11,169 research outputs found
Exact two-terminal reliability of some directed networks
The calculation of network reliability in a probabilistic context has long
been an issue of practical and academic importance. Conventional approaches
(determination of bounds, sums of disjoint products algorithms, Monte Carlo
evaluations, studies of the reliability polynomials, etc.) only provide
approximations when the network's size increases, even when nodes do not fail
and all edges have the same reliability p. We consider here a directed, generic
graph of arbitrary size mimicking real-life long-haul communication networks,
and give the exact, analytical solution for the two-terminal reliability. This
solution involves a product of transfer matrices, in which individual
reliabilities of edges and nodes are taken into account. The special case of
identical edge and node reliabilities (p and rho, respectively) is addressed.
We consider a case study based on a commonly-used configuration, and assess the
influence of the edges being directed (or not) on various measures of network
performance. While the two-terminal reliability, the failure frequency and the
failure rate of the connection are quite similar, the locations of complex
zeros of the two-terminal reliability polynomials exhibit strong differences,
and various structure transitions at specific values of rho. The present work
could be extended to provide a catalog of exactly solvable networks in terms of
reliability, which could be useful as building blocks for new and improved
bounds, as well as benchmarks, in the general case
Exact solutions for the two- and all-terminal reliabilities of the Brecht-Colbourn ladder and the generalized fan
The two- and all-terminal reliabilities of the Brecht-Colbourn ladder and the
generalized fan have been calculated exactly for arbitrary size as well as
arbitrary individual edge and node reliabilities, using transfer matrices of
dimension four at most. While the all-terminal reliabilities of these graphs
are identical, the special case of identical edge () and node ()
reliabilities shows that their two-terminal reliabilities are quite distinct,
as demonstrated by their generating functions and the locations of the zeros of
the reliability polynomials, which undergo structural transitions at
A century of enzyme kinetics. Should we believe in the Km and vmax estimates?
The application of the quasi-steady-state approximation (QSSA) in biochemical kinetics allows the reduction of a complex biochemical system with an initial fast transient into a simpler system. The simplified system yields insights into the behavior of the biochemical reaction, and analytical approximations can be obtained to determine its kinetic parameters. However, this process can lead to inaccuracies due to the inappropriate application of the QSSA. Here we present a number of approximate solutions and determine in which regions of the initial enzyme and substrate concentration parameter space they are valid. In particular, this illustrates that experimentalists must be careful to use the correct approximation appropriate to the initial conditions within the parameter space
Forecasts of non-Gaussian parameter spaces using Box-Cox transformations
Forecasts of statistical constraints on model parameters using the Fisher
matrix abound in many fields of astrophysics. The Fisher matrix formalism
involves the assumption of Gaussianity in parameter space and hence fails to
predict complex features of posterior probability distributions. Combining the
standard Fisher matrix with Box-Cox transformations, we propose a novel method
that accurately predicts arbitrary posterior shapes. The Box-Cox
transformations are applied to parameter space to render it approximately
multivariate Gaussian, performing the Fisher matrix calculation on the
transformed parameters. We demonstrate that, after the Box-Cox parameters have
been determined from an initial likelihood evaluation, the method correctly
predicts changes in the posterior when varying various parameters of the
experimental setup and the data analysis, with marginally higher computational
cost than a standard Fisher matrix calculation. We apply the Box-Cox-Fisher
formalism to forecast cosmological parameter constraints by future weak
gravitational lensing surveys. The characteristic non-linear degeneracy between
matter density parameter and normalisation of matter density fluctuations is
reproduced for several cases, and the capabilities of breaking this degeneracy
by weak lensing three-point statistics is investigated. Possible applications
of Box-Cox transformations of posterior distributions are discussed, including
the prospects for performing statistical data analysis steps in the transformed
Gaussianised parameter space.Comment: 14 pages, 7 figures; minor changes to match version published in
MNRA
- …