35,418 research outputs found
A sufficient condition for the existence of fractional -critical covered graphs
In data transmission networks, the availability of data transmission is
equivalent to the existence of the fractional factor of the corresponding graph
which is generated by the network. Research on the existence of fractional
factors under specific network structures can help scientists design and
construct networks with high data transmission rates. A graph is called a
fractional -covered graph if for any , admits a
fractional -factor covering . A graph is called a fractional
-critical covered graph if after removing any vertices of , the
resulting graph of is a fractional -covered graph. In this paper, we
verify that if a graph of order satisfies
,
and
, then is a
fractional -critical covered graph, where
be two functions such that for all ,
which is a generalization of Zhou's previous result [S. Zhou, Some new
sufficient conditions for graphs to have fractional -factors, International
Journal of Computer Mathematics 88(3)(2011)484--490].Comment: 1
Fractional total colourings of graphs of high girth
Reed conjectured that for every epsilon>0 and Delta there exists g such that
the fractional total chromatic number of a graph with maximum degree Delta and
girth at least g is at most Delta+1+epsilon. We prove the conjecture for
Delta=3 and for even Delta>=4 in the following stronger form: For each of these
values of Delta, there exists g such that the fractional total chromatic number
of any graph with maximum degree Delta and girth at least g is equal to
Delta+1
Graph cluster randomization: network exposure to multiple universes
A/B testing is a standard approach for evaluating the effect of online
experiments; the goal is to estimate the `average treatment effect' of a new
feature or condition by exposing a sample of the overall population to it. A
drawback with A/B testing is that it is poorly suited for experiments involving
social interference, when the treatment of individuals spills over to
neighboring individuals along an underlying social network. In this work, we
propose a novel methodology using graph clustering to analyze average treatment
effects under social interference. To begin, we characterize graph-theoretic
conditions under which individuals can be considered to be `network exposed' to
an experiment. We then show how graph cluster randomization admits an efficient
exact algorithm to compute the probabilities for each vertex being network
exposed under several of these exposure conditions. Using these probabilities
as inverse weights, a Horvitz-Thompson estimator can then provide an effect
estimate that is unbiased, provided that the exposure model has been properly
specified.
Given an estimator that is unbiased, we focus on minimizing the variance.
First, we develop simple sufficient conditions for the variance of the
estimator to be asymptotically small in n, the size of the graph. However, for
general randomization schemes, this variance can be lower bounded by an
exponential function of the degrees of a graph. In contrast, we show that if a
graph satisfies a restricted-growth condition on the growth rate of
neighborhoods, then there exists a natural clustering algorithm, based on
vertex neighborhoods, for which the variance of the estimator can be upper
bounded by a linear function of the degrees. Thus we show that proper cluster
randomization can lead to exponentially lower estimator variance when
experimentally measuring average treatment effects under interference.Comment: 9 pages, 2 figure
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