20,602 research outputs found
Probabilistic Properties of Highly Connected Random Geometric Graphs
In this paper we study the probabilistic properties of reliable networks of minimal total edge lengths. We study reliability in terms of k-edge-connectivity in graphs in d-dimensional space. We show this problem fits into Yukich’s framework for Euclidean functionals for arbitrary k, dimension d and distant-power gradient p, with p < d. With this framework several theorems on the convergence of optimal solutions follow. We apply Yukich’s framework for functionals so that we can use partitioning algorithms that rapidly compute near-optimal solutions on typical examples. These results are then extended to optimal k-edge-connected power assignment graphs, where we assign power to vertices and charge per vertex. The network can be modelled as a wireless network
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
(Quantum) Space-Time as a Statistical Geometry of Lumps in Random Networks
In the following we undertake to describe how macroscopic space-time (or
rather, a microscopic protoform of it) is supposed to emerge as a
superstructure of a web of lumps in a stochastic discrete network structure. As
in preceding work (mentioned below), our analysis is based on the working
philosophy that both physics and the corresponding mathematics have to be
genuinely discrete on the primordial (Planck scale) level. This strategy is
concretely implemented in the form of \tit{cellular networks} and \tit{random
graphs}. One of our main themes is the development of the concept of
\tit{physical (proto)points} or \tit{lumps} as densely entangled subcomplexes
of the network and their respective web, establishing something like
\tit{(proto)causality}. It may perhaps be said that certain parts of our
programme are realisations of some early ideas of Menger and more recent ones
sketched by Smolin a couple of years ago. We briefly indicate how this
\tit{two-story-concept} of \tit{quantum} space-time can be used to encode the
(at least in our view) existing non-local aspects of quantum theory without
violating macroscopic space-time causality.Comment: 35 pages, Latex, under consideration by CQ
On the Distribution of Random Geometric Graphs
Random geometric graphs (RGGs) are commonly used to model networked systems
that depend on the underlying spatial embedding. We concern ourselves with the
probability distribution of an RGG, which is crucial for studying its random
topology, properties (e.g., connectedness), or Shannon entropy as a measure of
the graph's topological uncertainty (or information content). Moreover, the
distribution is also relevant for determining average network performance or
designing protocols. However, a major impediment in deducing the graph
distribution is that it requires the joint probability distribution of the
distances between nodes randomly distributed in a bounded
domain. As no such result exists in the literature, we make progress by
obtaining the joint distribution of the distances between three nodes confined
in a disk in . This enables the calculation of the probability
distribution and entropy of a three-node graph. For arbitrary , we derive a
series of upper bounds on the graph entropy; in particular, the bound involving
the entropy of a three-node graph is tighter than the existing bound which
assumes distances are independent. Finally, we provide numerical results on
graph connectedness and the tightness of the derived entropy bounds.Comment: submitted to the IEEE International Symposium on Information Theory
201
Random geometric complexes
We study the expected topological properties of Cech and Vietoris-Rips
complexes built on i.i.d. random points in R^d. We find higher dimensional
analogues of known results for connectivity and component counts for random
geometric graphs. However, higher homology H_k is not monotone when k > 0. In
particular for every k > 0 we exhibit two thresholds, one where homology passes
from vanishing to nonvanishing, and another where it passes back to vanishing.
We give asymptotic formulas for the expectation of the Betti numbers in the
sparser regimes, and bounds in the denser regimes. The main technical
contribution of the article is in the application of discrete Morse theory in
geometric probability.Comment: 26 pages, 3 figures, final revisions, to appear in Discrete &
Computational Geometr
- …