Document

**Published in:** LIPIcs, Volume 257, 2nd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2023)

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.

Shunhao Oh, Dana Randall, and Andréa W. Richa. Adaptive Collective Responses to Local Stimuli in Anonymous Dynamic Networks. In 2nd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 257, pp. 6:1-6:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{oh_et_al:LIPIcs.SAND.2023.6, author = {Oh, Shunhao and Randall, Dana and Richa, Andr\'{e}a W.}, title = {{Adaptive Collective Responses to Local Stimuli in Anonymous Dynamic Networks}}, booktitle = {2nd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2023)}, pages = {6:1--6:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-275-4}, ISSN = {1868-8969}, year = {2023}, volume = {257}, editor = {Doty, David and Spirakis, Paul}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAND.2023.6}, URN = {urn:nbn:de:0030-drops-179424}, doi = {10.4230/LIPIcs.SAND.2023.6}, annote = {Keywords: Dynamic networks, adaptive stimuli, foraging, self-organizing particle systems, programmable matter} }

Document

Brief Announcement

**Published in:** LIPIcs, Volume 246, 36th International Symposium on Distributed Computing (DISC 2022)

The foraging problem asks how a collective of particles with limited computational, communication and movement capabilities can autonomously compress around a food source and disperse when the food is depleted or shifted, which may occur at arbitrary times. We would like the particles to iteratively self-organize, using only local interactions, to correctly gather whenever a food particle remains in a position long enough and search if no food particle has existed recently. Unlike previous approaches, these search and gather phases should be self-induced so as to be indefinitely repeatable as the food evolves, with microscopic changes to the food triggering macroscopic, system-wide phase transitions. We present a stochastic foraging algorithm based on a phase change in the fixed magnetization Ising model from statistical physics: Our algorithm is the first to leverage self-induced phase changes as an algorithmic tool. A key component of our algorithm is a careful token passing mechanism ensuring a dispersion broadcast wave will always outpace a compression wave.

Shunhao Oh, Dana Randall, and Andréa W. Richa. Brief Announcement: Foraging in Particle Systems via Self-Induced Phase Changes. In 36th International Symposium on Distributed Computing (DISC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 246, pp. 51:1-51:3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{oh_et_al:LIPIcs.DISC.2022.51, author = {Oh, Shunhao and Randall, Dana and Richa, Andr\'{e}a W.}, title = {{Brief Announcement: Foraging in Particle Systems via Self-Induced Phase Changes}}, booktitle = {36th International Symposium on Distributed Computing (DISC 2022)}, pages = {51:1--51:3}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-255-6}, ISSN = {1868-8969}, year = {2022}, volume = {246}, editor = {Scheideler, Christian}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2022.51}, URN = {urn:nbn:de:0030-drops-172423}, doi = {10.4230/LIPIcs.DISC.2022.51}, annote = {Keywords: Foraging, self-organized particle systems, compression, phase changes} }

Document

RANDOM

**Published in:** LIPIcs, Volume 245, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)

We present local distributed, stochastic algorithms for alignment in self-organizing particle systems (SOPS) on two-dimensional lattices, where particles occupy unique sites on the lattice, and particles can make spatial moves to neighboring sites if they are unoccupied. Such models are abstractions of programmable matter, composed of individual computational particles with limited memory, strictly local communication abilities, and modest computational capabilities. We consider oriented particle systems, where particles are assigned a vector pointing in one of q directions, and each particle can compute the angle between its direction and the direction of any neighboring particle, although without knowledge of global orientation with respect to a fixed underlying coordinate system. Particles move stochastically, with each particle able to either modify its direction or make a local spatial move along a lattice edge during a move. We consider two settings: (a) where particle configurations must remain simply connected at all times and (b) where spatial moves are unconstrained and configurations can disconnect.
Our algorithms are inspired by the Potts model and its planar oriented variant known as the planar Potts model or clock model from statistical physics. We prove that for any q ≥ 2, by adjusting a single parameter, these self-organizing particle systems can be made to collectively align along a single dominant direction (analogous to a solid or ordered state) or remain non-aligned, in which case the fraction of particles oriented along any direction is nearly equal (analogous to a gaseous or disordered state). In the connected SOPS setting, we allow for two distinct parameters, one controlling the ferromagnetic attraction between neighboring particles (regardless of orientation) and the other controlling the preference of neighboring particles to align. We show that with appropriate settings of the input parameters, we can achieve compression and expansion, controlling how tightly gathered the particles are, as well as alignment or nonalignment, producing a single dominant orientation or not. While alignment is known for the Potts and clock models at sufficiently low temperatures, our proof in the SOPS setting are significantly more challenging because the particles make spatial moves, not all sites are occupied, and the total number of particles is fixed.

