Shape of diffusion and size of monochromatic region of a two-dimensional spin system

Omidvar, Hamed; Franceschetti, Massimo

doi:10.1145/3188745.3188836

Cited by 5 publications

(23 citation statements)

References 27 publications

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“…The general strategy of the proof follows the one provided in [24] -we show that while the spread of the θ-affected nodes (i.e., nodes on which a θ-particle would be unstable) reaches the origin, the θ-affected nodes are still at distances at least exponential in N from the origin. Once the origin is reached, the unstable particles around it will w.h.p.…”

Section: Proof Outlinementioning

confidence: 79%

“…In a recent work by the authors [24], a shape theorem for the spread of "affected" nodes -namely nodes on which a particle would be unstable in exactly one of its states -has been provided. This theorem gives a precise geometrical description of the set of affected nodes at any given time of the system evolution, and is a consequence of a concentration bound that was developed for the spreading time.…”

Section: Introductionmentioning

confidence: 99%

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Improved Intolerance Intervals and Size Bounds for a Schelling-Type Spin System

Omidvar¹,

Franceschetti²

2018

Preprint

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We consider a Schelling model of self-organized segregation in an open system that is equivalent to a zero-temperature Ising model with Glauber dynamics, or an Asynchronous Cellular Automaton (ACA) with extended Moore neighborhoods. Previous work has shown that if the intolerance parameter of the model τ ∈ (∼ 0.488, ∼ 0.512) \ {1/2}, then for a sufficiently large neighborhood of interaction N , any particle will end up in an exponentially large monochromatic region almost surely.This paper extends the above result to the interval τ ∈ (∼ 0.433, ∼ 0.567) \ {1/2}. We also improve the bounds on the size of the monochromatic region by exponential factors in N . Finally, we show that when particles are placed on the infinite lattice Z 2 rather than on a flat torus, for the values of τ mentioned above, sufficiently large N , and after a sufficiently long evolution time, any particle is contained in a large monochromatic region of size exponential in N , almost surely. The new proof, critically relies on a novel geometric construction related to the formation of the monochromatic region.

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Section: Proof Outlinementioning

confidence: 79%

Section: Introductionmentioning

confidence: 99%

Improved Intolerance Intervals and Size Bounds for a Schelling-Type Spin System

Omidvar¹,

Franceschetti²

2018

Preprint

Self Cite

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show abstract

Section: Introductionmentioning

confidence: 99%

“…Close to our model is the work by Omidvar and Franceschetti [34,35], who initiated an analysis of the size of monochrome regions in the so called Schelling Spin Systems. Agents of two different types are placed on a grid [34] and a geometric graph [35], respectively. Then independent and identical Poisson clocks are assigned to all agents and, every time a clock rings, the state of the corresponding agent is flipped if and only if the agent is discontent w.r.t.…”

Section: Introductionmentioning

confidence: 99%

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The Flip Schelling Process on Random Geometric and Erdös-Rényi Graphs

Bläsius,

Friedrich,

Krejca

et al. 2021

Preprint

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Schelling's classical segregation model gives a coherent explanation for the wide-spread phenomenon of residential segregation. We consider an agent-based saturated open-city variant, the Flip-Schelling-Process (FSP), in which agents, placed on a graph, have one out of two types and, based on the predominant type in their neighborhood, decide whether to changes their types; similar to a new agent arriving as soon as another agent leaves the vertex.We investigate the probability that an edge {u, v} is monochrome, i.e., that both vertices u and v have the same type in the FSP, and we provide a general framework for analyzing the influence of the underlying graph topology on residential segregation. In particular, for two adjacent vertices, we show that a highly decisive common neighborhood, i.e., a common neighborhood where the absolute value of the difference between the number of vertices with different types is high, supports segregation and moreover, that large common neighborhoods are more decisive.As an application, we study the expected behavior of the FSP on two common random graph models with and without geometry: (1) For random geometric graphs, we show that the existence of an edge {u, v} makes a highly decisive common neighborhood for u and v more likely. Based on this, we prove the existence of a constant c > 0 such that the expected fraction of monochrome edges after the FSP is at least 1/2 + c. (2) For Erdős-Rényi graphs we show that large common neighborhoods are unlikely and that the expected fraction of monochrome edges after the FSP is at most 1/2 + o (1). Our results indicate that the cluster structure of the underlying graph has a significant impact on the obtained segregation strength.

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Topological influence and locality in swap schelling games

Bilò

Lenzner

et al. 2022

Auton Agent Multi-Agent Syst

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Residential segregation is a wide-spread phenomenon that can be observed in almost every major city. In these urban areas residents with different racial or socioeconomic background tend to form homogeneous clusters. Schelling’s famous agent-based model for residential segregation explains how such clusters can form even if all agents are tolerant, i.e., if they agree to live in mixed neighborhoods. For segregation to occur, all it needs is a slight bias towards agents preferring similar neighbors. Very recently, Schelling’s model has been investigated from a game-theoretic point of view with selfish agents that strategically select their residential location. In these games, agents can improve on their current location by performing a location swap with another agent who is willing to swap. We significantly deepen these investigations by studying the influence of the underlying topology modeling the residential area on the existence of equilibria, the Price of Anarchy and on the dynamic properties of the resulting strategic multi-agent system. Moreover, as a new conceptual contribution, we also consider the influence of locality, i.e., if the location swaps are restricted to swaps of neighboring agents. We give improved almost tight bounds on the Price of Anarchy for arbitrary underlying graphs and we present (almost) tight bounds for regular graphs, paths and cycles. Moreover, we give almost tight bounds for grids, which are commonly used in empirical studies. For grids we also show that locality has a severe impact on the game dynamics.

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Shape of diffusion and size of monochromatic region of a two-dimensional spin system

Cited by 5 publications

References 27 publications

Improved Intolerance Intervals and Size Bounds for a Schelling-Type Spin System

Improved Intolerance Intervals and Size Bounds for a Schelling-Type Spin System

The Flip Schelling Process on Random Geometric and Erdös-Rényi Graphs

Topological influence and locality in swap schelling games

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