196 research outputs found
The empirical process on Gaussian spherical harmonics
We establish weak convergence of the empirical process on the spherical
harmonics of a Gaussian random field in the presence of an unknown angular
power spectrum. This result suggests various Gaussianity tests with an
asymptotic justification. The issue of testing for Gaussianity on isotropic
spherical random fields has recently received strong empirical attention in the
cosmological literature, in connection with the statistical analysis of cosmic
microwave background radiation
On the Voting Time of the Deterministic Majority Process
In the deterministic binary majority process we are given a simple graph
where each node has one out of two initial opinions. In every round, every node
adopts the majority opinion among its neighbors. By using a potential argument
first discovered by Goles and Olivos (1980), it is known that this process
always converges in rounds to a two-periodic state in which every node
either keeps its opinion or changes it in every round.
It has been shown by Frischknecht, Keller, and Wattenhofer (2013) that the
bound on the convergence time of the deterministic binary majority
process is indeed tight even for dense graphs. However, in many graphs such as
the complete graph, from any initial opinion assignment, the process converges
in just a constant number of rounds.
By carefully exploiting the structure of the potential function by Goles and
Olivos (1980), we derive a new upper bound on the convergence time of the
deterministic binary majority process that accounts for such exceptional cases.
We show that it is possible to identify certain modules of a graph in order
to obtain a new graph with the property that the worst-case
convergence time of is an upper bound on that of . Moreover, even
though our upper bound can be computed in linear time, we show that, given an
integer , it is NP-hard to decide whether there exists an initial opinion
assignment for which it takes more than rounds to converge to the
two-periodic state.Comment: full version of brief announcement accepted at DISC'1
Simple Dynamics for Plurality Consensus
We study a \emph{Plurality-Consensus} process in which each of anonymous
agents of a communication network initially supports an opinion (a color chosen
from a finite set ). Then, in every (synchronous) round, each agent can
revise his color according to the opinions currently held by a random sample of
his neighbors. It is assumed that the initial color configuration exhibits a
sufficiently large \emph{bias} towards a fixed plurality color, that is,
the number of nodes supporting the plurality color exceeds the number of nodes
supporting any other color by additional nodes. The goal is having the
process to converge to the \emph{stable} configuration in which all nodes
support the initial plurality. We consider a basic model in which the network
is a clique and the update rule (called here the \emph{3-majority dynamics}) of
the process is the following: each agent looks at the colors of three random
neighbors and then applies the majority rule (breaking ties uniformly).
We prove that the process converges in time with high probability, provided that .
We then prove that our upper bound above is tight as long as . This fact implies an exponential time-gap between the
plurality-consensus process and the \emph{median} process studied by Doerr et
al. in [ACM SPAA'11].
A natural question is whether looking at more (than three) random neighbors
can significantly speed up the process. We provide a negative answer to this
question: In particular, we show that samples of polylogarithmic size can speed
up the process by a polylogarithmic factor only.Comment: Preprint of journal versio
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