41 research outputs found
A law of large numbers for weighted plurality
Consider an election between k candidates in which each voter votes randomly
(but not necessarily independently) and suppose that there is a single
candidate that every voter prefers (in the sense that each voter is more likely
to vote for this special candidate than any other candidate). Suppose we have a
voting rule that takes all of the votes and produces a single outcome and
suppose that each individual voter has little effect on the outcome of the
voting rule. If the voting rule is a weighted plurality, then we show that with
high probability, the preferred candidate will win the election. Conversely, we
show that this statement fails for all other reasonable voting rules.
This result is an extension of H\"aggstr\"om, Kalai and Mossel, who proved
the above in the case k=2
Robust dimension free isoperimetry in Gaussian space
We prove the first robust dimension free isoperimetric result for the
standard Gaussian measure and the corresponding boundary measure
in . The main result in the theory of Gaussian
isoperimetry (proven in the 1970s by Sudakov and Tsirelson, and independently
by Borell) states that if then the surface area of is
bounded by the surface area of a half-space with the same measure,
. Our results imply in particular that if
satisfies and
then there exists a half-space
such that for an absolute constant . Since the
Gaussian isoperimetric result was established, only recently a robust version
of the Gaussian isoperimetric result was obtained by Cianchi et al., who showed
that for some function with
no effective bounds. Compared to the results of Cianchi et al., our results
have optimal (i.e., no) dependence on the dimension, but worse dependence on .Comment: Published at http://dx.doi.org/10.1214/13-AOP860 in the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Standard Simplices and Pluralities are Not the Most Noise Stable
The Standard Simplex Conjecture and the Plurality is Stablest Conjecture are
two conjectures stating that certain partitions are optimal with respect to
Gaussian and discrete noise stability respectively. These two conjectures are
natural generalizations of the Gaussian noise stability result by Borell (1985)
and the Majority is Stablest Theorem (2004). Here we show that the standard
simplex is not the most stable partition in Gaussian space and that Plurality
is not the most stable low influence partition in discrete space for every
number of parts , for every value of the noise and for
every prescribed measures for the different parts as long as they are not all
equal to . Our results do not contradict the original statements of the
Plurality is Stablest and Standard Simplex Conjectures in their original
statements concerning partitions to sets of equal measure. However, they
indicate that if these conjectures are true, their veracity and their proofs
will crucially rely on assuming that the sets are of equal measures, in stark
contrast to Borell's result, the Majority is Stablest Theorem and many other
results in isoperimetric theory. Given our results it is natural to ask for
(conjectured) partitions achieving the optimum noise stability.Comment: 14 page
Consistency Thresholds for the Planted Bisection Model
The planted bisection model is a random graph model in which the nodes are
divided into two equal-sized communities and then edges are added randomly in a
way that depends on the community membership. We establish necessary and
sufficient conditions for the asymptotic recoverability of the planted
bisection in this model. When the bisection is asymptotically recoverable, we
give an efficient algorithm that successfully recovers it. We also show that
the planted bisection is recoverable asymptotically if and only if with high
probability every node belongs to the same community as the majority of its
neighbors.
Our algorithm for finding the planted bisection runs in time almost linear in
the number of edges. It has three stages: spectral clustering to compute an
initial guess, a "replica" stage to get almost every vertex correct, and then
some simple local moves to finish the job. An independent work by Abbe,
Bandeira, and Hall establishes similar (slightly weaker) results but only in
the case of logarithmic average degree.Comment: latest version contains an erratum, addressing an error pointed out
by Jan van Waai