1,109 research outputs found

    Equational Reasonings in Wireless Network Gossip Protocols

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    Gossip protocols have been proposed as a robust and efficient method for disseminating information throughout large-scale networks. In this paper, we propose a compositional analysis technique to study formal probabilistic models of gossip protocols expressed in a simple probabilistic timed process calculus for wireless sensor networks. We equip the calculus with a simulation theory to compare probabilistic protocols that have similar behaviour up to a certain tolerance. The theory is used to prove a number of algebraic laws which revealed to be very effective to estimate the performances of gossip networks, with and without communication collisions, and randomised gossip networks. Our simulation theory is an asymmetric variant of the weak bisimulation metric that maintains most of the properties of the original definition. However, our asymmetric version is particularly suitable to reason on protocols in which the systems under consideration are not approximately equivalent, as in the case of gossip protocols

    An Upper Bound on the Convergence Time for Quantized Consensus of Arbitrary Static Graphs

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    We analyze a class of distributed quantized consensus algorithms for arbitrary static networks. In the initial setting, each node in the network has an integer value. Nodes exchange their current estimate of the mean value in the network, and then update their estimation by communicating with their neighbors in a limited capacity channel in an asynchronous clock setting. Eventually, all nodes reach consensus with quantized precision. We analyze the expected convergence time for the general quantized consensus algorithm proposed by Kashyap et al \cite{Kashyap}. We use the theory of electric networks, random walks, and couplings of Markov chains to derive an O(N3logN)O(N^3\log N) upper bound for the expected convergence time on an arbitrary graph of size NN, improving on the state of art bound of O(N5)O(N^5) for quantized consensus algorithms. Our result is not dependent on graph topology. Example of complete graphs is given to show how to extend the analysis to graphs of given topology.Comment: to appear in IEEE Trans. on Automatic Control, January, 2015. arXiv admin note: substantial text overlap with arXiv:1208.078

    Lossy gossip and composition of metrics

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    We study the monoid generated by n-by-n distance matrices under tropical (or min-plus) multiplication. Using the tropical geometry of the orthogonal group, we prove that this monoid is a finite polyhedral fan of dimension n(n-1)/2, and we compute the structure of this fan for n up to 5. The monoid captures gossip among n gossipers over lossy phone lines, and contains the gossip monoid over ordinary phone lines as a submonoid. We prove several new results about this submonoid, as well. In particular, we establish a sharp bound on chains of calls in each of which someone learns something new.Comment: Minor textual edits, final versio
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