Hridesh Kedia, Shunhao Oh, and Dana Randall. Local Stochastic Algorithms for Alignment in Self-Organizing Particle Systems. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 14:1-14:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{kedia_et_al:LIPIcs.APPROX/RANDOM.2022.14, author = {Kedia, Hridesh and Oh, Shunhao and Randall, Dana}, title = {{Local Stochastic Algorithms for Alignment in Self-Organizing Particle Systems}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)}, pages = {14:1--14:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-249-5}, ISSN = {1868-8969}, year = {2022}, volume = {245}, editor = {Chakrabarti, Amit and Swamy, Chaitanya}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2022.14}, URN = {urn:nbn:de:0030-drops-171367}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2022.14}, annote = {Keywords: Self-organizing particle systems, alignment, Markov chains, active matter} }

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Invited Talk

**Published in:** LIPIcs, Volume 154, 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)

The field of randomized algorithms has benefitted greatly from insights from statistical physics. We give examples in two distinct settings. The first is in the context of Markov chain Monte Carlo algorithms, which have become ubiquitous across science and engineering as a means of exploring large configuration spaces. One of the most striking discoveries was the realization that many natural Markov chains undergo phase transitions, whereby they are efficient for some parameter settings and then suddenly become inefficient as a parameter of the system is slowly modified. The second is in the context of distributed algorithms for programmable matter. Self-organizing particle systems based on statistical models with phase changes have been used to achieve basic tasks involving coordination, movement, and conformation in a fully distributed, local setting. We briefly describe these two settings to demonstrate how computing and statistical physics together provide powerful insights that apply across multiple domains.

Dana Randall. Statistical Physics and Algorithms (Invited Talk). In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 1:1-1:6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{randall:LIPIcs.STACS.2020.1, author = {Randall, Dana}, title = {{Statistical Physics and Algorithms}}, booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)}, pages = {1:1--1:6}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-140-5}, ISSN = {1868-8969}, year = {2020}, volume = {154}, editor = {Paul, Christophe and Bl\"{a}ser, Markus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.1}, URN = {urn:nbn:de:0030-drops-118624}, doi = {10.4230/LIPIcs.STACS.2020.1}, annote = {Keywords: Markov chains, mixing times, phase transitions, programmable matter} }

Document

RANDOM

**Published in:** LIPIcs, Volume 145, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)

The six-vertex model in statistical physics is a weighted generalization of the ice model on Z^2 (i.e., Eulerian orientations) and the zero-temperature three-state Potts model (i.e., proper three-colorings). The phase diagram of the model represents its physical properties and suggests where local Markov chains will be efficient. In this paper, we analyze the mixing time of Glauber dynamics for the six-vertex model in the ordered phases. Specifically, we show that for all Boltzmann weights in the ferroelectric phase, there exist boundary conditions such that local Markov chains require exponential time to converge to equilibrium. This is the first rigorous result bounding the mixing time of Glauber dynamics in the ferroelectric phase. Our analysis demonstrates a fundamental connection between correlated random walks and the dynamics of intersecting lattice path models (or routings). We analyze the Glauber dynamics for the six-vertex model with free boundary conditions in the antiferroelectric phase and significantly extend the region for which local Markov chains are known to be slow mixing. This result relies on a Peierls argument and novel properties of weighted non-backtracking walks.

