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Elchanan Mossel

Publications and source records attributed to Elchanan Mossel.

At least 91 records · Page 5Linked to original sources
arXiv2021

A Phase Transition in Arrow's Theorem

Arrow's Theorem concerns a fundamental problem in social choice theory: given the individual preferences of members of a group, how can they be aggregated to form rational group preferences? Arrow showed that in an election between three or more candidates, there are situations where any voting rule satisfying a small list of natural "fairness" axioms must produce an apparently irrational intransitive outcome. Furthermore, quantitative versions of Arrow's Theorem in the literature show that when voters choose rankings in an i.i.d.\ fashion, the outcome is intransitive with non-negligible probability. It is natural to ask if such a quantitative version of Arrow's Theorem holds for non-i.i.d.\ models. To answer this question, we study Arrow's Theorem under a natural non-i.i.d.\ model of voters inspired by canonical models in statistical physics; indeed, a version of this model was previously introduced by Raffaelli and Marsili in the physics literature. This model has a parameter, temperature, that prescribes the correlation between different voters. We show that the behavior of Arrow's Theorem in this model undergoes a striking phase transition: in the entire high temperature regime of the model, a Quantitative Arrow's Theorem holds showing that the probability of paradox for any voting rule satisfying the axioms is non-negligible; this is tight because the probability of paradox under pairwise majority goes to zero when approaching the critical temperature, and becomes exponentially small in the number of voters beyond it. We prove this occurs in another natural model of correlated voters and conjecture this phenomena is quite general.

math.PR↗
arXiv2021

A Geometric Model of Opinion Polarization

We introduce a simple, geometric model of opinion polarization. It is a model of political persuasion, as well as marketing and advertising, utilizing social values. It focuses on the interplay between different topics and persuasion efforts. We demonstrate that societal opinion polarization often arises as an unintended byproduct of influencers attempting to promote a product or idea. We discuss a number of mechanisms for the emergence of polarization involving one or more influencers, sending messages strategically, heuristically, or randomly. We also examine some computational aspects of choosing the most effective means of influencing agents, and the effects of those strategic considerations on polarization.

cs.SI↗
arXiv2021

Information Spread with Error Correction

We study the process of information dispersal in a network with communication errors and local error-correction. Specifically we consider a simple model where a single bit of information initially known to a single source is dispersed through the network, and communication errors lead to differences in the agents' opinions on this information. Naturally, such errors can very quickly make the communication completely unreliable, and in this work we study to what extent this unreliability can be mitigated by local error-correction, where nodes periodically correct their opinion based on the opinion of (some subset of) their neighbors. We analyze how the error spreads in the "early stages" of information dispersal by monitoring the average opinion, i.e., the fraction of agents that have the correct information among all nodes that hold an opinion at a given time. Our main results show that even with significant effort in error-correction, tiny amounts of noise can lead the average opinion to be nearly uncorrelated with the truth in early stages. We also propose some local methods to help agents gauge when the information they have has stabilized.

cs.SI↗
arXiv2021

Approximate polymorphisms

For a function $g\colon\{0,1\}^m\to\{0,1\}$, a function $f\colon \{0,1\}^n\to\{0,1\}$ is called a $g$-polymorphism if their actions commute: $f(g(\mathsf{row}_1(Z)),\ldots,g(\mathsf{row}_n(Z))) = g(f(\mathsf{col}_1(Z)),\ldots,f(\mathsf{col}_m(Z)))$ for all $Z\in\{0,1\}^{n\times m}$. The function $f$ is called an approximate polymorphism if this equality holds with probability close to $1$, when $Z$ is sampled uniformly. We study the structure of exact polymorphisms as well as approximate polymorphisms. Our results include: - We prove that an approximate polymorphism $f$ must be close to an exact polymorphism; - We give a characterization of exact polymorphisms, showing that besides trivial cases, only the functions $g = \mathsf{AND}, \mathsf{XOR}, \mathsf{OR}, \mathsf{NXOR}$ admit non-trivial exact polymorphisms. We also study the approximate polymorphism problem in the list-decoding regime (i.e., when the probability equality holds is not close to $1$, but is bounded away from some value). We show that if $f(x \land y) = f(x) \land f(y)$ with probability larger than $s_\land \approx 0.815$ then $f$ correlates with some low-degree character, and $s_\land$ is the optimal threshold for this property. Our result generalize the classical linearity testing result of Blum, Luby and Rubinfeld, that in this language showed that the approximate polymorphisms of $g = \mathsf{XOR}$ are close to XOR's, as well as a recent result of Filmus, Lifshitz, Minzer and Mossel, showing that the approximate polymorphisms of AND can only be close to AND functions.

