16,666 research outputs found
A Weighted Correlation Index for Rankings with Ties
Understanding the correlation between two different scores for the same set
of items is a common problem in information retrieval, and the most commonly
used statistics that quantifies this correlation is Kendall's . However,
the standard definition fails to capture that discordances between items with
high rank are more important than those between items with low rank. Recently,
a new measure of correlation based on average precision has been proposed to
solve this problem, but like many alternative proposals in the literature it
assumes that there are no ties in the scores. This is a major deficiency in a
number of contexts, and in particular while comparing centrality scores on
large graphs, as the obvious baseline, indegree, has a very large number of
ties in web and social graphs. We propose to extend Kendall's definition in a
natural way to take into account weights in the presence of ties. We prove a
number of interesting mathematical properties of our generalization and
describe an algorithm for its computation. We also validate the
usefulness of our weighted measure of correlation using experimental data
Tackling information asymmetry in networks: a new entropy-based ranking index
Information is a valuable asset for agents in socio-economic systems, a
significant part of the information being entailed into the very network of
connections between agents. The different interlinkages patterns that agents
establish may, in fact, lead to asymmetries in the knowledge of the network
structure; since this entails a different ability of quantifying relevant
systemic properties (e.g. the risk of financial contagion in a network of
liabilities), agents capable of providing a better estimate of (otherwise)
unaccessible network properties, ultimately have a competitive advantage. In
this paper, we address for the first time the issue of quantifying the
information asymmetry arising from the network topology. To this aim, we define
a novel index - InfoRank - intended to measure the quality of the information
possessed by each node, computing the Shannon entropy of the ensemble
conditioned on the node-specific information. Further, we test the performance
of our novel ranking procedure in terms of the reconstruction accuracy of the
(unaccessible) network structure and show that it outperforms other popular
centrality measures in identifying the "most informative" nodes. Finally, we
discuss the socio-economic implications of network information asymmetry.Comment: 12 pages, 8 figure
Medial/skeletal linking structures for multi-region configurations
We consider a generic configuration of regions, consisting of a collection of
distinct compact regions in which may be
either smooth regions disjoint from the others or regions which meet on their
piecewise smooth boundaries in a generic way. We introduce a
skeletal linking structure for the collection of regions which simultaneously
captures the regions' individual shapes and geometric properties as well as the
"positional geometry" of the collection. The linking structure extends in a
minimal way the individual "skeletal structures" on each of the regions,
allowing us to significantly extend the mathematical methods introduced for
single regions to the configuration.
We prove for a generic configuration of regions the existence of a special
type of Blum linking structure which builds upon the Blum medial axes of the
individual regions. This requires proving several transversality theorems for
certain associated "multi-distance" and "height-distance" functions for such
configurations. We show that by relaxing the conditions on the Blum linking
structures we obtain the more general class of skeletal linking structures
which still capture the geometric properties.
In addition to yielding geometric invariants which capture the shapes and
geometry of individual regions, the linking structures are used to define
invariants which measure positional properties of the configuration such as:
measures of relative closeness of neighboring regions and relative significance
of the individual regions for the configuration. These invariants, which are
computed by formulas involving "skeletal linking integrals" on the internal
skeletal structures, are then used to construct a "tiered linking graph," which
identifies subconfigurations and provides a hierarchical ordering of the
regions.Comment: 135 pages, 36 figures. Version to appear in Memoirs of the Amer.
Math. So
Educational commitment and social networking: The power of informal networks
The lack of an engaging pedagogy and the highly competitive atmosphere in
introductory science courses tend to discourage students from pursuing science,
technology, engineering, and mathematics (STEM) majors. Once in a STEM field,
academic and social integration has been long thought to be important for
students' persistence. Yet, it is rarely investigated. In particular, the
relative impact of in-class and out-of-class interactions remains an open
issue. Here, we demonstrate that, surprisingly, for students whose grades fall
in the "middle of the pack," the out-of-class network is the most significant
predictor of persistence. To do so, we use logistic regression combined with
Akaike's information criterion to assess in- and out-of-class networks, grades,
and other factors. For students with grades at the very top (and bottom), final
grade, unsurprisingly, is the best predictor of persistence---these students
are likely already committed (or simply restricted from continuing) so they
persist (or drop out). For intermediate grades, though, only out-of-class
closeness---a measure of one's immersion in the network---helps predict
persistence. This does not negate the need for in-class ties. However, it
suggests that, in this cohort, only students that get past the convenient
in-class interactions and start forming strong bonds outside of class are or
become committed to their studies. Since many students are lost through
attrition, our results suggest practical routes for increasing students'
persistence in STEM majors.Comment: 12 pages, 2 figures, 8 tables, 6 pages of Supplementary Material
Pranab Kumar Sen: Life and works
In this article, we describe briefly the highlights and various
accomplishments in the personal as well as the academic life of Professor
Pranab Kumar Sen.Comment: Published in at http://dx.doi.org/10.1214/193940307000000013 the IMS
Collections (http://www.imstat.org/publications/imscollections.htm) by the
Institute of Mathematical Statistics (http://www.imstat.org
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