Phase transitions in Phylogeny

We apply the theory of markov random fields on trees to derive a phase transition in the number of samples needed in order to reconstruct phylogenies. We consider the Cavender-Farris-Neyman model of evolution on trees, where all the inner nodes have degree at least 3, and the net transition on each edge is bounded by e. Motivated by a conjecture by M. Steel, we show that if 2 (1 - 2 e) (1 - 2e) > 1, then for balanced trees, the topology of the underlying tree, having n leaves, can be reconstructed from O(log n) samples (characters) at the leaves. On the other hand, we show that if 2 (1 - 2 e) (1 - 2 e) < 1, then there exist topologies which require at least poly(n) samples for reconstruction. Our results are the first rigorous results to establish the role of phase transitions for markov random fields on trees as studied in probability, statistical physics and information theory to the study of phylogenies in mathematical biology.Comment: To appear in Transactions of the AM

Robust reconstruction on trees is determined by the second eigenvalue

Consider a Markov chain on an infinite tree T=(V,E) rooted at \rho. In such a chain, once the initial root state \sigma(\rho) is chosen, each vertex iteratively chooses its state from the one of its parent by an application of a Markov transition rule (and all such applications are independent). Let \mu_j denote the resulting measure for \sigma(\rho)=j. The resulting measure \mu_j is defined on configurations \sigma=(\sigma(x))_{x\in V}\in A^V, where A is some finite set. Let \mu_j^n denote the restriction of \mu to the sigma-algebra generated by the variables \sigma(x), where x is at distance exactly n from \rho. Letting \alpha_n=max_{i,j\in A}d_{TV}(\mu_i^n,\mu_j^n), where d_{TV} denotes total variation distance, we say that the reconstruction problem is solvable if lim inf_{n\to\infty}\alpha_n>0. Reconstruction solvability roughly means that the nth level of the tree contains a nonvanishing amount of information on the root of the tree as n\to\infty. In this paper we study the problem of robust reconstruction. Let \nu be a nondegenerate distribution on A and \epsilon >0. Let \sigma be chosen according to \mu_j^n and \sigma' be obtained from \sigma by letting for each node independently, \sigma(v)=\sigma'(v) with probability 1-\epsilon and \sigma'(v) be an independent sample from \nu otherwise. We denote by \mu_j^n[\nu,\epsilon ] the resulting measure on \sigma'. The measure \mu_j^n[\nu,\epsilon ] is a perturbation of the measure \mu_j^n.Comment: Published at http://dx.doi.org/10.1214/009117904000000153 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Complete Characterization of Functions Satisfying the Conditions of Arrow's Theorem

Arrow's theorem implies that a social choice function satisfying Transitivity, the Pareto Principle (Unanimity) and Independence of Irrelevant Alternatives (IIA) must be dictatorial. When non-strict preferences are allowed, a dictatorial social choice function is defined as a function for which there exists a single voter whose strict preferences are followed. This definition allows for many different dictatorial functions. In particular, we construct examples of dictatorial functions which do not satisfy Transitivity and IIA. Thus Arrow's theorem, in the case of non-strict preferences, does not provide a complete characterization of all social choice functions satisfying Transitivity, the Pareto Principle, and IIA. The main results of this article provide such a characterization for Arrow's theorem, as well as for follow up results by Wilson. In particular, we strengthen Arrow's and Wilson's result by giving an exact if and only if condition for a function to satisfy Transitivity and IIA (and the Pareto Principle). Additionally, we derive formulas for the number of functions satisfying these conditions.Comment: 11 pages, 1 figur

On the mixing time of simple random walk on the super critical percolation cluster

We study the robustness under perturbations of mixing times, by studying mixing times of random walks in percolation clusters inside boxes in $\Z^d$. We show that for $d \geq 2$ and $p > p_c(\Z^d)$, the mixing time of simple random walk on the largest cluster inside $\{-n,...,n\}^d$ is $\Theta(n^2)$ - thus the mixing time is robust up to constant factor

Sharp Thresholds for Monotone Non Boolean Functions and Social Choice Theory

A key fact in the theory of Boolean functions $f : \{0,1\}^n \to \{0,1\}$ is that they often undergo sharp thresholds. For example: if the function $f : \{0,1\}^n \to \{0,1\}$ is monotone and symmetric under a transitive action with \E_p[f] = \eps and \E_q[f] = 1-\eps then $q-p \to 0$ as $n \to \infty$. Here \E_p denotes the product probability measure on $\{0,1\}^n$ where each coordinate takes the value $1$ independently with probability $p$. The fact that symmetric functions undergo sharp thresholds is important in the study of random graphs and constraint satisfaction problems as well as in social choice.In this paper we prove sharp thresholds for monotone functions taking values in an arbitrary finite sets. We also provide examples of applications of the results to social choice and to random graph problems. Among the applications is an analog for Condorcet's jury theorem and an indeterminacy result for a large class of social choice functions

Coin flipping from a cosmic source: On error correction of truly random bits

We study a problem related to coin flipping, coding theory, and noise sensitivity. Consider a source of truly random bits x \in \bits^n, and $k$ parties, who have noisy versions of the source bits y^i \in \bits^n, where for all $i$ and $j$, it holds that \Pr[y^i_j = x_j] = 1 - \eps, independently for all $i$ and $j$. That is, each party sees each bit correctly with probability $1-\epsilon$, and incorrectly (flipped) with probability $\epsilon$, independently for all bits and all parties. The parties, who cannot communicate, wish to agree beforehand on {\em balanced} functions f_i : \bits^n \to \bits such that $\Pr[f_1(y^1) = ... = f_k(y^k)]$ is maximized. In other words, each party wants to toss a fair coin so that the probability that all parties have the same coin is maximized. The functions $f_i$ may be thought of as an error correcting procedure for the source $x$. When $k=2,3$ no error correction is possible, as the optimal protocol is given by $f_i(x^i) = y^i_1$. On the other hand, for large values of $k$, better protocols exist. We study general properties of the optimal protocols and the asymptotic behavior of the problem with respect to $k$, $n$ and \eps. Our analysis uses tools from probability, discrete Fourier analysis, convexity and discrete symmetrization