Self-organized Segregation on the Grid

Abstract

We consider an agent-based model in which two types of agents interact locally over a graph and have a common intolerance threshold $\tau$ for changing their types with exponentially distributed waiting times. The model is equivalent to an unperturbed Schelling model of self-organized segregation, an Asynchronous Cellular Automata (ACA) with extended Moore neighborhoods, or a zero-temperature Ising model with Glauber dynamics, 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 formation of large segregated regions of agents of a single type from the known size $\epsilon>0$ to size $\approx 0.134$. Namely, we show that for $0.433 < \tau < 1/2$ (and by symmetry $1/2<\tau<0.567$), 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 large segregated regions to size $\approx 0.312$ 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 $0.344 < \tau \leq 0.433$ (and by symmetry for $0.567 \leq \tau<0.656$) the expected size of the largest almost segregated region containing an arbitrary agent is exponential in the size of the neighborhood. 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 tolerance considered

Improved intolerance intervals and size bounds for a Schelling-type spin system

“…The next lemma, which is a restatement of lemma 8 in [28], 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 [27] 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 [28] 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.…”

“…By the following lemma (which is a restatement of lemma 11 in [28]), an N-block is a good block w.h.p. Since the number of different particles in a good block is balanced enough for sufficiently large N, a node whose entire neighborhood is contained in a good block cannot be a θ-affected node (we assume throughout that N is large enough such that this is the case).…”

“…We now review some results and definitions from a prior work [28]. In the current paper, one of the goals is to identify a better configuration, compared to the one constructed in [28], that can trigger a cascading process leading to monochromatic balls w.h.p. Hence we start with the following proposition which is proposition 1 in [28].…”

“…In the current paper, one of the goals is to identify a better configuration, compared to the one constructed in [28], that can trigger a cascading process leading to monochromatic balls w.h.p. Hence we start with the following proposition which is proposition 1 in [28]. Let N (u) be the neighborhood of an arbitrary particle u containing N particles.…”