cs.DM↗
arXiv2021

Regular graphs with linearly many triangles

A $d$-regular graph on $n$ nodes has at most $T_{\max} = \frac{n}{3} \tbinom{d}{2}$ triangles. We compute the leading asymptotics of the probability that a large random $d$-regular graph has at least $c \cdot T_{\max}$ triangles, and provide a strong structural description of such graphs. When $d$ is fixed, we show that such graphs typically consist of many disjoint $d+1$-cliques and an almost triangle-free part. When $d$ is allowed to grow with $n$, we show that such graphs typically consist of $d+o(d)$ sized almost cliques together with an almost triangle-free part. This confirms a conjecture of Collet and Eckmann from 2002 and considerably strengthens their observation that the triangles cannot be totally scattered in typical instances of regular graphs with many triangles.

math.CO↗
arXiv2021

Learning to Sample from Censored Markov Random Fields

We study learning Censor Markov Random Fields (abbreviated CMRFs). These are Markov Random Fields where some of the nodes are censored (not observed). We present an algorithm for learning high-temperature CMRFs within o(n) transportation distance. Crucially our algorithm makes no assumption about the structure of the graph or the number or location of the observed nodes. We obtain stronger results for high girth high-temperature CMRFs as well as computational lower bounds indicating that our results can not be qualitatively improved.

cs.LG↗
arXiv2021

Robust testing of low-dimensional functions

A natural problem in high-dimensional inference is to decide if a classifier $f:\mathbb{R}^n \rightarrow \{-1,1\}$ depends on a small number of linear directions of its input data. Call a function $g: \mathbb{R}^n \rightarrow \{-1,1\}$, a linear $k$-junta if it is completely determined by some $k$-dimensional subspace of the input space. A recent work of the authors showed that linear $k$-juntas are testable. Thus there exists an algorithm to distinguish between: 1. $f: \mathbb{R}^n \rightarrow \{-1,1\}$ which is a linear $k$-junta with surface area $s$, 2. $f$ is $ε$-far from any linear $k$-junta with surface area $(1+ε)s$, where the query complexity of the algorithm is independent of the ambient dimension $n$. Following the surge of interest in noise-tolerant property testing, in this paper we prove a noise-tolerant (or robust) version of this result. Namely, we give an algorithm which given any $c>0$, $ε>0$, distinguishes between 1. $f: \mathbb{R}^n \rightarrow \{-1,1\}$ has correlation at least $c$ with some linear $k$-junta with surface area $s$. 2. $f$ has correlation at most $c-ε$ with any linear $k$-junta with surface area at most $s$. The query complexity of our tester is $k^{\mathsf{poly}(s/ε)}$. Using our techniques, we also obtain a fully noise tolerant tester with the same query complexity for any class $\mathcal{C}$ of linear $k$-juntas with surface area bounded by $s$. As a consequence, we obtain a fully noise tolerant tester with query complexity $k^{O(\mathsf{poly}(\log k/ε))}$ for the class of intersection of $k$-halfspaces (for constant $k$) over the Gaussian space. Our query complexity is independent of the ambient dimension $n$. Previously, no non-trivial noise tolerant testers were known even for a single halfspace.

cs.CC↗
arXiv2020

The Probability of Intransitivity in Dice and Close Elections

We study the phenomenon of intransitivity in models of dice and voting. First, we follow a recent thread of research for $n$-sided dice with pairwise ordering induced by the probability, relative to $1/2$, that a throw from one die is higher than the other. We build on a recent result of Polymath showing that three dice with i.i.d. faces drawn from the uniform distribution on $\{1,\ldots,n\}$ and conditioned on the average of faces equal to $(n+1)/2$ are intransitive with asymptotic probability $1/4$. We show that if dice faces are drawn from a non-uniform continuous mean zero distribution conditioned on the average of faces equal to $0$, then three dice are transitive with high probability. We also extend our results to stationary Gaussian dice, whose faces, for example, can be the fractional Brownian increments with Hurst index $H\in(0,1)$. Second, we pose an analogous model in the context of Condorcet voting. We consider $n$ voters who rank $k$ alternatives independently and uniformly at random. The winner between each two alternatives is decided by a majority vote based on the preferences. We show that in this model, if all pairwise elections are close to tied, then the asymptotic probability of obtaining any tournament on the $k$ alternatives is equal to $2^{-k(k-1)/2}$, which markedly differs from known results in the model without conditioning. We also explore the Condorcet voting model where methods other than simple majority are used for pairwise elections. We investigate some natural definitions of "close to tied" for general functions and exhibit an example where the distribution over tournaments is not uniform under those definitions.