Matthew Fahrbach and Dana Randall. Slow Mixing of Glauber Dynamics for the Six-Vertex Model in the Ordered Phases. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 37:1-37:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{fahrbach_et_al:LIPIcs.APPROX-RANDOM.2019.37, author = {Fahrbach, Matthew and Randall, Dana}, title = {{Slow Mixing of Glauber Dynamics for the Six-Vertex Model in the Ordered Phases}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)}, pages = {37:1--37:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-125-2}, ISSN = {1868-8969}, year = {2019}, volume = {145}, editor = {Achlioptas, Dimitris and V\'{e}gh, L\'{a}szl\'{o} A.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2019.37}, URN = {urn:nbn:de:0030-drops-112523}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2019.37}, annote = {Keywords: Correlated random walk, Markov chain Monte Carlo, Six-vertex model} }

Document

RANDOM

**Published in:** LIPIcs, Volume 145, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)

We present and rigorously analyze the behavior of a distributed, stochastic algorithm for separation and integration in self-organizing particle systems, an abstraction of programmable matter. Such systems are composed of individual computational particles with limited memory, strictly local communication abilities, and modest computational power. We consider heterogeneous particle systems of two different colors and prove that these systems can collectively separate into different color classes or integrate, indifferent to color. We accomplish both behaviors with the same fully distributed, local, stochastic algorithm. Achieving separation or integration depends only on a single global parameter determining whether particles prefer to be next to other particles of the same color or not; this parameter is meant to represent external, environmental influences on the particle system. The algorithm is a generalization of a previous distributed, stochastic algorithm for compression (PODC '16) that can be viewed as a special case of separation where all particles have the same color. It is significantly more challenging to prove that the desired behavior is achieved in the heterogeneous setting, however, even in the bichromatic case we focus on. This requires combining several new techniques, including the cluster expansion from statistical physics, a new variant of the bridging argument of Miracle, Pascoe and Randall (RANDOM '11), the high-temperature expansion of the Ising model, and careful probabilistic arguments.

Sarah Cannon, Joshua J. Daymude, Cem Gökmen, Dana Randall, and Andréa W. Richa. A Local Stochastic Algorithm for Separation in Heterogeneous Self-Organizing Particle Systems. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 54:1-54:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{cannon_et_al:LIPIcs.APPROX-RANDOM.2019.54, author = {Cannon, Sarah and Daymude, Joshua J. and G\"{o}kmen, Cem and Randall, Dana and Richa, Andr\'{e}a W.}, title = {{A Local Stochastic Algorithm for Separation in Heterogeneous Self-Organizing Particle Systems}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)}, pages = {54:1--54:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-125-2}, ISSN = {1868-8969}, year = {2019}, volume = {145}, editor = {Achlioptas, Dimitris and V\'{e}gh, L\'{a}szl\'{o} A.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2019.54}, URN = {urn:nbn:de:0030-drops-112696}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2019.54}, annote = {Keywords: Markov chains, Programmable matter, Cluster expansion} }

