2,989 research outputs found
Distributed Estimation of Graph Spectrum
In this paper, we develop a two-stage distributed algorithm that enables
nodes in a graph to cooperatively estimate the spectrum of a matrix
associated with the graph, which includes the adjacency and Laplacian matrices
as special cases. In the first stage, the algorithm uses a discrete-time linear
iteration and the Cayley-Hamilton theorem to convert the problem into one of
solving a set of linear equations, where each equation is known to a node. In
the second stage, if the nodes happen to know that is cyclic, the algorithm
uses a Lyapunov approach to asymptotically solve the equations with an
exponential rate of convergence. If they do not know whether is cyclic, the
algorithm uses a random perturbation approach and a structural controllability
result to approximately solve the equations with an error that can be made
small. Finally, we provide simulation results that illustrate the algorithm.Comment: 15 pages, 2 figure
Controlled Hopwise Averaging: Bandwidth/Energy-Efficient Asynchronous Distributed Averaging for Wireless Networks
This paper addresses the problem of averaging numbers across a wireless
network from an important, but largely neglected, viewpoint: bandwidth/energy
efficiency. We show that existing distributed averaging schemes have several
drawbacks and are inefficient, producing networked dynamical systems that
evolve with wasteful communications. Motivated by this, we develop Controlled
Hopwise Averaging (CHA), a distributed asynchronous algorithm that attempts to
"make the most" out of each iteration by fully exploiting the broadcast nature
of wireless medium and enabling control of when to initiate an iteration. We
show that CHA admits a common quadratic Lyapunov function for analysis, derive
bounds on its exponential convergence rate, and show that they outperform the
convergence rate of Pairwise Averaging for some common graphs. We also
introduce a new way to apply Lyapunov stability theory, using the Lyapunov
function to perform greedy, decentralized, feedback iteration control. Finally,
through extensive simulation on random geometric graphs, we show that CHA is
substantially more efficient than several existing schemes, requiring far fewer
transmissions to complete an averaging task.Comment: 33 pages, 4 figure
Difference independence of the Riemann zeta function
It is proved that the Riemann zeta function does not satisfy any nontrivial
algebraic difference equation whose coefficients are meromorphic functions
with Nevanlinna characteristic satisfying as Comment: To appear in Acta Arithmetic
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