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 $\gamma_n$ and the corresponding boundary measure $\gamma_n^+$ in $\mathbb {R}^n$. The main result in the theory of Gaussian isoperimetry (proven in the 1970s by Sudakov and Tsirelson, and independently by Borell) states that if $\gamma_n(A)=1/2$ then the surface area of $A$ is bounded by the surface area of a half-space with the same measure, $\gamma_n^+(A)\leq(2\pi)^{-1/2}$. Our results imply in particular that if $A\subset \mathbb {R}^n$ satisfies $\gamma_n(A)=1/2$ and $\gamma_n^+(A)\leq(2\pi)^{-1/2}+\delta$ then there exists a half-space $B\subset \mathbb {R}^n$ such that $\gamma_n(A\Delta B)\leq C\smash{\log^{-1/2}}(1/\delta)$ for an absolute constant $C$. 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 $\gamma_n(A\Delta B)\le C(n)\sqrt{\delta}$ for some function $C(n)$ 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 $\delta$.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 $k \geq 3$, for every value $\rho \neq 0$ of the noise and for every prescribed measures for the different parts as long as they are not all equal to $1/k$. 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