Exact solution for a metapopulation version of Schelling's model.

Journal Article (Journal Article)

In 1971, Schelling introduced a model in which families move if they have too many neighbors of the opposite type. In this paper, we will consider a metapopulation version of the model in which a city is divided into N neighborhoods, each of which has L houses. There are ρNL red families and ρNL blue families for some ρ < 1/2. Families are happy if there are ≤ ρ(c)L families of the opposite type in their neighborhood and unhappy otherwise. Each family moves to each vacant house at rates that depend on their happiness at their current location and that of their destination. Our main result is that if neighborhoods are large, then there are critical values ρ(b) < ρ(d) < ρ(c), so that for ρ < ρ(b), the two types are distributed randomly in equilibrium. When ρ > ρ(b), a new segregated equilibrium appears; for ρ(b) < ρ < ρ(d), there is bistability, but when ρ increases past ρ(d) the random state is no longer stable. When ρ(c) is small enough, the random state will again be the stationary distribution when ρ is close to 1/2. If so, this is preceded by a region of bistability.

Full Text

Duke Authors

Cited Authors

  • Durrett, R; Zhang, Y

Published Date

  • September 2014

Published In

Volume / Issue

  • 111 / 39

Start / End Page

  • 14036 - 14041

PubMed ID

  • 25225367

Pubmed Central ID

  • PMC4191761

Electronic International Standard Serial Number (EISSN)

  • 1091-6490

International Standard Serial Number (ISSN)

  • 0027-8424

Digital Object Identifier (DOI)

  • 10.1073/pnas.1414915111


  • eng