Rentian scaling for the measurement of optimal embedding of complex networks into physical space

© The authors 2016. Published by Oxford University Press. All rights reserved. The London Underground is one of the largest, oldest and most widely used systems of public transitin the world. Transportation in London is constantly challenged to expand and adapt its system to meet thechanging requirements of London's populace while maintaining a cost-effective and efficient network. Previousstudies have described this system using concepts from graph theory, reporting network diagnosticsand core-periphery architecture. These studies provide information about the basic structure and efficiencyof this network; however, the question of system optimization in the context of evolving demands is seldominvestigated. In this paper we examined the cost effectiveness of the topological-physical embedding ofthe Tube using estimations of the topological dimension, wiring length and Rentian scaling, an isometricscaling relationship between the number of elements and connections in a system. We measured theseproperties in both two- and three-dimensional embeddings of the networks into Euclidean space, as well asbetween two time points, and across the densely interconnected core and sparsely interconnected periphery.While the two- and three-dimensional representations of the present-day Tube exhibit Rentian scalingrelationships between nodes and edges of the system, the overall network is approximately cost-efficientlyembedded into its physical environment in two dimensions, but not in three. We further investigated anotable disparity in the topology of the network's local core versus its more extended periphery, suggestingan underlying relationship between meso-scale structure and physical embedding. The collectivefindings from this study, including changes in Rentian scaling over time, provide evidence for differentialembedding efficiency in planned versus self-organized networks. These findings suggest that conceptsof optimal physical embedding can be applied more broadly to other physical systems whose links areembedded in a well-defined space, and whose topology is constrained by a cost function that minimizeslink lengths within that space.