We consider programmable matter as a collection of simple computational elements (or particles) with limited (constantsize) memory that self-organize to solve system-wide problems of movement, configuration, and coordination. Here, we focus on the compression problem, in which the particle system gathers as tightly together as possible, as in a sphere or its equivalent in the presence of some underlying geometry. More specifically, we seek fully distributed, local, and asynchronous algorithms that lead the system to converge to a configuration with small perimeter. We present a Markov chain based algorithm that solves the compression problem under the geometric amoebot model, for particle systems that begin in a connected configuration with no holes. The algorithm takes as input a bias parameter λ, where λ > 1 corresponds to particles favoring inducing more lattice triangles within the particle system. We show that for all λ > 5, there is a constant α > 1 such that at stationarity with all but exponentially small probability the particles are α-compressed, meaning the perimeter of the system configuration is at most α • pmin, where pmin is the minimum possible perimeter of the particle system. We additionally prove that the same algorithm can be used for expansion for small values of λ; in particular, for all 0 < λ < √ 2, there is a constant β < 1 such that at stationarity, with all but an ex-* A full version of this paper, including omitted proofs, is

Abstract. We consider programmable matter that consists of computationally limited devices (called particles) that are able to self-organize in order to achieve some collective goal without the need for central control or external intervention. We use the geometric amoebot model to describe such self-organizing particle systems, which defines how particles can actively move and communicate with one another. In this paper, we present an efficient local-control algorithm which solves the leader election problem in O(n) asynchronous rounds with high probability, where n is the number of particles in the system. Our algorithm relies only on local information -particles do not have unique identifiers, any knowledge of n, or any sort of global coordinate system -and requires only constant memory per particle.

Extreme polarization can undermine democracy by making compromise impossible and transforming politics into a zero-sum game. “Ideological polarization”—the extent to which political views are widely dispersed—is already strong among elites, but less so among the general public [N. McCarty, Polarization: What Everyone Needs to Know, 2019, pp. 50–68]. Strong mutual distrust and hostility between Democrats and Republicans in the United States, combined with the elites’ already strong ideological polarization, could lead to increasing ideological polarization among the public. The paper addresses two questions: 1) Is there a level of ideological polarization above which polarization feeds upon itself to become a runaway process? 2) If so, what policy interventions could prevent such dangerous positive feedback loops? To explore these questions, we present an agent-based model of ideological polarization that differentiates between the tendency for two actors to interact (“exposure”) and how they respond when interactions occur, positing that interaction between similar actors reduces their difference, while interaction between dissimilar actors increases their difference. Our analysis explores the effects on polarization of different levels of tolerance to other views, responsiveness to other views, exposure to dissimilar actors, multiple ideological dimensions, economic self-interest, and external shocks. The results suggest strategies for preventing, or at least slowing, the development of extreme polarization.

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