16,678 research outputs found
Percolation-like Scaling Exponents for Minimal Paths and Trees in the Stochastic Mean Field Model
In the mean field (or random link) model there are points and inter-point
distances are independent random variables. For and in the
limit, let (maximum number of steps
in a path whose average step-length is ). The function
is analogous to the percolation function in percolation theory:
there is a critical value at which becomes
non-zero, and (presumably) a scaling exponent in the sense
. Recently developed probabilistic
methodology (in some sense a rephrasing of the cavity method of Mezard-Parisi)
provides a simple albeit non-rigorous way of writing down such functions in
terms of solutions of fixed-point equations for probability distributions.
Solving numerically gives convincing evidence that . A parallel
study with trees instead of paths gives scaling exponent . The new
exponents coincide with those found in a different context (comparing optimal
and near-optimal solutions of mean-field TSP and MST) and reinforce the
suggestion that these scaling exponents determine universality classes for
optimization problems on random points.Comment: 19 page
Matrices of forests, analysis of networks, and ranking problems
The matrices of spanning rooted forests are studied as a tool for analysing
the structure of networks and measuring their properties. The problems of
revealing the basic bicomponents, measuring vertex proximity, and ranking from
preference relations / sports competitions are considered. It is shown that the
vertex accessibility measure based on spanning forests has a number of
desirable properties. An interpretation for the stochastic matrix of
out-forests in terms of information dissemination is given.Comment: 8 pages. This article draws heavily from arXiv:math/0508171.
Published in Proceedings of the First International Conference on Information
Technology and Quantitative Management (ITQM 2013). This version contains
some corrections and addition
Detecting a Currency's Dominance or Dependence using Foreign Exchange Network Trees
In a system containing a large number of interacting stochastic processes,
there will typically be many non-zero correlation coefficients. This makes it
difficult to either visualize the system's inter-dependencies, or identify its
dominant elements. Such a situation arises in Foreign Exchange (FX) which is
the world's biggest market. Here we develop a network analysis of these
correlations using Minimum Spanning Trees (MSTs). We show that not only do the
MSTs provide a meaningful representation of the global FX dynamics, but they
also enable one to determine momentarily dominant and dependent currencies. We
find that information about a country's geographical ties emerges from the raw
exchange-rate data. Most importantly from a trading perspective, we discuss how
to infer which currencies are `in play' during a particular period of time
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