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
2112753256
This paper studies local algorithms for autonomous robot systems, namely, algorithms that use only information of the positions of a bounded number of their nearest neighbors. The paper focuses on the spreading problem. It defines measures for the quality of spreading, presents a local algorithm for the one-dimensional spreading problem, proves its convergence to the equally spaced configuration and discusses its convergence rate in the synchronous and semi-synchronous settings. It then presents a local algorithm achieving the exact equally spaced configuration in finite time in the synchronous setting, and proves it is time optimal for local algorithms. Finally, the paper also proposes a possible algorithm for the two-dimensional case and presents partial simulation results of its effectiveness. 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. 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. Given an n × m nonnegative matrix A = (a ij ) and positive integral vectors r e ℜ n and c e Âm having a common one-norm h, the (r,c)-scaling problem is to obtain positive diagonal matrices X and Y, if they exist, such that XAY has row and column sums equal to r and c, respectively. The entropy minimization problem corresponding to A is to find an n × m matrix z = (z ij ) having the same zero pattern as A, the sum of whose entries is a given number h, its row and column sums are within given integral vectors of lower and upper bounds, and such that the entropy function consisting of the sum of the terms z ij ln (z ij a ij ) is minimized. When the lower and upper bounds coincide, matrix scaling and entropy minimization are closely related. In this paper we present several complexity bounds for the e-approximate (r,c)-scaling problem, polynomial in n,m,h, 1 e, and ln V v, where V and v are the largest and the smallest positive entries of A, respectively. These bounds, although not polynomial in ln(1 1 e), not only provide alternative complexities for the polynomial time algorithms, but could result in better overall complexities. In particular, our theoretical analysis supports the practicality of the well-known RAS algorithm. In our analysis we obtain bounds on the norm of scaling vectors which will be used in deriving not only some of the above complexities, but also a complexity for square nonnegative matrices having positive permanent. In particular, our results extend, nontrivially, many bounds for the doubly stochastic scaling of square nonnegative matrices previously given by Kalantari and Khachiyan to the case of general (r,c)-scaling. Finally, we study a more general entropy minimization problem where row and column sums are constrained to lie in prescribed intervals, and the sum of all entries is also prescribed. Balinski and Demange described an RAS type algorithm for its continuous version, but did not analyze its complexity. We show that this algorithm produces an 1 e-approximate solution within complexity polynomial in n, m, h, V v and 1 1 e.