895 research outputs found
Reachability of Consensus and Synchronizing Automata
We consider the problem of determining the existence of a sequence of
matrices driving a discrete-time consensus system to consensus. We transform
this problem into one of the existence of a product of the transition
(stochastic) matrices that has a positive column. We then generalize some
results from automata theory to sets of stochastic matrices. We obtain as a
main result a polynomial-time algorithm to decide the existence of a sequence
of matrices achieving consensus.Comment: Update after revie
Average Consensus in the Presence of Delays and Dynamically Changing Directed Graph Topologies
Classical approaches for asymptotic convergence to the global average in a
distributed fashion typically assume timely and reliable exchange of
information between neighboring components of a given multi-component system.
These assumptions are not necessarily valid in practical settings due to
varying delays that might affect transmissions at different times, as well as
possible changes in the underlying interconnection topology (e.g., due to
component mobility). In this work, we propose protocols to overcome these
limitations. We first consider a fixed interconnection topology (captured by a
- possibly directed - graph) and propose a discrete-time protocol that can
reach asymptotic average consensus in a distributed fashion, despite the
presence of arbitrary (but bounded) delays in the communication links. The
protocol requires that each component has knowledge of the number of its
outgoing links (i.e., the number of components to which it sends information).
We subsequently extend the protocol to also handle changes in the underlying
interconnection topology and describe a variety of rather loose conditions
under which the modified protocol allows the components to reach asymptotic
average consensus. The proposed algorithms are illustrated via examples.Comment: 37 page
Products of Generalized Stochastic Sarymsakov Matrices
In the set of stochastic, indecomposable, aperiodic (SIA) matrices, the class
of stochastic Sarymsakov matrices is the largest known subset (i) that is
closed under matrix multiplication and (ii) the infinitely long left-product of
the elements from a compact subset converges to a rank-one matrix. In this
paper, we show that a larger subset with these two properties can be derived by
generalizing the standard definition for Sarymsakov matrices. The
generalization is achieved either by introducing an "SIA index", whose value is
one for Sarymsakov matrices, and then looking at those stochastic matrices with
larger SIA indices, or by considering matrices that are not even SIA. Besides
constructing a larger set, we give sufficient conditions for generalized
Sarymsakov matrices so that their products converge to rank-one matrices. The
new insight gained through studying generalized Sarymsakov matrices and their
products has led to a new understanding of the existing results on consensus
algorithms and will be helpful for the design of network coordination
algorithms
A stabilization theorem for dynamics of continuous opinions
A stabilization theorem for processes of opinion dynamics is presented. The
theorem is applicable to a wide class of models of continuous opinion dynamics
based on averaging (like the models of Hegselmann-Krause and
Weisbuch-Deffuant). The analysis detects self-confidence as a driving force of
stabilization.Comment: 7 pages, no figures, first time presented at First Bonzenfreies
Colloquium on Market Dynamics and Quantitative Economics, Sep 9/10 200
Approximate Consensus in Highly Dynamic Networks: The Role of Averaging Algorithms
In this paper, we investigate the approximate consensus problem in highly
dynamic networks in which topology may change continually and unpredictably. We
prove that in both synchronous and partially synchronous systems, approximate
consensus is solvable if and only if the communication graph in each round has
a rooted spanning tree, i.e., there is a coordinator at each time. The striking
point in this result is that the coordinator is not required to be unique and
can change arbitrarily from round to round. Interestingly, the class of
averaging algorithms, which are memoryless and require no process identifiers,
entirely captures the solvability issue of approximate consensus in that the
problem is solvable if and only if it can be solved using any averaging
algorithm. Concerning the time complexity of averaging algorithms, we show that
approximate consensus can be achieved with precision of in a
coordinated network model in synchronous
rounds, and in rounds when
the maximum round delay for a message to be delivered is . While in
general, an upper bound on the time complexity of averaging algorithms has to
be exponential, we investigate various network models in which this exponential
bound in the number of nodes reduces to a polynomial bound. We apply our
results to networked systems with a fixed topology and classical benign fault
models, and deduce both known and new results for approximate consensus in
these systems. In particular, we show that for solving approximate consensus, a
complete network can tolerate up to 2n-3 arbitrarily located link faults at
every round, in contrast with the impossibility result established by Santoro
and Widmayer (STACS '89) showing that exact consensus is not solvable with n-1
link faults per round originating from the same node
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