math.PR↗
arXiv2020

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.

math.PR↗
arXiv2020

Rational Groupthink

We study how long-lived rational agents learn from repeatedly observing a private signal and each others' actions. With normal signals, a group of any size learns more slowly than just four agents who directly observe each others' private signals in each period. Similar results apply to general signal structures. We identify rational groupthink---in which agents ignore their private signals and choose the same action for long periods of time---as the cause of this failure of information aggregation.

cs.GT↗
arXiv2020

Efficient Reconstruction of Stochastic Pedigrees

We introduce a new algorithm called {\sc Rec-Gen} for reconstructing the genealogy or \textit{pedigree} of an extant population purely from its genetic data. We justify our approach by giving a mathematical proof of the effectiveness of {\sc Rec-Gen} when applied to pedigrees from an idealized generative model that replicates some of the features of real-world pedigrees. Our algorithm is iterative and provides an accurate reconstruction of a large fraction of the pedigree while having relatively low \emph{sample complexity}, measured in terms of the length of the genetic sequences of the population. We propose our approach as a prototype for further investigation of the pedigree reconstruction problem toward the goal of applications to real-world examples. As such, our results have some conceptual bearing on the increasingly important issue of genomic privacy.

cs.DS↗
arXiv2020

Distributed Corruption Detection in Networks

We consider the problem of distributed corruption detection in networks. In this model, each vertex of a directed graph is either truthful or corrupt. Each vertex reports the type (truthful or corrupt) of each of its outneighbors. If it is truthful, it reports the truth, whereas if it is corrupt, it reports adversarially. This model, first considered by Preparata, Metze, and Chien in 1967, motivated by the desire to identify the faulty components of a digital system by having the other components checking them, became known as the PMC model. The main known results for this model characterize networks in which \emph{all} corrupt (that is, faulty) vertices can be identified, when there is a known upper bound on their number. We are interested in networks in which the identity of a \emph{large fraction} of the vertices can be identified. It is known that in the PMC model, in order to identify all corrupt vertices when their number is $t$, all indegrees have to be at least $t$. In contrast, we show that in $d$ regular-graphs with strong expansion properties, a $1-O(1/d)$ fraction of the corrupt vertices, and a $1-O(1/d)$ fraction of the truthful vertices can be identified, whenever there is a majority of truthful vertices. We also observe that if the graph is very far from being a good expander, namely, if the deletion of a small set of vertices splits the graph into small components, then no corruption detection is possible even if most of the vertices are truthful. Finally, we discuss the algorithmic aspects and the computational hardness of the problem.

math.CO↗
arXiv2020

Broadcasting on Random Directed Acyclic Graphs

We study a generalization of the well-known model of broadcasting on trees. Consider a directed acyclic graph (DAG) with a unique source vertex $X$, and suppose all other vertices have indegree $d\geq 2$. Let the vertices at distance $k$ from $X$ be called layer $k$. At layer $0$, $X$ is given a random bit. At layer $k\geq 1$, each vertex receives $d$ bits from its parents in layer $k-1$, which are transmitted along independent binary symmetric channel edges, and combines them using a $d$-ary Boolean processing function. The goal is to reconstruct $X$ with probability of error bounded away from $1/2$ using the values of all vertices at an arbitrarily deep layer. This question is closely related to models of reliable computation and storage, and information flow in biological networks. In this paper, we analyze randomly constructed DAGs, for which we show that broadcasting is only possible if the noise level is below a certain degree and function dependent critical threshold. For $d\geq 3$, and random DAGs with layer sizes $Ω(\log k)$ and majority processing functions, we identify the critical threshold. For $d=2$, we establish a similar result for NAND processing functions. We also prove a partial converse for odd $d\geq 3$ illustrating that the identified thresholds are impossible to improve by selecting different processing functions if the decoder is restricted to using a single vertex. Finally, for any noise level, we construct explicit DAGs (using expander graphs) with bounded degree and layer sizes $Θ(\log k)$ admitting reconstruction. In particular, we show that such DAGs can be generated in deterministic quasi-polynomial time or randomized polylogarithmic time in the depth. These results portray a doubly-exponential advantage for storing a bit in DAGs compared to trees, where $d=1$ but layer sizes must grow exponentially with depth in order to enable broadcasting.

