1,049 research outputs found
Isomorphism of Hilbert modules over stably finite C*-algebras
It is shown that if A is a stably finite C*-algebra and E is a countably
generated Hilbert A-module, then E gives rise to a compact element of the Cuntz
semigroup if and only if E is algebraically finitely generated and projective.
It follows that if E and F are equivalent in the sense of Coward, Elliott and
Ivanescu (CEI) and E is algebraically finitely generated and projective, then E
and F are isomorphic. In contrast to this, we exhibit two CEI-equivalent
Hilbert modules over a stably finite C*-algebra that are not isomorphic
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
Quantifying Link Stability in Ad Hoc Wireless Networks Subject to Ornstein-Uhlenbeck Mobility
The performance of mobile ad hoc networks in general and that of the routing
algorithm, in particular, can be heavily affected by the intrinsic dynamic
nature of the underlying topology. In this paper, we build a new
analytical/numerical framework that characterizes nodes' mobility and the
evolution of links between them. This formulation is based on a stationary
Markov chain representation of link connectivity. The existence of a link
between two nodes depends on their distance, which is governed by the mobility
model. In our analysis, nodes move randomly according to an Ornstein-Uhlenbeck
process using one tuning parameter to obtain different levels of randomness in
the mobility pattern. Finally, we propose an entropy-rate-based metric that
quantifies link uncertainty and evaluates its stability. Numerical results show
that the proposed approach can accurately reflect the random mobility in the
network and fully captures the link dynamics. It may thus be considered a
valuable performance metric for the evaluation of the link stability and
connectivity in these networks.Comment: 6 pages, 4 figures, Submitted to IEEE International Conference on
Communications 201
On an integral identity
We give three elementary proofs of a nice equality of definite integrals,
which arises from the theory of bivariate hypergeometric functions, and has
connections with irrationality proofs in number theory. We furthermore provide
a generalization together with an equally elementary proof and discuss some
consequences.Comment: 6 page
TreeExplorer: a coding algorithm for rooted trees with application to wireless and ad hoc routing
Routing tables in ad hoc and wireless routing protocols can be represented
using rooted trees. The constant need for communication and storage of these
trees in routing protocols demands an efficient rooted tree coding algorithm.
This efficiency is defined in terms of the average code length, and the
optimality of the algorithm is measured by comparing the average code length
with the entropy of the source. In this work, TreeExplorer is introduced as an
easy-to-implement and nearly optimal algorithm for coding rooted tree
structures. This method utilizes the number of leaves of the tree as an
indicator for choosing the best method of coding. We show how TreeExplorer can
improve existing routing protocols for ad hoc and wireless systems, which
normally entails a significant communication overhead
Statistical Properties of Transmissions Subject to Rayleigh Fading and Ornstein-Uhlenbeck Mobility
In this paper, we derive closed-form expressions for significant statistical
properties of the link signal-to-noise ratio (SNR) and the separation distance
in mobile ad hoc networks subject to Ornstein-Uhlenbeck (OU) mobility and
Rayleigh fading. In these systems, the SNR is a critical parameter as it
directly influences link performance. In the absence of signal fading, the
distribution of the link SNR depends exclusively on the squared distance
between nodes, which is governed by the mobility model. In our analysis, nodes
move randomly according to an Ornstein-Uhlenbeck process, using one tuning
parameter to control the temporal dependency in the mobility pattern. We derive
a complete statistical description of the squared distance and show that it
forms a stationary Markov process. Then, we compute closed-form expressions for
the probability density function (pdf), the cumulative distribution function
(cdf), the bivariate pdf, and the bivariate cdf of the link SNR. Next, we
introduce small-scale fading, modelled by a Rayleigh random variable, and
evaluate the pdf of the link SNR for rational path loss exponents. The validity
of our theoretical analysis is verified by extensive simulation studies. The
results presented in this work can be used to quantify link uncertainty and
evaluate stability in mobile ad hoc wireless systems
A Value of Information Framework for Latent Variable Models
In this paper, a general value of information (VoI) framework is formalised
for latent variable models. In particular, the mutual information between the
current status at the source node and the observed noisy measurements at the
destination node is used to evaluate the information value, which gives the
theoretical interpretation of the reduction in uncertainty in the current
status given that we have measurements of the latent process. Moreover, the VoI
expression for a hidden Markov model is obtained in this setting. Numerical
results are provided to show the relationship between the VoI and the
traditional age of information (AoI) metric, and the VoI of Markov and hidden
Markov models are analysed for the particular case when the latent process is
an Ornstein-Uhlenbeck process. While the contributions of this work are
theoretical, the proposed VoI framework is general and useful in designing
wireless systems that support timely, but noisy, status updates in the physical
world.Comment: 6 pages, 7 figure
Self-Organization Scheme for Balanced Routing in Large-Scale Multi-Hop Networks
We propose a self-organization scheme for cost-effective and load-balanced
routing in multi-hop networks. To avoid overloading nodes that provide
favourable routing conditions, we assign each node with a cost function that
penalizes high loads. Thus, finding routes to sink nodes is formulated as an
optimization problem in which the global objective function strikes a balance
between route costs and node loads. We apply belief propagation (its min-sum
version) to solve the network optimization problem and obtain a distributed
algorithm whereby the nodes collectively discover globally optimal routes by
performing low-complexity computations and exchanging messages with their
neighbours. We prove that the proposed method converges to the global optimum
after a finite number of local exchanges of messages. Finally, we demonstrate
numerically our framework's efficacy in balancing the node loads and study the
trade-off between load reduction and total cost minimization
Graph Compression with Side Information at the Decoder
In this paper, we study the problem of graph compression with side
information at the decoder. The focus is on the situation when an unlabelled
graph (which is also referred to as a structure) is to be compressed or is
available as side information. For correlated Erd\H{o}s-R\'enyi weighted random
graphs, we give a precise characterization of the smallest rate at which a
labelled graph or its structure can be compressed with aid of a correlated
labelled graph or its structure at the decoder. We approach this problem by
using the entropy-spectrum framework and establish some convergence results for
conditional distributions involving structures, which play a key role in the
construction of an optimal encoding and decoding scheme. Our proof essentially
uses the fact that, in the considered correlated Erd\H{o}s-R\'enyi model, the
structure retains most of the information about the labelled graph.
Furthermore, we consider the case of unweighted graphs and present how the
optimal decoding can be done using the notion of graph alignment.Comment: 21 pages, 2 figures, submitted to the IEEE Journal on Selected Areas
in Information Theor
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