
Avoidable Vertices and Edges in Graphs
A vertex in a graph is simplicial if its neighborhood forms a clique. We...
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Volume Doubling Condition and a Local Poincaré Inequality on Unweighted Random Geometric Graphs
The aim of this paper is to establish two fundamental measuremetric pro...
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Reconfiguration of graph minors
Under the reconfiguration framework, we consider the various ways that a...
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Convergence and Hardness of Strategic Schelling Segregation
The phenomenon of residential segregation was captured by Schelling's fa...
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Graphical Construction of Spatial Gibbs Random Graphs
We present a Spatial Gibbs Random Graphs Model that incorporates the int...
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Connectivity and Structure in Large Networks
Large reallife complex networks are often modeled by various random gra...
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Perfectly Secure Communication, based on GraphTopological Addressing in UniqueNeighborhood Networks
We consider network graphs G=(V,E) in which adjacent nodes share common ...
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The Flip Schelling Process on Random Geometric and ErdösRényi Graphs
Schelling's classical segregation model gives a coherent explanation for the widespread phenomenon of residential segregation. We consider an agentbased saturated opencity 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 ErdösRényi 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.
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