cs.IT↗
arXiv2019

Being Corrupt Requires Being Clever, But Detecting Corruption Doesn't

We consider a variation of the problem of corruption detection on networks posed by Alon, Mossel, and Pemantle '15. In this model, each vertex of a graph can be either truthful or corrupt. Each vertex reports about the types (truthful or corrupt) of all its neighbors to a central agency, where truthful nodes report the true types they see and corrupt nodes report adversarially. The central agency aggregates these reports and attempts to find a single truthful node. Inspired by real auditing networks, we pose our problem for arbitrary graphs and consider corruption through a computational lens. We identify a key combinatorial parameter of the graph $m(G)$, which is the minimal number of corrupted agents needed to prevent the central agency from identifying a single truthful node. We give an efficient (in fact, linear time) algorithm for the central agency to identify a truthful node that is successful whenever the number of corrupt nodes is less than $m(G)/2$. On the other hand, we prove that for any constant $α> 1$, it is NP-hard to find a subset of nodes $S$ in $G$ such that corrupting $S$ prevents the central agency from finding one truthful node and $|S| \leq αm(G)$, assuming the Small Set Expansion Hypothesis (Raghavendra and Steurer, STOC '10). We conclude that being corrupt requires being clever, while detecting corruption does not. Our main technical insight is a relation between the minimum number of corrupt nodes required to hide all truthful nodes and a certain notion of vertex separability for the underlying graph. Additionally, this insight lets us design an efficient algorithm for a corrupt party to decide which graphs require the fewest corrupted nodes, up to a multiplicative factor of $O(\log n)$.

cs.SI↗
arXiv2019

Parallels Between Phase Transitions and Circuit Complexity?

In many natural average-case problems, there are or there are believed to be critical values in the parameter space where the structure of the space of solutions changes in a fundamental way. These phase transitions are often believed to coincide with drastic changes in the computational complexity of the associated problem. In this work, we study the circuit complexity of inference in the broadcast tree model, which has important applications in phylogenetic reconstruction and close connections to community detection. We establish a number of qualitative connections between phase transitions and circuit complexity in this model. Specifically, we show that there is a $\mathbf{TC}^0$ circuit that competes with the Bayes optimal predictor in some range of parameters above the Kesten-Stigum bound. We also show that there is a $16$ label broadcast tree model beneath the Kesten-Stigum bound in which it is possible to accurately guess the label of the root, but beating random guessing is $\mathbf{NC}^1$-hard on average. The key to locating phase transitions is often to study some intrinsic notions of complexity associated with belief propagation \--- e.g. where do linear statistics fail, or when is the posterior sensitive to noise? Ours is the first work to study the complexity of belief propagation in a way that is grounded in circuit complexity.

cs.CC↗
arXiv2019

AND Testing and Robust Judgement Aggregation

A function $f\colon\{0,1\}^n\to \{0,1\}$ is called an approximate AND-homomorphism if choosing ${\bf x},{\bf y}\in\{0,1\}^n$ randomly, we have that $f({\bf x}\land {\bf y}) = f({\bf x})\land f({\bf y})$ with probability at least $1-ε$, where $x\land y = (x_1\land y_1,\ldots,x_n\land y_n)$. We prove that if $f\colon \{0,1\}^n \to \{0,1\}$ is an approximate AND-homomorphism, then $f$ is $δ$-close to either a constant function or an AND function, where $δ(ε) \to 0$ as $ε\to0$. This improves on a result of Nehama, who proved a similar statement in which $δ$ depends on $n$. Our theorem implies a strong result on judgement aggregation in computational social choice. In the language of social choice, our result shows that if $f$ is $ε$-close to satisfying judgement aggregation, then it is $δ(ε)$-close to an oligarchy (the name for the AND function in social choice theory). This improves on Nehama's result, in which $δ$ decays polynomially with $n$. Our result follows from a more general one, in which we characterize approximate solutions to the eigenvalue equation $\mathrm T f = λg$, where $\mathrm T$ is the downwards noise operator $\mathrm T f(x) = \mathbb{E}_{\bf y}[f(x \land {\bf y})]$, $f$ is $[0,1]$-valued, and $g$ is $\{0,1\}$-valued. We identify all exact solutions to this equation, and show that any approximate solution in which $\mathrm T f$ and $λg$ are close is close to an exact solution.

cs.DM↗
arXiv2019

Social learning equilibria

We consider a large class of social learning models in which a group of agents face uncertainty regarding a state of the world, share the same utility function, observe private signals, and interact in a general dynamic setting. We introduce Social Learning Equilibria, a static equilibrium concept that abstracts away from the details of the given extensive form, but nevertheless captures the corresponding asymptotic equilibrium behavior. We establish general conditions for agreement, herding, and information aggregation in equilibrium, highlighting a connection between agreement and information aggregation.

math.ST↗