11,122 research outputs found
Nonuniform random geometric graphs with location-dependent radii
We propose a distribution-free approach to the study of random geometric
graphs. The distribution of vertices follows a Poisson point process with
intensity function , where , and is a
probability density function on . A vertex located at
connects via directed edges to other vertices that are within a cut-off
distance . We prove strong law results for (i) the critical cut-off
function so that almost surely, the graph does not contain any node with
out-degree zero for sufficiently large and (ii) the maximum and minimum
vertex degrees. We also provide a characterization of the cut-off function for
which the number of nodes with out-degree zero converges in distribution to a
Poisson random variable. We illustrate this result for a class of densities
with compact support that have at most polynomial rates of decay to zero.
Finally, we state a sufficient condition for an enhanced version of the above
graph to be almost surely connected eventually.Comment: Published in at http://dx.doi.org/10.1214/11-AAP823 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Spatial networks with wireless applications
Many networks have nodes located in physical space, with links more common
between closely spaced pairs of nodes. For example, the nodes could be wireless
devices and links communication channels in a wireless mesh network. We
describe recent work involving such networks, considering effects due to the
geometry (convex,non-convex, and fractal), node distribution,
distance-dependent link probability, mobility, directivity and interference.Comment: Review article- an amended version with a new title from the origina
Shaping bursting by electrical coupling and noise
Gap-junctional coupling is an important way of communication between neurons
and other excitable cells. Strong electrical coupling synchronizes activity
across cell ensembles. Surprisingly, in the presence of noise synchronous
oscillations generated by an electrically coupled network may differ
qualitatively from the oscillations produced by uncoupled individual cells
forming the network. A prominent example of such behavior is the synchronized
bursting in islets of Langerhans formed by pancreatic \beta-cells, which in
isolation are known to exhibit irregular spiking. At the heart of this
intriguing phenomenon lies denoising, a remarkable ability of electrical
coupling to diminish the effects of noise acting on individual cells.
In this paper, we derive quantitative estimates characterizing denoising in
electrically coupled networks of conductance-based models of square wave
bursting cells. Our analysis reveals the interplay of the intrinsic properties
of the individual cells and network topology and their respective contributions
to this important effect. In particular, we show that networks on graphs with
large algebraic connectivity or small total effective resistance are better
equipped for implementing denoising. As a by-product of the analysis of
denoising, we analytically estimate the rate with which trajectories converge
to the synchronization subspace and the stability of the latter to random
perturbations. These estimates reveal the role of the network topology in
synchronization. The analysis is complemented by numerical simulations of
electrically coupled conductance-based networks. Taken together, these results
explain the mechanisms underlying synchronization and denoising in an important
class of biological models
Pregeometric Concepts on Graphs and Cellular Networks as Possible Models of Space-Time at the Planck-Scale
Starting from the working hypothesis that both physics and the corresponding
mathematics have to be described by means of discrete concepts on the
Planck-scale, one of the many problems one has to face is to find the discrete
protoforms of the building blocks of continuum physics and mathematics. In the
following we embark on developing such concepts for irregular structures like
(large) graphs or networks which are intended to emulate (some of) the generic
properties of the presumed combinatorial substratum from which continuum
physics is assumed to emerge as a coarse grained and secondary model theory. We
briefly indicate how various concepts of discrete (functional) analysis and
geometry can be naturally constructed within this framework, leaving a larger
portion of the paper to the systematic developement of dimensional concepts and
their properties, which may have a possible bearing on various branches of
modern physics beyond quantum gravity.Comment: 16 pages, Invited paper to appear in the special issue of the Journal
of Chaos, Solitons and Fractals on: "Superstrings, M, F, S ... Theory" (M.S.
El Naschie, C. Castro, Editors
Sampling-based Algorithms for Optimal Motion Planning
During the last decade, sampling-based path planning algorithms, such as
Probabilistic RoadMaps (PRM) and Rapidly-exploring Random Trees (RRT), have
been shown to work well in practice and possess theoretical guarantees such as
probabilistic completeness. However, little effort has been devoted to the
formal analysis of the quality of the solution returned by such algorithms,
e.g., as a function of the number of samples. The purpose of this paper is to
fill this gap, by rigorously analyzing the asymptotic behavior of the cost of
the solution returned by stochastic sampling-based algorithms as the number of
samples increases. A number of negative results are provided, characterizing
existing algorithms, e.g., showing that, under mild technical conditions, the
cost of the solution returned by broadly used sampling-based algorithms
converges almost surely to a non-optimal value. The main contribution of the
paper is the introduction of new algorithms, namely, PRM* and RRT*, which are
provably asymptotically optimal, i.e., such that the cost of the returned
solution converges almost surely to the optimum. Moreover, it is shown that the
computational complexity of the new algorithms is within a constant factor of
that of their probabilistically complete (but not asymptotically optimal)
counterparts. The analysis in this paper hinges on novel connections between
stochastic sampling-based path planning algorithms and the theory of random
geometric graphs.Comment: 76 pages, 26 figures, to appear in International Journal of Robotics
Researc
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