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The use of data-random graphs in statistical testing of spatial patterns isintroduced recently. In this approach, a random directed graph is constructedfrom the data using the relative positions of the points from various classes.Different random graphs result from different definitions of the proximityregion associated with each data point and different graph statistics can beemployed for pattern testing. The approach used in this article is based onunderlying graphs of a family of data-random digraphs which is determined by afamily of parameterized proximity maps. The relative edge density of the AND-and OR-underlying graphs is used as the summary statistic, providing analternative to the relative arc density and domination number of the digraphemployed previously. Properly scaled, relative edge density of the underlyinggraphs is a U-statistic, facilitating analytic study of its asymptoticdistribution using standard U-statistic central limit theory. The approach isillustrated with an application to the testing of bivariate spatial clusteringpatterns of segregation and association. Knowledge of the asymptoticdistribution allows evaluation of the Pitman asymptotic efficiency, henceselection of the proximity map parameter to optimize efficiency. Asymptoticefficiency and Monte Carlo simulation analysis indicate that the AND-underlyingversion is better (in terms of power and efficiency) for the segregationalternative, while the OR-underlying version is better for the associationalternative. The approach presented here is also valid for data in higherdimensions.

Relative Edge Density of the Underlying Graphs Based on  Proportional-Edge Proximity Catch Digraphs for Testing Bivariate Spatial  Patterns (Technical Report)