  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.
