366 research outputs found
Connectivity of sparse Bluetooth networks
Consider a random geometric graph defined on n vertices uniformly distributed in the d-dimensional unit torus. Two vertices are connected if their distance is less than a “visibility radius ” rn. We consider Bluetooth networks that are locally sparsified random geometric graphs. Each vertex selects c of its neighbors in the random geometric graph at random and connects only to the selected points. We show that if the visibility radius is at least of the order of n−(1−δ)/d for some δ> 0, then a constant value of c is sufficient for the graph to be connected, with high probability. It suffices to take c ≥ √ (1 + ɛ)/δ + K for any positive ɛ where K is a constant depending on d only. On the other hand, with c ≤ √ (1 − ɛ)/δ, the graph is disconnected, with high probability. 1 Introduction an
Long and short paths in uniform random recursive dags
In a uniform random recursive k-dag, there is a root, 0, and each node in
turn, from 1 to n, chooses k uniform random parents from among the nodes of
smaller index. If S_n is the shortest path distance from node n to the root,
then we determine the constant \sigma such that S_n/log(n) tends to \sigma in
probability as n tends to infinity. We also show that max_{1 \le i \le n}
S_i/log(n) tends to \sigma in probability.Comment: 16 page
Devroye Inequality for a Class of Non-Uniformly Hyperbolic Dynamical Systems
In this paper, we prove an inequality, which we call "Devroye inequality",
for a large class of non-uniformly hyperbolic dynamical systems (M,f). This
class, introduced by L.-S. Young, includes families of piece-wise hyperbolic
maps (Lozi-like maps), scattering billiards (e.g., planar Lorentz gas),
unimodal and H{\'e}non-like maps. Devroye inequality provides an upper bound
for the variance of observables of the form K(x,f(x),...,f^{n-1}(x)), where K
is any separately Holder continuous function of n variables. In particular, we
can deal with observables which are not Birkhoff averages. We will show in
\cite{CCS} some applications of Devroye inequality to statistical properties of
this class of dynamical systems.Comment: Corrected version; To appear in Nonlinearit
Traveling Waves, Front Selection, and Exact Nontrivial Exponents in a Random Fragmentation Problem
We study a random bisection problem where an initial interval of length x is
cut into two random fragments at the first stage, then each of these two
fragments is cut further, etc. We compute the probability P_n(x) that at the
n-th stage, each of the 2^n fragments is shorter than 1. We show that P_n(x)
approaches a traveling wave form, and the front position x_n increases as
x_n\sim n^{\beta}{\rho}^n for large n. We compute exactly the exponents
\rho=1.261076... and \beta=0.453025.... as roots of transcendental equations.
We also solve the m-section problem where each interval is broken into m
fragments. In particular, the generalized exponents grow as \rho_m\approx
m/(\ln m) and \beta_m\approx 3/(2\ln m) in the large m limit. Our approach
establishes an intriguing connection between extreme value statistics and
traveling wave propagation in the context of the fragmentation problem.Comment: 4 pages Revte
Stationary probability density of stochastic search processes in global optimization
A method for the construction of approximate analytical expressions for the
stationary marginal densities of general stochastic search processes is
proposed. By the marginal densities, regions of the search space that with high
probability contain the global optima can be readily defined. The density
estimation procedure involves a controlled number of linear operations, with a
computational cost per iteration that grows linearly with problem size
Simulation of truncated normal variables
We provide in this paper simulation algorithms for one-sided and two-sided
truncated normal distributions. These algorithms are then used to simulate
multivariate normal variables with restricted parameter space for any
covariance structure.Comment: This 1992 paper appeared in 1995 in Statistics and Computing and the
gist of it is contained in Monte Carlo Statistical Methods (2004), but I
receive weekly requests for reprints so here it is
Robustness and Generalization
We derive generalization bounds for learning algorithms based on their
robustness: the property that if a testing sample is "similar" to a training
sample, then the testing error is close to the training error. This provides a
novel approach, different from the complexity or stability arguments, to study
generalization of learning algorithms. We further show that a weak notion of
robustness is both sufficient and necessary for generalizability, which implies
that robustness is a fundamental property for learning algorithms to work
Understanding Search Trees via Statistical Physics
We study the random m-ary search tree model (where m stands for the number of
branches of a search tree), an important problem for data storage in computer
science, using a variety of statistical physics techniques that allow us to
obtain exact asymptotic results. In particular, we show that the probability
distributions of extreme observables associated with a random search tree such
as the height and the balanced height of a tree have a traveling front
structure. In addition, the variance of the number of nodes needed to store a
data string of a given size N is shown to undergo a striking phase transition
at a critical value of the branching ratio m_c=26. We identify the mechanism of
this phase transition, show that it is generic and occurs in various other
problems as well. New results are obtained when each element of the data string
is a D-dimensional vector. We show that this problem also has a phase
transition at a critical dimension, D_c= \pi/\sin^{-1}(1/\sqrt{8})=8.69363...Comment: 11 pages, 8 .eps figures included. Invited contribution to
STATPHYS-22 held at Bangalore (India) in July 2004. To appear in the
proceedings of STATPHYS-2
Extreme Value Statistics and Traveling Fronts: Various Applications
An intriguing connection between extreme value statistics and traveling
fronts has been found recently in a number of diverse problems. In this brief
review we outline a few such problems and consider their various applications.Comment: A brief review (6 pages, 2 figures) to appear in Physica A as part of
the proceedings of Statphys-Kolkata IV (2002
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