16,184 research outputs found
Expected propagation time for probabilistic zero forcing
Zero forcing is a coloring process on a graph that was introduced more than fifteen years ago in several different applications. The goal is to color all the vertices blue by repeated use of a (deterministic) color change rule. Probabilistic zero forcing was introduced by Kang and Yi in [Bull. Inst. Combin. Appl. 67 (2013), 9–16] and yields a discrete dynamical system, which is a better model for some applications. Since in a connected graph any one vertex can eventually color the entire graph blue using probabilistic zero forcing, the expected time to do this is a natural parameter to study. We determine expected propagation time exactly for paths and cycles, establish the asymptotic value for stars, and present asymptotic upper and lower bounds for any graph in terms of its radius and order. We apply these results to obtain values and bounds on ℓ-round probabilistic zero forcing and confidence levels for propagation time
Using Markov chains to determine expected propagation time for probabilistic zero forcing
Zero forcing is a coloring game played on a graph where each vertex is
initially colored blue or white and the goal is to color all the vertices blue
by repeated use of a (deterministic) color change rule starting with as few
blue vertices as possible. Probabilistic zero forcing yields a discrete
dynamical system governed by a Markov chain. Since in a connected graph any one
vertex can eventually color the entire graph blue using probabilistic zero
forcing, the expected time to do this studied. Given a Markov transition matrix
for a probabilistic zero forcing process, we establish an exact formula for
expected propagation time. We apply Markov chains to determine bounds on
expected propagation time for various families of graphs
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Investigating the impact of remotely sensed precipitation and hydrologic model uncertainties on the ensemble streamflow forecasting
In the past few years sequential data assimilation (SDA) methods have emerged as the best possible method at hand to properly treat all sources of error in hydrological modeling. However, very few studies have actually implemented SDA methods using realistic input error models for precipitation. In this study we use particle filtering as a SDA method to propagate input errors through a conceptual hydrologic model and quantify the state, parameter and streamflow uncertainties. Recent progress in satellite-based precipitation observation techniques offers an attractive option for considering spatiotemporal variation of precipitation. Therefore, we use the PERSIANN-CCS precipitation product to propagate input errors through our hydrologic model. Some uncertainty scenarios are set up to incorporate and investigate the impact of the individual uncertainty sources from precipitation, parameters and also combined error sources on the hydrologic response. Also probabilistic measure are used to quantify the quality of ensemble prediction. Copyright 2006 by the American Geophysical Union
A Comparison between the Zero Forcing Number and the Strong Metric Dimension of Graphs
The \emph{zero forcing number}, , of a graph is the minimum
cardinality of a set of black vertices (whereas vertices in are
colored white) such that is turned black after finitely many
applications of "the color-change rule": a white vertex is converted black if
it is the only white neighbor of a black vertex. The \emph{strong metric
dimension}, , of a graph is the minimum among cardinalities of all
strong resolving sets: is a \emph{strong resolving set} of
if for any , there exists an such that either
lies on an geodesic or lies on an geodesic. In this paper, we
prove that for a connected graph , where is
the cycle rank of . Further, we prove the sharp bound
when is a tree or a unicyclic graph, and we characterize trees
attaining . It is easy to see that can be
arbitrarily large for a tree ; we prove that and
show that the bound is sharp.Comment: 8 pages, 5 figure
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