Self-organized Segregation on the Grid

Abstract

We consider an agent-based model with exponentially distributed waiting times in which two types of agents interact locally over a graph, and based on this interaction and on the value of a common intolerance threshold τ , decide whether to change their types. This is equivalent to a zero-temperature Ising model with Glauber dynamics, an Asynchronous Cellular Automaton (ACA) with extended Moore neighborhoods, or a Schelling model of self-organized segregation in an open system, and has applications in the analysis of social and biological networks, and spin glasses systems. Some rigorous results were recently obtained in the theoretical computer science literature, and this work provides several extensions. We enlarge the intolerance interval leading to the expected formation of large segregated regions of agents of a single type from the known size > 0 to size ≈ 0.134. Namely, we show that for 0.433 < τ < 1/2 (and by symmetry 1/2 < τ < 0.567), the expected size of the largest segregated region containing an arbitrary agent is exponential in the size of the neighborhood. We further extend the interval leading to expected large segregated regions to size ≈ 0.312 considering "almost segregated" regions, namely regions where the ratio of the number of agents of one type and the number of agents of the other type vanishes quickly as the size of the neighborhood grows. In this case, we show that for 0.344 < τ ≤ 0.433 (and by symmetry for 0.567 ≤ τ < 0.656) the expected size of the largest almost segregated region containing an arbitrary agent is exponential in the size of the neighborhood. This behavior is reminiscent of supercritical percolation, where small clusters of empty sites can be observed within any sufficiently large region of the occupied percolation cluster. The exponential bounds that we provide also imply that complete segregation, where agents of a single type cover the whole grid, does not occur with high probability for p = 1/2 and the range of intolerance considered.

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

“…By a region of expansion we mean a region of expansion of only one type (θ). The next lemma, which is a restatement of Lemma 8 in [22], shows that as long as τ ∈ (τ * , 1/2), w.h.p. a monochromatic block on G w can make an exponentially large area monochromatic.…”

“…In the two-dimensional model case, Immorlica et al [21] have shown for the Glauber dynamics that for τ ∈ (1/2 − , 1/2) the expected size of the monochromatic region is exponential in the size of the neighborhood. This -size interval has been enlarged in a previous work by the authors [22] to 0.433 < τ < 1/2 (and by symmetry 1/2 < τ < 0.567). In the same work, the interval has been further extended to 0.344 < τ ≤ 0.433 (and by symmetry for 0.567 ≤ τ < 0.656) considering "almost monochromatic" regions, namely regions where the ratio of the number of particles in one state and the number of particles in the other state quickly vanishes as the size of the neighborhood grows.…”

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

“…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.…”

“…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.…”

“…The commonly used underlying topology for modeling the residential areas are (toroidal) grid graphs [10,29,34], regular graphs [10,15,19], paths [10,30], cycles [3,5,6,11,45] and trees [1,10,20,30]. Considering the influence of the given topology that models the residential area regarding, e.g., the existence of stable states and convergence behavior leads to phenomena like non-existence of stable states [19,20], non-convergence to stable states [10,15,19], and high-mixing times [9,26].…”