6,635 research outputs found
Critical percolation of free product of groups
In this article we study percolation on the Cayley graph of a free product of
groups.
The critical probability of a free product of groups
is found as a solution of an equation involving only the expected subcritical
cluster size of factor groups . For finite groups these
equations are polynomial and can be explicitly written down. The expected
subcritical cluster size of the free product is also found in terms of the
subcritical cluster sizes of the factors. In particular, we prove that
for the Cayley graph of the modular group (with the
standard generators) is , the unique root of the polynomial
in the interval .
In the case when groups can be "well approximated" by a sequence of
quotient groups, we show that the critical probabilities of the free product of
these approximations converge to the critical probability of
and the speed of convergence is exponential. Thus for residually finite groups,
for example, one can restrict oneself to the case when each free factor is
finite.
We show that the critical point, introduced by Schonmann,
of the free product is just the minimum of for the factors
Complex networks: new trends for the analysis of brain connectivity
Today, the human brain can be studied as a whole. Electroencephalography,
magnetoencephalography, or functional magnetic resonance imaging techniques
provide functional connectivity patterns between different brain areas, and
during different pathological and cognitive neuro-dynamical states. In this
Tutorial we review novel complex networks approaches to unveil how brain
networks can efficiently manage local processing and global integration for the
transfer of information, while being at the same time capable of adapting to
satisfy changing neural demands.Comment: Tutorial paper to appear in the Int. J. Bif. Chao
Quantitative estimates of discrete harmonic measures
A theorem of Bourgain states that the harmonic measure for a domain in
is supported on a set of Hausdorff dimension strictly less than
\cite{Bourgain}. We apply Bourgain's method to the discrete case, i.e., to the
distribution of the first entrance point of a random walk into a subset of , . By refining the argument, we prove that for all \b>0 there
exists \rho (d,\b)N(d,\b), any , and any | \{y\in\Z^d\colon \nu_{A,x}(y)
\geq n^{-\b} \}| \leq n^{\rho(d,\b)}, where denotes the
probability that is the first entrance point of the simple random walk
starting at into . Furthermore, must converge to as \b \to
\infty.Comment: 16 pages, 2 figures. Part (B) of the theorem is ne
Fast computation by block permanents of cumulative distribution functions of order statistics from several populations
The joint cumulative distribution function for order statistics arising from
several different populations is given in terms of the distribution function of
the populations. The computational cost of the formula in the case of two
populations is still exponential in the worst case, but it is a dramatic
improvement compared to the general formula by Bapat and Beg. In the case when
only the joint distribution function of a subset of the order statistics of
fixed size is needed, the complexity is polynomial, for the case of two
populations.Comment: 21 pages, 3 figure
Finding Exogenous Variables in Data with Many More Variables than Observations
Many statistical methods have been proposed to estimate causal models in
classical situations with fewer variables than observations (p<n, p: the number
of variables and n: the number of observations). However, modern datasets
including gene expression data need high-dimensional causal modeling in
challenging situations with orders of magnitude more variables than
observations (p>>n). In this paper, we propose a method to find exogenous
variables in a linear non-Gaussian causal model, which requires much smaller
sample sizes than conventional methods and works even when p>>n. The key idea
is to identify which variables are exogenous based on non-Gaussianity instead
of estimating the entire structure of the model. Exogenous variables work as
triggers that activate a causal chain in the model, and their identification
leads to more efficient experimental designs and better understanding of the
causal mechanism. We present experiments with artificial data and real-world
gene expression data to evaluate the method.Comment: A revised version of this was published in Proc. ICANN201
Generative rules of Drosophila locomotor behavior as a candidate homology across phyla
The discovery of shared behavioral processes across phyla is a significant step in the establishment of a comparative study of behavior. We use immobility as an origin and reference for the measurement of fly locomotor behavior; speed, walking direction and trunk orientation as the degrees of freedom shaping this behavior; and cocaine as the parameter inducing progressive transitions in and out of immobility. We characterize and quantify the generative rules that shape Drosophila locomotor behavior, bringing about a gradual buildup of kinematic degrees of freedom during the transition from immobility to normal behavior, and the opposite narrowing down into immobility. Transitions into immobility unfold via sequential enhancement and then elimination of translation, curvature and finally rotation. Transitions out of immobility unfold by progressive addition of these degrees of freedom in the opposite order. The same generative rules have been found in vertebrate locomotor behavior in several contexts (pharmacological manipulations, ontogeny, social interactions) involving transitions in-and-out of immobility. Recent claims for deep homology between arthropod central complex and vertebrate basal ganglia provide an opportunity to examine whether the rules we report also share common descent. Our approach prompts the discovery of behavioral homologies, contributing to the elusive problem of behavioral evolution
Generative discriminative models for multivariate inference and statistical mapping in medical imaging
This paper presents a general framework for obtaining interpretable
multivariate discriminative models that allow efficient statistical inference
for neuroimage analysis. The framework, termed generative discriminative
machine (GDM), augments discriminative models with a generative regularization
term. We demonstrate that the proposed formulation can be optimized in closed
form and in dual space, allowing efficient computation for high dimensional
neuroimaging datasets. Furthermore, we provide an analytic estimation of the
null distribution of the model parameters, which enables efficient statistical
inference and p-value computation without the need for permutation testing. We
compared the proposed method with both purely generative and discriminative
learning methods in two large structural magnetic resonance imaging (sMRI)
datasets of Alzheimer's disease (AD) (n=415) and Schizophrenia (n=853). Using
the AD dataset, we demonstrated the ability of GDM to robustly handle
confounding variations. Using Schizophrenia dataset, we demonstrated the
ability of GDM to handle multi-site studies. Taken together, the results
underline the potential of the proposed approach for neuroimaging analyses.Comment: To appear in MICCAI 2018 proceeding
Lognormal scale invariant random measures
In this article, we consider the continuous analog of the celebrated
Mandelbrot star equation with lognormal weights. Mandelbrot introduced this
equation to characterize the law of multiplicative cascades. We show existence
and uniqueness of measures satisfying the aforementioned continuous equation;
these measures fall under the scope of the Gaussian multiplicative chaos theory
developed by J.P. Kahane in 1985 (or possibly extensions of this theory). As a
by product, we also obtain an explicit characterization of the covariance
structure of these measures. We also prove that qualitative properties such as
long-range independence or isotropy can be read off the equation.Comment: 31 pages; Probability Theory and Related Fields (2012) electronic
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