Segregation patterns for non-homogeneous locations in Schellings model

Abstract

We study a variant of the classical spatial proximity Schelling’s segregation model, including a function that breaks the homogeneity over the land. This weighting function represents objective and subjective assessments or valuations of the territory. It justifies why the agents give importance to their neighbors according to the locations they occupy. In this new model, agents belong to two ethnic groups and react when a proportion of their neighbors weighted with the land function is above a tolerance parameter. We show that a smooth behavior of the weighting function, with few maxima and minima, gives rise to large-scale segregation. In contrast, highly oscillating weights generate more fragmented patterns, with smaller clusters. We present computational simulations of this phenomenon and relate them to several American cities’ segregation patterns. Also, we characterize the equilibria of the model as minimizers of a weighted, discrete, Laplacian eigenvalue problem, derived from a Hamiltonian or total energy of a particle system. This framework allows proving that total segregation results for the particular condition of a weighting function with a single minimum, with two clusters of maximum size. Besides, we predict the place where the clusters will appear. These analytical results agree with computational simulations.