The Flip Schelling Process on Random Geometric and ErdősŰRényi Graphs Thomas Bläsius  ↸ Karlsruhe Institute of Technology, Karslruhe, Germany Tobias Friedrich  ↸ Hasso Plattner Institute, University of Potsdam, Potsdam, Germany Martin S. Krejca  Sorbonne Université, CNRS, LIP6, France Louise Molitor  ↸ Hasso Plattner Institute, University of Potsdam, Potsdam, Germany Abstract SchellingŠs 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 inĆuence 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 diference between the number of vertices with diferent 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 ErdősŰRényi 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 signiĄcant impact on the obtained segregation strength. 2012 ACM Subject ClassiĄcation Theory of computation → Network formation; Theory of compu- tation → Random network models Keywords and phrases Agent-based Model, Schelling Segregation, Spin System Funding Martin S. Krejca: This work was supported by the Paris Île-de-France Region. Acknowledgements We want to thank Thomas Sauerwald for the discussions on random walks. arXiv:2102.09856v1 [math.PR] 19 Feb 2021