125,041 research outputs found
A simple axiomatization of the median procedure on median graphs
A profile = (x1, ..., xk), of length k, in a finite connected graph G is a sequence
of vertices of G, with repetitions allowed. A median x of is a vertex for which
the sum of the distances from x to the vertices in the profile is minimum. The
median function finds the set of all medians of a profile. Medians are important in
location theory and consensus theory. A median graph is a graph for which every
profile of length 3 has a unique median. Median graphs are well studied. They
arise in many arenas, and have many applications.
We establish a succinct axiomatic characterization of the median procedure on
median graphs. This is a simplification of the characterization given by McMorris,
Mulder and Roberts [17] in 1998. We show that the median procedure can be characterized
on the class of all median graphs with only three simple and intuitively
appealing axioms: anonymity, betweenness and consistency. We also extend a key
result of the same paper, characterizing the median function for profiles of even
length on median graphs
Marc Barbut au pays des médianes
Mathématique/Théorie des treillis Classification AMS : 06 - Order, lattices, ordered algebraic structures/06B - Lattices 05 - Combinatorics For finite fields/05C - Graph theory for applications of graphs 91 - Game theory, economics, social and behavioral sciences/91B - Mathematical economics for econometrics/91B14 - Social choice URL des Documents de travail : http://centredeconomiesorbonne.univ-paris1.fr/bandeau-haut/document-de-travail/Documents de travail du Centre d'Economie de la Sorbonne 2013.39 - ISSN : 1955-611XThe notion of median originally appeared in Statistics was introduced more later in Algebra and Combinatorics. Marc Barbut was the first to develop the link between these two notions of median. I present his precursory works linking the metric medians and the algebraic medians of a distributive lattice and using these links within the framework of the "median procedure" in data analysis. I also give a short survey on the development of the - more general - theory of "median spaces" and I mention some problems about the median procedure.La notion de médiane apparue d'abord en statistique (notamment sous forme métrique) l'a été ensuite en algèbre et combinatoire. Marc Barbut a été le premier à développer le lien entre ces deux aspects. Je présente ses travaux précurseurs reliant les médianes métriques et les médianes latticielles d'un treillis distributif et utilisant leurs liens dans le cadre d'une " procédure médiane " en analyse des données. Je fais aussi un bref survol du développement de la théorie (plus générale) des " espaces à médianes " et des problèmes posés par la procédure médiane
One-step Estimation of Networked Population Size: Respondent-Driven Capture-Recapture with Anonymity
Population size estimates for hidden and hard-to-reach populations are
particularly important when members are known to suffer from disproportion
health issues or to pose health risks to the larger ambient population in which
they are embedded. Efforts to derive size estimates are often frustrated by a
range of factors that preclude conventional survey strategies, including social
stigma associated with group membership or members' involvement in illegal
activities.
This paper extends prior research on the problem of network population size
estimation, building on established survey/sampling methodologies commonly used
with hard-to-reach groups. Three novel one-step, network-based population size
estimators are presented, to be used in the context of uniform random sampling,
respondent-driven sampling, and when networks exhibit significant clustering
effects. Provably sufficient conditions for the consistency of these estimators
(in large configuration networks) are given. Simulation experiments across a
wide range of synthetic network topologies validate the performance of the
estimators, which are seen to perform well on a real-world location-based
social networking data set with significant clustering. Finally, the proposed
schemes are extended to allow them to be used in settings where participant
anonymity is required. Systematic experiments show favorable tradeoffs between
anonymity guarantees and estimator performance.
Taken together, we demonstrate that reasonable population estimates can be
derived from anonymous respondent driven samples of 250-750 individuals, within
ambient populations of 5,000-40,000. The method thus represents a novel and
cost-effective means for health planners and those agencies concerned with
health and disease surveillance to estimate the size of hidden populations.
Limitations and future work are discussed in the concluding section
Nice labeling problem for event structures: a counterexample
In this note, we present a counterexample to a conjecture of Rozoy and
Thiagarajan from 1991 (called also the nice labeling problem) asserting that
any (coherent) event structure with finite degree admits a labeling with a
finite number of labels, or equivalently, that there exists a function such that an event structure with degree
admits a labeling with at most labels. Our counterexample is based on
the Burling's construction from 1965 of 3-dimensional box hypergraphs with
clique number 2 and arbitrarily large chromatic numbers and the bijection
between domains of event structures and median graphs established by
Barth\'elemy and Constantin in 1993
Sparse Median Graphs Estimation in a High Dimensional Semiparametric Model
In this manuscript a unified framework for conducting inference on complex
aggregated data in high dimensional settings is proposed. The data are assumed
to be a collection of multiple non-Gaussian realizations with underlying
undirected graphical structures. Utilizing the concept of median graphs in
summarizing the commonality across these graphical structures, a novel
semiparametric approach to modeling such complex aggregated data is provided
along with robust estimation of the median graph, which is assumed to be
sparse. The estimator is proved to be consistent in graph recovery and an upper
bound on the rate of convergence is given. Experiments on both synthetic and
real datasets are conducted to illustrate the empirical usefulness of the
proposed models and methods
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