6,530 research outputs found
Modeling the variability of rankings
For better or for worse, rankings of institutions, such as universities,
schools and hospitals, play an important role today in conveying information
about relative performance. They inform policy decisions and budgets, and are
often reported in the media. While overall rankings can vary markedly over
relatively short time periods, it is not unusual to find that the ranks of a
small number of "highly performing" institutions remain fixed, even when the
data on which the rankings are based are extensively revised, and even when a
large number of new institutions are added to the competition. In the present
paper, we endeavor to model this phenomenon. In particular, we interpret as a
random variable the value of the attribute on which the ranking should ideally
be based. More precisely, if items are to be ranked then the true, but
unobserved, attributes are taken to be values of independent and
identically distributed variates. However, each attribute value is observed
only with noise, and via a sample of size roughly equal to , say. These
noisy approximations to the true attributes are the quantities that are
actually ranked. We show that, if the distribution of the true attributes is
light-tailed (e.g., normal or exponential) then the number of institutions
whose ranking is correct, even after recalculation using new data and even
after many new institutions are added, is essentially fixed. Formally, is
taken to be of order for any fixed , and the number of institutions
whose ranking is reliable depends very little on .Comment: Published in at http://dx.doi.org/10.1214/10-AOS794 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Tight Upper Bounds for Streett and Parity Complementation
Complementation of finite automata on infinite words is not only a
fundamental problem in automata theory, but also serves as a cornerstone for
solving numerous decision problems in mathematical logic, model-checking,
program analysis and verification. For Streett complementation, a significant
gap exists between the current lower bound and upper
bound , where is the state size, is the number of
Streett pairs, and can be as large as . Determining the complexity
of Streett complementation has been an open question since the late '80s. In
this paper show a complementation construction with upper bound for and for ,
which matches well the lower bound obtained in \cite{CZ11a}. We also obtain a
tight upper bound for parity complementation.Comment: Corrected typos. 23 pages, 3 figures. To appear in the 20th
Conference on Computer Science Logic (CSL 2011
On the limiting behavior of parameter-dependent network centrality measures
We consider a broad class of walk-based, parameterized node centrality
measures for network analysis. These measures are expressed in terms of
functions of the adjacency matrix and generalize various well-known centrality
indices, including Katz and subgraph centrality. We show that the parameter can
be "tuned" to interpolate between degree and eigenvector centrality, which
appear as limiting cases. Our analysis helps explain certain correlations often
observed between the rankings obtained using different centrality measures, and
provides some guidance for the tuning of parameters. We also highlight the
roles played by the spectral gap of the adjacency matrix and by the number of
triangles in the network. Our analysis covers both undirected and directed
networks, including weighted ones. A brief discussion of PageRank is also
given.Comment: First 22 pages are the paper, pages 22-38 are the supplementary
material
Bayesian nonparametric models for ranked data
We develop a Bayesian nonparametric extension of the popular Plackett-Luce
choice model that can handle an infinite number of choice items. Our framework
is based on the theory of random atomic measures, with the prior specified by a
gamma process. We derive a posterior characterization and a simple and
effective Gibbs sampler for posterior simulation. We develop a time-varying
extension of our model, and apply it to the New York Times lists of weekly
bestselling books.Comment: NIPS - Neural Information Processing Systems (2012
Existence of an infinite particle limit of stochastic ranking process
We study a stochastic particle system which models the time evolution of the
ranking of books by online bookstores (e.g., Amazon). In this system, particles
are lined in a queue. Each particle jumps at random jump times to the top of
the queue, and otherwise stays in the queue, being pushed toward the tail every
time another particle jumps to the top. In an infinite particle limit, the
random motion of each particle between its jumps converges to a deterministic
trajectory. (This trajectory is actually observed in the ranking data on web
sites.) We prove that the (random) empirical distribution of this particle
system converges to a deterministic space-time dependent distribution. A core
of the proof is the law of large numbers for {\it dependent} random variables
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