5 research outputs found
Asymptotic Consensus Without Self-Confidence
This paper studies asymptotic consensus in systems in which agents do not
necessarily have self-confidence, i.e., may disregard their own value during
execution of the update rule. We show that the prevalent hypothesis of
self-confidence in many convergence results can be replaced by the existence of
aperiodic cores. These are stable aperiodic subgraphs, which allow to virtually
store information about an agent's value distributedly in the network. Our
results are applicable to systems with message delays and memory loss.Comment: 13 page
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
How to decide consensus? A combinatorial necessary and sufficient condition and a proof that consensus is decidable but NP-hard
A set of stochastic matrices is a consensus set if for every
sequence of matrices whose elements belong to
and every initial state , the sequence of states defined by converges to a vector whose entries are all identical.
In this paper, we introduce an "avoiding set condition" for compact sets of
matrices and prove in our main theorem that this explicit combinatorial
condition is both necessary and sufficient for consensus. We show that several
of the conditions for consensus proposed in the literature can be directly
derived from the avoiding set condition. The avoiding set condition is easy to
check with an elementary algorithm, and so our result also establishes that
consensus is algorithmically decidable. Direct verification of the avoiding set
condition may require more than a polynomial time number of operations. This is
however likely to be the case for any consensus checking algorithm since we
also prove in this paper that unless , consensus cannot be decided in
polynomial time