The impact of geometry on monochrome regions in the flip Schelling process

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