Clustering and Mixing Times for Segregation Models on ℤ<sup>2</sup>

Bhakta, Prateek; Miracle, Sarah; Randall, Dana

doi:10.1137/1.9781611973402.24

Cited by 32 publications

(42 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Since by (19) for all m ≥ ρ , the probability of having a monochromatic region of size m 2 containing u has at most a double exponentially small probability, the tail of the remaining series in (20) converges to a constant, while for sufficiently large N the sum of the first ρ − ρ − 1 terms is smaller than the first term of (20), and the proof is complete.…”

Section: By Lemma 8 We Havementioning

confidence: 83%

Self-organized Segregation on the Grid

Omidvar

Franceschetti

2017

Proceedings of the ACM Symposium on Principles of Distributed Computing

View full text Add to dashboard Cite

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.

show abstract

Section: By Lemma 8 We Havementioning

confidence: 83%

Self-organized Segregation on the Grid

Omidvar

Franceschetti

2017

Proceedings of the ACM Symposium on Principles of Distributed Computing

View full text Add to dashboard Cite

show abstract

“…The convergence time was considered by Mobius & Rosenblat [28] who observe that the Markov chain analyzed in [37,38,39] has a very high mixing time. Bhakta et al [7] show in the two-dimensional grid case a dichotomy in mixing times for high τ and very low τ values.…”

Section: Related Workmentioning

confidence: 98%

Convergence and Hardness of Strategic Schelling Segregation

Echzell

Friedrich

Lenzner

et al. 2019

Lecture Notes in Computer Science

View full text Add to dashboard Cite

The phenomenon of residential segregation was captured by Schelling's famous segregation model where two types of agents are placed on a grid and an agent is content with her location if the fraction of her neighbors which have the same type as her is at least τ , for some 0 < τ < 1. Discontent agents simply swap their location with a randomly chosen other discontent agent or jump to a random empty cell.We analyze a generalized game-theoretic model of Schelling segregation which allows more than two agent types and more general underlying graphs modeling the residential area. For this we show that both aspects heavily influence the dynamic properties and the tractability of finding an optimal placement. We map the boundary of when improving response dynamics (IRD), i.e., the natural approach for finding equilibrium states, are guaranteed to converge. For this we prove several sharp threshold results where guaranteed IRD convergence suddenly turns into the strongest possible non-convergence result: a violation of weak acyclicity. In particular, we show such threshold results also for Schelling's original model, which is in contrast to the standard assumption in many empirical papers. Furthermore, we show that in case of convergence, IRD find an equilibrium in O(m) steps, where m is the number of edges in the underlying graph and show that this bound is met in empirical simulations starting from random initial agent placements.

show abstract

“…These values S t , depend upon Ω(R t ) which will be proportions of the overall count from all macrostate sizes in Equation (9). This is because each non-redundant permutation of the labels will correspond to a particular value of R,…”

Section: Estimating the Entropy Of The Schelling Model From The Micromentioning

confidence: 99%

“…Approaches to finding a spatial measure which captures obscure cases is an ongoing area of research and can be relied upon for specified classes of patterns [7] produces a physically inspired model for the gradient of the cluster formations so that the arrangements over the iterations resemble that in other physical phenomena but these states are still assess qualitatively by inspection as noted in [8]. From the methodology of [9] a representation for certain 'patterns' (specific macroscopic configurations) are considered for their size examining a range of predefined scenarios. The approach also employs a Markov chain with Metropolis transition probabilities for the change of states between and although very interesting it does deviate from the archetypical Schelling model paradigm.…”

Section: Introductionmentioning

confidence: 99%

Examining the Schelling Model Simulation through an Estimation of Its Entropy

Mantzaris

Marich

Halfman

2018

Entropy

View full text Add to dashboard Cite

The Schelling model of segregation allows for a general description of residential movements in an environment modeled by a lattice. The key factor is that occupants change positions until they are surrounded by a designated minimum number of similarly labeled residents. An analogy to the Ising model has been made in previous research, primarily due the assumption of state changes being dependent upon the adjacent cell positions. This allows for concepts produced in statistical mechanics to be applied to the Schelling model. Here is presented a methodology to estimate the entropy of the model for different states of the simulation. A Monte Carlo estimate is obtained for the set of macrostates defined as the different aggregate homogeneity satisfaction values across all residents, which allows for the entropy value to be produced for each state. This produces a trace of the estimated entropy value for the states of the lattice configurations to be displayed with each iteration. The results show that the initial random placements of residents have larger entropy values than the final states of the simulation when the overall homogeneity of the residential locality is increased.

show abstract

Clustering and Mixing Times for Segregation Models on ℤ²

Cited by 32 publications

References 21 publications

Self-organized Segregation on the Grid

Self-organized Segregation on the Grid

Convergence and Hardness of Strategic Schelling Segregation

Examining the Schelling Model Simulation through an Estimation of Its Entropy

Contact Info

Product

Resources

About

Clustering and Mixing Times for Segregation Models on ℤ2

Cited by 32 publications

References 21 publications

Self-organized Segregation on the Grid

Self-organized Segregation on the Grid

Convergence and Hardness of Strategic Schelling Segregation

Examining the Schelling Model Simulation through an Estimation of Its Entropy

Contact Info

Product

Resources

About

Clustering and Mixing Times for Segregation Models on ℤ²