Towards a classification of planar maps

Planar graphs and their spatial embedding-planar maps-are used in many different fields due to their ubiquity in the real world (leaf veins in biology, street patterns in urban studies, etc.) and are also fundamental objects in mathematics and combinatorics. These graphs have been well described in the literature, but we do not have so far a clear way to cluster them in different families. A typology of planar maps would be very useful and would allow to monitor their changes, to compare them with each other, or to correlate their structure with other properties. Using an algorithm which merges recursively the smallest areas in the graph with the largest ones, we plot the Gini coefficient of areas of cells and obtain a profile associated to each network. We test the relevance of these 'Gini profiles' on simulated networks and on real street networks of Barcelona (Spain), New York City (USA), Tokyo (Japan), and discuss their main properties. We also apply this method to the case of Paris (France) at different dates which allows us to follow the structural changes of this system. Finally, we discuss the important ingredient of spatial heterogeneity of real-world planar graphs and test some ideas on Manhattan and Tokyo. Our results show that the Gini profile encodes various informations about the structure of the corresponding planar map and represents a good candidate for constructing relevant classes of these objects.

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