7,032 research outputs found
Bounds on the Total Coefficient Size of Nullstellensatz Proofs of the Pigeonhole Principle and the Ordering Principle
In this paper, we investigate the total coefficient size of Nullstellensatz
proofs. We show that Nullstellensatz proofs of the pigeonhole principle on
pigeons require total coefficient size and that there exist
Nullstellensatz proofs of the ordering principle on elements with total
coefficient size
Detectability thresholds and optimal algorithms for community structure in dynamic networks
We study the fundamental limits on learning latent community structure in
dynamic networks. Specifically, we study dynamic stochastic block models where
nodes change their community membership over time, but where edges are
generated independently at each time step. In this setting (which is a special
case of several existing models), we are able to derive the detectability
threshold exactly, as a function of the rate of change and the strength of the
communities. Below this threshold, we claim that no algorithm can identify the
communities better than chance. We then give two algorithms that are optimal in
the sense that they succeed all the way down to this limit. The first uses
belief propagation (BP), which gives asymptotically optimal accuracy, and the
second is a fast spectral clustering algorithm, based on linearizing the BP
equations. We verify our analytic and algorithmic results via numerical
simulation, and close with a brief discussion of extensions and open questions.Comment: 9 pages, 3 figure
On the Construction of Radio Environment Maps for Cognitive Radio Networks
The Radio Environment Map (REM) provides an effective approach to Dynamic
Spectrum Access (DSA) in Cognitive Radio Networks (CRNs). Previous results on
REM construction show that there exists a tradeoff between the number of
measurements (sensors) and REM accuracy. In this paper, we analyze this
tradeoff and determine that the REM error is a decreasing and convex function
of the number of measurements (sensors). The concept of geographic entropy is
introduced to quantify this relationship. And the influence of sensor
deployment on REM accuracy is examined using information theory techniques. The
results obtained in this paper are applicable not only for the REM, but also
for wireless sensor network deployment.Comment: 6 pages, 7 figures, IEEE WCNC conferenc
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