1,848 research outputs found
Functional Bipartite Ranking: a Wavelet-Based Filtering Approach
It is the main goal of this article to address the bipartite ranking issue
from the perspective of functional data analysis (FDA). Given a training set of
independent realizations of a (possibly sampled) second-order random function
with a (locally) smooth autocorrelation structure and to which a binary label
is randomly assigned, the objective is to learn a scoring function s with
optimal ROC curve. Based on linear/nonlinear wavelet-based approximations, it
is shown how to select compact finite dimensional representations of the input
curves adaptively, in order to build accurate ranking rules, using recent
advances in the ranking problem for multivariate data with binary feedback.
Beyond theoretical considerations, the performance of the learning methods for
functional bipartite ranking proposed in this paper are illustrated by
numerical experiments
Spectral and Dynamical Properties in Classes of Sparse Networks with Mesoscopic Inhomogeneities
We study structure, eigenvalue spectra and diffusion dynamics in a wide class
of networks with subgraphs (modules) at mesoscopic scale. The networks are
grown within the model with three parameters controlling the number of modules,
their internal structure as scale-free and correlated subgraphs, and the
topology of connecting network. Within the exhaustive spectral analysis for
both the adjacency matrix and the normalized Laplacian matrix we identify the
spectral properties which characterize the mesoscopic structure of sparse
cyclic graphs and trees. The minimally connected nodes, clustering, and the
average connectivity affect the central part of the spectrum. The number of
distinct modules leads to an extra peak at the lower part of the Laplacian
spectrum in cyclic graphs. Such a peak does not occur in the case of
topologically distinct tree-subgraphs connected on a tree. Whereas the
associated eigenvectors remain localized on the subgraphs both in trees and
cyclic graphs. We also find a characteristic pattern of periodic localization
along the chains on the tree for the eigenvector components associated with the
largest eigenvalue equal 2 of the Laplacian. We corroborate the results with
simulations of the random walk on several types of networks. Our results for
the distribution of return-time of the walk to the origin (autocorrelator)
agree well with recent analytical solution for trees, and it appear to be
independent on their mesoscopic and global structure. For the cyclic graphs we
find new results with twice larger stretching exponent of the tail of the
distribution, which is virtually independent on the size of cycles. The
modularity and clustering contribute to a power-law decay at short return
times
Communities in Networks
We survey some of the concepts, methods, and applications of community
detection, which has become an increasingly important area of network science.
To help ease newcomers into the field, we provide a guide to available
methodology and open problems, and discuss why scientists from diverse
backgrounds are interested in these problems. As a running theme, we emphasize
the connections of community detection to problems in statistical physics and
computational optimization.Comment: survey/review article on community structure in networks; published
version is available at
http://people.maths.ox.ac.uk/~porterm/papers/comnotices.pd
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