179,089 research outputs found
A Deterministic Equivalent for the Analysis of Non-Gaussian Correlated MIMO Multiple Access Channels
Large dimensional random matrix theory (RMT) has provided an efficient
analytical tool to understand multiple-input multiple-output (MIMO) channels
and to aid the design of MIMO wireless communication systems. However, previous
studies based on large dimensional RMT rely on the assumption that the transmit
correlation matrix is diagonal or the propagation channel matrix is Gaussian.
There is an increasing interest in the channels where the transmit correlation
matrices are generally nonnegative definite and the channel entries are
non-Gaussian. This class of channel models appears in several applications in
MIMO multiple access systems, such as small cell networks (SCNs). To address
these problems, we use the generalized Lindeberg principle to show that the
Stieltjes transforms of this class of random matrices with Gaussian or
non-Gaussian independent entries coincide in the large dimensional regime. This
result permits to derive the deterministic equivalents (e.g., the Stieltjes
transform and the ergodic mutual information) for non-Gaussian MIMO channels
from the known results developed for Gaussian MIMO channels, and is of great
importance in characterizing the spectral efficiency of SCNs.Comment: This paper is the revision of the original manuscript titled "A
Deterministic Equivalent for the Analysis of Small Cell Networks". We have
revised the original manuscript and reworked on the organization to improve
the presentation as well as readabilit
Zig-zag Sort: A Simple Deterministic Data-Oblivious Sorting Algorithm Running in O(n log n) Time
We describe and analyze Zig-zag Sort--a deterministic data-oblivious sorting
algorithm running in O(n log n) time that is arguably simpler than previously
known algorithms with similar properties, which are based on the AKS sorting
network. Because it is data-oblivious and deterministic, Zig-zag Sort can be
implemented as a simple O(n log n)-size sorting network, thereby providing a
solution to an open problem posed by Incerpi and Sedgewick in 1985. In
addition, Zig-zag Sort is a variant of Shellsort, and is, in fact, the first
deterministic Shellsort variant running in O(n log n) time. The existence of
such an algorithm was posed as an open problem by Plaxton et al. in 1992 and
also by Sedgewick in 1996. More relevant for today, however, is the fact that
the existence of a simple data-oblivious deterministic sorting algorithm
running in O(n log n) time simplifies the inner-loop computation in several
proposed oblivious-RAM simulation methods (which utilize AKS sorting networks),
and this, in turn, implies simplified mechanisms for privacy-preserving data
outsourcing in several cloud computing applications. We provide both
constructive and non-constructive implementations of Zig-zag Sort, based on the
existence of a circuit known as an epsilon-halver, such that the constant
factors in our constructive implementations are orders of magnitude smaller
than those for constructive variants of the AKS sorting network, which are also
based on the use of epsilon-halvers.Comment: Appearing in ACM Symp. on Theory of Computing (STOC) 201
Accelerated growth in outgoing links in evolving networks: deterministic vs. stochastic picture
In several real-world networks like the Internet, WWW etc., the number of
links grow in time in a non-linear fashion. We consider growing networks in
which the number of outgoing links is a non-linear function of time but new
links between older nodes are forbidden. The attachments are made using a
preferential attachment scheme. In the deterministic picture, the number of
outgoing links at any time is taken as where is
the number of nodes present at that time. The continuum theory predicts a power
law decay of the degree distribution: , while the degree of the node introduced at time is given by
when the
network is evolved till time . Numerical results show a growth in the degree
distribution for small values at any non-zero . In the stochastic
picture, is a random variable. As long as is time-dependent, e.g.,
when follows a distribution . The behaviour
of changes significantly as is varied: for , the
network has a scale-free distribution belonging to the BA class as predicted by
the mean field theory, for smaller values of it shows different
behaviour. Characteristic features of the clustering coefficients in both
models have also been discussed.Comment: Revised text, references added, to be published in PR
Population density equations for stochastic processes with memory kernels
We present a method for solving population density equations (PDEs)–-a mean-field technique describing homogeneous populations of uncoupled neurons—where the populations can be subject to non-Markov noise for arbitrary distributions of jump sizes. The method combines recent developments in two different disciplines that traditionally have had limited interaction: computational neuroscience and the theory of random networks. The method uses a geometric binning scheme, based on the method of characteristics, to capture the deterministic neurodynamics of the population, separating the deterministic and stochastic process cleanly. We can independently vary the choice of the deterministic model and the model for the stochastic process, leading to a highly modular numerical solution strategy. We demonstrate this by replacing the master equation implicit in many formulations of the PDE formalism by a generalization called the generalized Montroll-Weiss equation—a recent result from random network theory—describing a random walker subject to transitions realized by a non-Markovian process. We demonstrate the method for leaky- and quadratic-integrate and fire neurons subject to spike trains with Poisson and gamma-distributed interspike intervals. We are able to model jump responses for both models accurately to both excitatory and inhibitory input under the assumption that all inputs are generated by one renewal process
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