Schelling's classical segregation model gives a coherent explanation for the wide-spread phenomenon of residential segregation. We consider an agent-based saturated open-city variant, the Flip Schelling Process (FSP), in which agents, placed on a graph, have one out of two types and, based on the predominant type in their neighborhood, decide whether to changes their types; similar to a new agent arriving as soon as another agent leaves the vertex. We investigate the probability that an edge $\{u,v\}$ is monochrome, i.e., that both vertices $u$ and $v$ have the same type in the FSP, and we provide a general framework for analyzing the influence of the underlying graph topology on residential segregation. In particular, for two adjacent vertices, we show that a highly decisive common neighborhood, i.e., a common neighborhood where the absolute value of the difference between the number of vertices with different types is high, supports segregation and moreover, that large common neighborhoods are more decisive. As an application, we study the expected behavior of the FSP on two common random graph models with and without geometry: (1) For random geometric graphs, we show that the existence of an edge $\{u,v\}$ makes a highly decisive common neighborhood for $u$ and $v$ more likely. Based on this, we prove the existence of a constant $c > 0$ such that the expected fraction of monochrome edges after the FSP is at least $1/2 + c$. (2) For Erd\"os-R\'enyi graphs we show that large common neighborhoods are unlikely and that the expected fraction of monochrome edges after the FSP is at most $1/2 + o(1)$. Our results indicate that the cluster structure of the underlying graph has a significant impact on the obtained segregation strength.

中文翻译：

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随机几何图和Erdös-Rényi图的翻转Schelling过程

Schelling的经典隔离模型为住宅隔离的广泛现象提供了连贯的解释。我们考虑基于代理的饱和开放城市变体Flip Schelling Process（FSP），在该变量中，放置在图形上的代理具有两种类型中的一种，并根据其附近的主要类型来决定是否更改他们的类型；类似于新的特工一旦另一个特工离开顶点就到达。我们研究了边$ \ {u，v \} $是单色的可能性，即，顶点$ u $和$ v $在FSP中具有相同的类型，并且我们提供了一个通用框架来分析居民隔离的基础图拓扑。特别是，对于两个相邻的顶点，我们显示出一个高度决定性的公共邻域，即 一个普通邻域，其中不同类型的顶点数量之间的差异的绝对值很高，它支持隔离，此外，大的公共邻域更具决定性。作为一种应用，我们研究了带有和不带有几何图形的两个常见随机图形模型上FSP的预期行为：（1）对于随机几何图形，我们表明边$ \ {u，v \} $的存在使得具有决定性的高度共同性社区，因此更有可能产生$ u $和$ v $。基于此，我们证明存在常数$ c> 0 $，使得FSP之后的单色边缘的预期分数至少为$ 1/2 + c $。（2）对于Erd \“ os-R \'enyi图，我们显示出大的公共邻域是不太可能的，并且FSP之后的单色边缘的预期比例最多为$ 1/2 + o（1）$。