Document

**Published in:** LIPIcs, Volume 110, 29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018)

Many physical models undergo phase transitions as some parameter of the system is varied. This phenomenon has bearing on the convergence times for local Markov chains walking among the configurations of the physical system. One of the most basic examples of this phenomenon is the ferromagnetic Ising model on an n x n square lattice region Lambda with mixed boundary conditions. For this spin system, if we fix the spins on the top and bottom sides of the square to be + and the left and right sides to be -, a standard Peierls argument based on energy shows that below some critical temperature t_c, any local Markov chain M requires time exponential in n to mix.
Spin glasses are magnetic alloys that generalize the Ising model by specifying the strength of nearest neighbor interactions on the lattice, including whether they are ferromagnetic or antiferromagnetic. Whenever a face of the lattice is bounded by an odd number of edges with ferromagnetic interactions, the face is considered frustrated because the local competing objectives cannot be simultaneously satisfied. We consider spin glasses with exactly four well-separated frustrated faces that are symmetric around the center of the lattice region under 90 degree rotations. We show that local Markov chains require exponential time for all spin glasses in this class. This class includes the ferromagnetic Ising model with mixed boundary conditions described above, where the frustrated faces are on the boundary. The standard Peierls argument breaks down when the frustrated faces are on the interior of Lambda and yields weaker results when they are on the boundary of Lambda but not near the corners. We show that there is a universal temperature T below which M will be slow for all spin glasses with four well-separated frustrated faces. Our argument shows that there is an exponentially small cut indicated by the free energy, carefully exploiting both entropy and energy to establish a small bottleneck in the state space to establish slow mixing.

David Gillman and Dana Randall. Slow Convergence of Ising and Spin Glass Models with Well-Separated Frustrated Vertices. In 29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 110, pp. 24:1-24:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{gillman_et_al:LIPIcs.AofA.2018.24, author = {Gillman, David and Randall, Dana}, title = {{Slow Convergence of Ising and Spin Glass Models with Well-Separated Frustrated Vertices}}, booktitle = {29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018)}, pages = {24:1--24:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-078-1}, ISSN = {1868-8969}, year = {2018}, volume = {110}, editor = {Fill, James Allen and Ward, Mark Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2018.24}, URN = {urn:nbn:de:0030-drops-89170}, doi = {10.4230/LIPIcs.AofA.2018.24}, annote = {Keywords: Mixing time, spin glass, Ising model, mixed boundary conditions, frustration} }

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Keynote Talk

**Published in:** LIPIcs, Volume 91, 31st International Symposium on Distributed Computing (DISC 2017)

Markov chain Monte Carlo methods have become ubiquitous across science and engineering to model dynamics and explore large sets of configurations. The idea is to perform a random walk among the configurations so that even though only a very small part of the space is visited, samples will be drawn from a desirable distribution. Over the last 20 years there have been tremendous advances in the design and analysis of efficient sampling algorithms for this purpose, building on insights from statistical physics. One of the striking discoveries has been the realization that many natural Markov chains undergo phase transitions, whereby they change from being efficient to inefficient as some parameter of the system is modified, also revealing interesting properties of the underlying random structures.
We will explore how phase transitions can provide valuable insights in three settings. First, they allow us to understand the limitations of certain classes of sampling algorithms, potentially leading to faster alternative approaches. Second, they reveal statistical properties of stationary distributions, giving insight into various interacting models. Example include colloids, or binary mixtures of molecules, segregation models, where individuals are more likely move when they are unhappy with their local demographics, and interacting particle systems from statistical physics. Last, they predict emergent phenomena that can be harnessed for the design of distributed algorithms for certain asynchronous models of programmable active matter. We will see how these three research threads are closely interrelated and inform one another.
The talk will take a random walk through some of the results included in the references.

Dana Randall. Phase Transitions and Emergent Phenomena in Random Structures and Algorithms (Keynote Talk). In 31st International Symposium on Distributed Computing (DISC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 91, pp. 3:1-3:2, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{randall:LIPIcs.DISC.2017.3, author = {Randall, Dana}, title = {{Phase Transitions and Emergent Phenomena in Random Structures and Algorithms}}, booktitle = {31st International Symposium on Distributed Computing (DISC 2017)}, pages = {3:1--3:2}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-053-8}, ISSN = {1868-8969}, year = {2017}, volume = {91}, editor = {Richa, Andr\'{e}a}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2017.3}, URN = {urn:nbn:de:0030-drops-80212}, doi = {10.4230/LIPIcs.DISC.2017.3}, annote = {Keywords: Markov chains, phase transitions, sampling, emergent phenomena, programmable matter} }