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We develop a framework for self-induced phase changes in programmable matter in which a collection of agents with limited computational and communication capabilities can collectively perform appropriate global tasks in response to local stimuli that dynamically appear and disappear. Agents are represented by vertices in a dynamic graph whose edge set changes over time, and stimuli are placed adversarially on the vertices of where each agent is only capable of recognizing a co-located stimulus. Agents communicate via token passing along edges to alert other agents to transition to an Aware state when stimuli are present and an Unaware state when the stimuli disappear. We present an Adaptive Stimuli Algorithm that can handle arbitrary adversarial stimulus dynamics, while an adversary (or the agents themselves) reconfigures the connections (edges) of over time in a controlled way. This algorithm can be used to solve the foraging problem on reconfigurable graphs where, in addition to food sources (stimuli) being discovered, removed, or shifted arbitrarily, we would like the agents to consistently self-organize, using only local interactions, such that if the food remains in a position long enough, the agents transition to a gather phase in which many collectively form a single large component with small perimeter around the food. Alternatively, if no food source has existed recently, the agents should undergo a self-induced collective phase change and switch to a search phase in which they distribute themselves randomly throughout the graph to search for food. Unlike previous approaches to foraging, this process is indefinitely repeatable, withstanding competing broadcast waves of state transition that may interfere with each other. Like a physical phase change, such as the ferromagnetic models underlying the gather and search algorithms used for foraging, microscopic changes in the environment trigger these macroscopic, system-wide transitions as agents share information and respond locally to get the desired collective response. 

The global steady state of a system in thermal equilibrium exponentially favors configurations with lesser energy. This principle is a powerful explanation of self-organization because energy is a local property of configurations. For nonequilibrium systems, there is no such property for which an analogous principle holds, hence no common explanation of the diverse forms of self-organization they exhibit. However, a flurry of recent empirical results has shown that a local property of configurations called “rattling” predicts the steady states of some nonequilibrium systems, leading to claims of a far-reaching principle of nonequilibrium self-organization. But for which nonequilibrium systems is rattling accurate, and why? We develop a theory of rattling in terms of Markov processes that gives simple and precise answers to these key questions. Our results show that rattling predicts a broader class of nonequilibrium steady states than has been claimed and for different reasons than have been suggested. Its predictions hold to an extent determined by the relative variance of, and correlation between, the local and global “parts” of a steady state. We show how these quantities characterize the local-global relationships of various random walks on random graphs, spin-glass dynamics, and models of animal collective behavior. Surprisingly, we find that the core idea of rattling is so general as to apply to equilibrium and nonequilibrium systems alike. 

Local interactions of uncoordinated individuals produce the collective behaviors of many biological systems, inspiring much of the current research in programmable matter. A striking example is the spontaneous assembly of fire ants into "bridges" comprising their own bodies to traverse obstacles and reach sources of food. Experiments and simulations suggest that, remarkably, these ants always form one bridge - instead of multiple, competing bridges - despite a lack of central coordination. We argue that the reliable formation of a single bridge does not require sophistication on behalf of the individuals by provably reproducing this behavior in a self-organizing particle system. We show that the formation of a single bridge by the particles is a statistical inevitability of their preferences to move in a particular direction, such as toward a food source, and their preference for more neighbors. Two parameters, η and β, reflect the strengths of these preferences and determine the Gibbs stationary measure of the corresponding particle system’s Markov chain dynamics. We show that a single bridge almost certainly forms when η and β are sufficiently large. Our proof introduces an auxiliary Markov chain, called an "occupancy chain," that captures only the significant, global changes to the system. Through the occupancy chain, we abstract away information about the motion of individual particles, but we gain a more direct means of analyzing their collective behavior. Such abstractions provide a promising new direction for understanding many other systems of programmable matter. 

Local interactions of uncoordinated individuals produce the collective behaviors of many biological systems, inspiring much of the current research in programmable matter. A striking example is the spontaneous assembly of fire ants into "bridges" comprising their own bodies to traverse obstacles and reach sources of food. Experiments and simulations suggest that, remarkably, these ants always form one bridge - instead of multiple, competing bridges - despite a lack of central coordination. We argue that the reliable formation of a single bridge does not require sophistication on behalf of the individuals by provably reproducing this behavior in a self-organizing particle system. We show that the formation of a single bridge by the particles is a statistical inevitability of their preferences to move in a particular direction, such as toward a food source, and their preference for more neighbors. Two parameters, η and β, reflect the strengths of these preferences and determine the Gibbs stationary measure of the corresponding particle system’s Markov chain dynamics. We show that a single bridge almost certainly forms when η and β are sufficiently large. Our proof introduces an auxiliary Markov chain, called an "occupancy chain," that captures only the significant, global changes to the system. Through the occupancy chain, we abstract away information about the motion of individual particles, but we gain a more direct means of analyzing their collective behavior. Such abstractions provide a promising new direction for understanding many other systems of programmable matter. 

