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We investigate two discrete models of excitable media on a one-dimensional integer lattice ℤ: the κ-color Cyclic Cellular Automaton (CCA) and the κ-color Firefly Cellular Automaton (FCA). In both models, sites are assigned uniformly random colors from ℤ/κℤ. Neighboring sites with colors within a specified interaction range r tend to synchronize their colors upon a particular local event of 'excitation'. We establish that there are three phases of CCA/FCA on ℤ as we vary the interaction range r. First, if r is too small (undercoupled), there are too many non-interacting pairs of colors, and the whole graph ℤ will be partitioned into non-interacting intervals of sites with no excitation within each interval. If r is within a sweet spot (critical), then we show the system clusters into ever-growing monochromatic intervals. For the critical interaction range r=⌊κ/2⌋, we show the density of edges of differing colors at time t is Θ(t−1/2) and each site excites Θ(t1/2) times up to time t. Lastly, if r is too large (overcoupled), then neighboring sites can excite each other and such 'defects' will generate waves of excitation at a constant rate so that each site will get excited at least at a linear rate. For the special case of FCA with r=⌊2/κ⌋+1, we show that every site will become (κ+1)-periodic eventually.
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A Heterogeneous Schelling Model for Wealth Disparity and its Effect on Segregation
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.
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- Award ID(s):
- 2106687
- PAR ID:
- 10425781
- Date Published:
- Journal Name:
- EAAMO '22: Equity and Access in Algorithms, Mechanisms, and Optimization
- Page Range / eLocation ID:
- 1 to 10
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1145/3551624.3555293
