28,298 research outputs found
Robustness of scale-free spatial networks
A growing family of random graphs is called robust if it retains a giant
component after percolation with arbitrary positive retention probability. We
study robustness for graphs, in which new vertices are given a spatial position
on the -dimensional torus and are connected to existing vertices with a
probability favouring short spatial distances and high degrees. In this model
of a scale-free network with clustering we can independently tune the power law
exponent of the degree distribution and the rate at which the
connection probability decreases with the distance of two vertices. We show
that the network is robust if , but fails to be robust if
. In the case of one-dimensional space we also show that the network is
not robust if . This implies that robustness of a
scale-free network depends not only on its power-law exponent but also on its
clustering features. Other than the classical models of scale-free networks our
model is not locally tree-like, and hence we need to develop novel methods for
its study, including, for example, a surprising application of the
BK-inequality.Comment: 34 pages, 4 figure
Mutual information rate and bounds for it
The amount of information exchanged per unit of time between two nodes in a
dynamical network or between two data sets is a powerful concept for analysing
complex systems. This quantity, known as the mutual information rate (MIR), is
calculated from the mutual information, which is rigorously defined only for
random systems. Moreover, the definition of mutual information is based on
probabilities of significant events. This work offers a simple alternative way
to calculate the MIR in dynamical (deterministic) networks or between two data
sets (not fully deterministic), and to calculate its upper and lower bounds
without having to calculate probabilities, but rather in terms of well known
and well defined quantities in dynamical systems. As possible applications of
our bounds, we study the relationship between synchronisation and the exchange
of information in a system of two coupled maps and in experimental networks of
coupled oscillators
Anomalous heat-kernel decay for random walk among bounded random conductances
We consider the nearest-neighbor simple random walk on , ,
driven by a field of bounded random conductances . The
conductance law is i.i.d. subject to the condition that the probability of
exceeds the threshold for bond percolation on . For
environments in which the origin is connected to infinity by bonds with
positive conductances, we study the decay of the -step return probability
. We prove that is bounded by a random
constant times in , while it is in and
in . By producing examples with anomalous heat-kernel
decay approaching we prove that the bound in is the
best possible. We also construct natural -dependent environments that
exhibit the extra factor in . See also math.PR/0701248.Comment: 22 pages. Includes a self-contained proof of isoperimetric inequality
for supercritical percolation clusters. Version to appear in AIHP +
additional correction
A rigorous but gentle introduction for economists
This open access textbook is the first to provide Business and Economics Ph.D. students with a precise and intuitive introduction to the formal backgrounds of modern financial theory. It explains Brownian motion, random processes, measures, and Lebesgue integrals intuitively, but without sacrificing the necessary mathematical formalism, making them accessible for readers with little or no previous knowledge of the field. It also includes mathematical definitions and the hidden stories behind the terms discussing why the theories are presented in specific ways.
Optimal Uncertainty Quantification
We propose a rigorous framework for Uncertainty Quantification (UQ) in which
the UQ objectives and the assumptions/information set are brought to the
forefront. This framework, which we call \emph{Optimal Uncertainty
Quantification} (OUQ), is based on the observation that, given a set of
assumptions and information about the problem, there exist optimal bounds on
uncertainties: these are obtained as values of well-defined optimization
problems corresponding to extremizing probabilities of failure, or of
deviations, subject to the constraints imposed by the scenarios compatible with
the assumptions and information. In particular, this framework does not
implicitly impose inappropriate assumptions, nor does it repudiate relevant
information. Although OUQ optimization problems are extremely large, we show
that under general conditions they have finite-dimensional reductions. As an
application, we develop \emph{Optimal Concentration Inequalities} (OCI) of
Hoeffding and McDiarmid type. Surprisingly, these results show that
uncertainties in input parameters, which propagate to output uncertainties in
the classical sensitivity analysis paradigm, may fail to do so if the transfer
functions (or probability distributions) are imperfectly known. We show how,
for hierarchical structures, this phenomenon may lead to the non-propagation of
uncertainties or information across scales. In addition, a general algorithmic
framework is developed for OUQ and is tested on the Caltech surrogate model for
hypervelocity impact and on the seismic safety assessment of truss structures,
suggesting the feasibility of the framework for important complex systems. The
introduction of this paper provides both an overview of the paper and a
self-contained mini-tutorial about basic concepts and issues of UQ.Comment: 90 pages. Accepted for publication in SIAM Review (Expository
Research Papers). See SIAM Review for higher quality figure
Unlacing the lace expansion: a survey to hypercube percolation
The purpose of this note is twofold. First, we survey the study of the
percolation phase transition on the Hamming hypercube {0,1}^m obtained in the
series of papers [9,10,11,24]. Secondly, we explain how this study can be
performed without the use of the so-called "lace-expansion" technique. To that
aim, we provide a novel simple proof that the triangle condition holds at the
critical probability. We hope that some of these techniques will be useful to
obtain non-perturbative proofs in the analogous, yet much more difficult study
on high-dimensional tori.Comment: 19 pages. arXiv admin note: text overlap with arXiv:1201.395
- …