We develop a framework for self-induced phase changes in programmable matter in which a collection of agents with limited computational and communication capabilities can collectively perform appropriate global tasks in response to local stimuli that dynamically appear and disappear. Agents reside on graph vertices, where each stimulus is only recognized locally, and agents communicate via token passing along edges to alert other agents to transition to an Aware state when stimuli are present and an Unaware state when the stimuli disappear. We present an Adaptive Stimuli Algorithm that is robust to competing waves of messages as multiple stimuli change, possibly adversarially. Moreover, in addition to handling arbitrary stimulus dynamics, the algorithm can handle agents reconfiguring the connections (edges) of the graph over time in a controlled way. As an application, we show how this Adaptive Stimuli Algorithm on reconfigurable graphs can be used to solve the foraging problem, where food sources may be discovered, removed, or shifted at arbitrary times. We would like the agents to consistently self-organize, using only local interactions, such that if the food remains in a position long enough, the agents transition to a gather phase in which many collectively form a single large component with small perimeter around the food. Alternatively, if no food source has existed recently, the agents should undergo a self-induced phase change and switch to a search phase in which they distribute themselves randomly throughout the lattice region to search for food. Unlike previous approaches to foraging, this process is indefinitely repeatable, withstanding competing waves of messages that may interfere with each other. Like a physical phase change, microscopic changes such as the deletion or addition of a single food source trigger these macroscopic, system-wide transitions as agents share information about the environment and respond locally to get the desired collective response. 

We develop a framework for self-induced phase changes in programmable matter in which a collection of agents with limited computational and communication capabilities can collectively perform appropriate global tasks in response to local stimuli that dynamically appear and disappear. Agents reside on graph vertices, where each stimulus is only recognized locally, and agents communicate via token passing along edges to alert other agents to transition to an Aware state when stimuli are present and an Unaware state when the stimuli disappear. We present an Adaptive Stimuli Algorithm that is robust to competing waves of messages as multiple stimuli change, possibly adversarially. Moreover, in addition to handling arbitrary stimulus dynamics, the algorithm can handle agents reconfiguring the connections (edges) of the graph over time in a controlled way. As an application, we show how this Adaptive Stimuli Algorithm on reconfigurable graphs can be used to solve the foraging problem, where food sources may be discovered, removed, or shifted at arbitrary times. We would like the agents to consistently self-organize, using only local interactions, such that if the food remains in a position long enough, the agents transition to a gather phase in which many collectively form a single large component with small perimeter around the food. Alternatively, if no food source has existed recently, the agents should undergo a self-induced phase change and switch to a search phase in which they distribute themselves randomly throughout the lattice region to search for food. Unlike previous approaches to foraging, this process is indefinitely repeatable, withstanding competing waves of messages that may interfere with each other. Like a physical phase change, microscopic changes such as the deletion or addition of a single food source trigger these macroscopic, system-wide transitions as agents share information about the environment and respond locally to get the desired collective response. 

The Schelling model of segregation was introduced in economics to show how micro-motives can influence macro-behavior. Agents on a lattice have two colors and try to move to a different location if the number of their neighbors with a different color exceeds some threshold. Simulations reveal that even such mild local color preferences, or homophily, are sufficient to cause segregation. In this work, we propose a stochastic generalization of the Schelling model, based on both race and wealth, to understand how carefully architected placement of incentives, such as urban infrastructure, might affect segregation. In our model, each agent is assigned one of two colors along with a label, rich or poor. Further, we designate certain vertices on the lattice as “urban sites,” providing civic infrastructure that most benefits the poorer population, thus incentivizing the occupation of such vertices by poor agents of either color. We look at the stationary distribution of a Markov process reflecting these preferences to understand the long-term effects. We prove that when incentives are large enough, we will have ”urbanization of poverty,” an observed effect whereby poor people tend to congregate on urban sites. Moreover, even when homophily preferences are very small, if the incentives are large and there is income inequality in the two-color classes, we can get racial segregation on urban sites but integration on non-urban sites. In contrast, we find an overall mitigation of segregation when the urban sites are distributed throughout the lattice and the incentives for urban sites exceed the homophily biases. We prove that in this case, no matter how strong homophily preferences are, it will be exponentially unlikely that a configuration chosen from stationarity will have large, homogeneous clusters of agents of either color, suggesting we will have racial integration with high probability. 

To audit political district maps for partisan gerrymandering, one may determine a baseline for the expected distribution of partisan outcomes by sampling an ensemble of maps. One approach to sampling is to use redistricting policy as a guide to precisely codify preferences between maps. Such preferences give rise to a probability distribution on the space of redistricting plans, and Metropolis-Hastings methods allow one to sample ensembles of maps from the specified distribution. Although these approaches have nice theoretical properties and have successfully detected gerrymandering in legal settings, sampling from commonly-used policy-driven distributions is often computationally difficult. As of yet, there is no algorithm that can be used off-the-shelf for checking maps under generic redistricting criteria. In this work, we mitigate the computational challenges in a Metropolized-sampling technique through a parallel tempering method combined with ReCom[11] and, for the first time, validate that such techniques are effective on these problems at the scale of statewide precinct graphs for more policy informed measures. We develop these improvements through the first case study of district plans in Georgia. Our analysis projects that any election in Georgia will reliably elect 9 Republicans and 5 Democrats under the enacted plan. This result is largely fixed even as public opinion shifts toward either party and the partisan outcome of the enacted plan does not respond to the will of the people. Only 0.12% of the ∼ 160K plans in our ensemble were similarly non-responsive. 

Robots have components that work together to accomplish a task. Colloids are particles, usually less than 100 µm, that are small enough that they do not settle out of solution. Colloidal robots are particles capable of functions such as sensing, computation, communication, locomotion and energy management that are all controlled by the particle itself. Their design and synthesis is an emerging area of interdisciplinary research drawing from materials science, colloid science, self-assembly, robophysics and control theory. Many colloidal robot systems approach synthetic versions of biological cells in autonomy and may find ultimate utility in bringing these specialized functions to previously inaccessible locations. This Perspective examines the emerging literature and highlights certain design principles and strategies towards the realization of colloidal robots.