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.
After writing the proof of Theorem , we became aware of two recent papers both containing Theorem in the synchronous case. @cite_14 , Hendrickx and Tsitsiklis showed that the agreement algorithm achieves asymptotic consensus under assumptions A, B with @math , D1 and the so-called cut-balance condition. In light of Proposition below, the latter condition turns out to correspond to the decentralized model. Independently, Touri and Nedi 'c @cite_7 established a general convergence result for the infinite product of random stochastic matrices, and to do that, they first proved that this result holds in the deterministic case; see Lemma 5 in @cite_7 . It is easy to see that this lemma actually coincides with our Theorem in the particular case of a synchronous multiagent system. In fact, our technique for proving the lemmas in Sections and infra specialized to @math , is similar to the one used for the proof of the deterministic result in @cite_7 .
@cite_14 @cite_7
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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. 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. This paper uses recently established properties of compositions of directed graphs together with results from the theory of nonhomogeneous Markov chains to derive worst case convergence rates for the headings of a group of mobile autonomous agents which arise in connection with the widely studied Vicsek consensus problem. The paper also uses graph-theoretic constructions to solve modified versions of the Vicsek problem in which there are measurement delays, asynchronous events, or a group leader. In all three cases the conditions under which consensus is achieved prove to be almost the same as the conditions under which consensus is achieved in the synchronous, delay-free, leaderless case. Mulmuley [Mul12a] recently gave an explicit version of Noether’s Normalization lemma for ring of invariants of matrices under simultaneous conjugation, under the conjecture that there are deterministic black-box algorithms for polynomial identity testing (PIT). He argued that this gives evidence that constructing such algorithms for PIT is beyond current techniques. In this work, we show this is not the case. That is, we improve Mulmuley’s reduction and correspondingly weaken the conjecture regarding PIT needed to give explicit Noether Normalization. We then observe that the weaker conjecture has recently been nearly settled by the authors ([FS12]), who gave quasipolynomial size hitting sets for the class of read-once oblivious algebraic branching programs (ROABPs). This gives the desired explicit Noether Normalization unconditionally, up to quasipolynomial factors. As a consequence of our proof we give a deterministic parallel polynomial-time algorithm for deciding if two matrix tuples have intersecting orbit closures, under simultaneous conjugation. We also study the strength of conjectures that Mulmuley requires to obtain similar results as ours. We prove that his conjectures are stronger, in the sense that the computational model he needs PIT algorithms for is equivalent to the well-known algebraic branching program (ABP) model, which is provably stronger than the ROABP model. Finally, we consider the depth-3 diagonal circuit model as defined by Saxena [Sax08], as PIT algorithms for this model also have implications in Mulmuley’s work. Previous work (such as [ASS12] and [FS12]) have given quasipolynomial size hitting sets for this model. In this work, we give a much simpler construction of such hitting sets, using techniques of Shpilka and Volkovich [SV09].