities of interest. This problem can be generalized in the context of convex metric spaces that extend the standard notion of convexity. In this paper we introduce and analyze a randomized gossip algorithm for solving the generalized consensus problem on convex metric spaces. We study the convergence properties of the algorithm using stochastic differential equations theory. We show that the dynamics of the distances between the states of the agents can be upper bounded by the dynamics of a stochastic differential equation driven by Poisson counters. In addition, we introduce instances of the generalized consensus algorithm for several examples of convex metric spaces together with numerical simulations.I. Introduction Distributed algorithms are found in applications related to sensor, peer-to-peer and ad-hoc networks. A particular distributed algorithm is the consensus (or agreement) algorithm, where a group of dynamic agents seek to agree upon certain quantities of interest by exchanging information among them, according to a set of rules. This problem can model many phenomena involving information exchange between agents such as cooperative control of vehicles, formation control, flocking, synchronization, parallel computing, etc. Distributed computation over networks has a long history in control theory starting with the work of Borkar [21]. In addition to network variability, nodes in sensor networks operate under limited computational, communication, and energy resources. These constraints have motivated the design of gossip algorithms, in which a node communicates with a randomly chosen neighbor. Studies of randomized gossip consensus algorithms can be found in [2], [22].In this paper we introduce and analyze a generalized randomized gossip algorithm for achieving consensus. The algorithm acts on convex metric spaces, that are metric spaces endowed with a convex structure. We show that under the given algorithm, the agents' states converge to consensus with probability one and in the r th mean sense. The convergence study is based on analyzing the dynamics of a set of stochastic differential equations driven by poisson counters. Additionally, for a particular network topology we investigate in more depth the rate of convergence of the first and second moment of the distances between the agents' states. We present instances of the generalized gossip algorithm for three convex

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“…where ξ i and ξ j were defined in (18). The following Corollary follows from the previous Lemma and describes the properties of matrix Q.…”

Section: Preliminary Resultsmentioning

confidence: 91%

“…This motivated the investigation of consensus algorithms under a stochastic framework [6], [10], [11], [18], [20], [21]. In addition to network variability, nodes in sensor networks operate under limited computational, communication, and energy resources.…”

Section: Introductionmentioning

confidence: 99%

A randomized gossip consensus algorithm on convex metric spaces

Matei

Somarakis

Baras

2012

2012 IEEE 51st IEEE Conference on Decision and Control (CDC)

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“…Lin et al [21,22] have studied the consensus problem in the context of formation control of autonomous vehicles and have demonstrated that formation stabilization to a certain point is only feasible 2 Complexity if the sensor digraph has a globally reachable node. The consensus problem has also been studied for switching communication topology [23,24], asynchronous consensus [25,26], high-dimensional consensus [27], consensus with sampled communication [28], and consensus with external disturbances and model uncertainties [29]. The research so far has focused more on the determination of the weakest and the simplest communication topology than the design of the information exchange protocol to achieve consensus under the given topology.…”

Section: Introductionmentioning

confidence: 99%

Finite-Time Consensus with a Time-Varying Reference State and Switching Topology

Wang¹,

Tian²,

Zhu³

et al. 2017

Complexity

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The finite-time consensus problem in the networks of multiple mobile agents is comprehensively investigated. In order to resolve this problem, a novel nonlinear information exchange protocol is proposed. The proposed protocol ensures that the states of the agents are converged to a weighted-average consensus in finite time if the communication topology is a weighted directed graph with a spanning tree and each strongly connected component is detail-balanced. Furthermore, the proposed protocol is also able to solve the finite-time consensus problem of networks with a switching topology. Finally, computer simulations are presented to demonstrate and validate the effectiveness of the theoretical analysis under the proposed protocol.

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“…The purely linear case g(s) ⌘ 1 of the system (2) appears in studies of consensus and synchronization algorithms on a random graph [24,36,19,37]. Related models also frequently demand a theoretical understanding how a (possibly dynamic) random network interaction structure a↵ects well-understood, deterministic behaviors such as phase transitions [1], consensus and synchronization [35,43], and the emergence of collective behavior in locust swarms [21].…”

Section: Introductionmentioning

confidence: 99%

Swarming on Random Graphs II

Brecht¹,

Sudakov

Bertozzi³

2014

J Stat Phys

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We consider an individual-based model where agents interact over a random network via firstorder dynamics that involve both attraction and repulsion. In the case of all-to-all coupling of agents in R d this system has a lowest energy state in which an equal number of agents occupy the vertices of the d-dimensional simplex. The purpose of this paper is to study the behavior of this model when the interaction between the N agents occurs according to an Erdős-Rényi random graph G(N, p) instead of all-to-all coupling. In particular, we study the e↵ect of randomness on the stability of these simplicial solutions, and provide rigorous results to demonstrate that stability of these solutions persists for probabilities greater than Np = O(log N ). In other words, only a relatively small number of interactions are required to maintain stability of the state. The results rely on basic probability arguments together with spectral properties of random graphs.

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Consensus Seeking Over Random Weighted Directed Graphs

Cited by 284 publications

References 16 publications

A randomized gossip consensus algorithm on convex metric spaces

A randomized gossip consensus algorithm on convex metric spaces

Finite-Time Consensus with a Time-Varying Reference State and Switching Topology

Swarming on Random Graphs II

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