D-summable fractal dimensions of complex networks. (February 2019) - Ramirez-Arellano, Aldo; Bermúdez-Gómez, Salvador; Hernández-Simón, Luis Manuel; Bory-Reyes, Juan
Highlights: The concept of d-summable sets is applied in complex networks to measure fractal dimension. The concept of d-summable sets has not yet been applied in complex networks. The d-summable information model fits the covering of the network best for several examples. Empirical evidence suggest that information dimension is not a universal model of real complex networks. Abstract: In past two decades a wide range of complex systems, spanning many different disciplines, have been structured in the form of networks. Network dimension is a crucial concept to understand not only network topology, but also dynamical processes on networks. From the perspective of the box covering, volume dimension, information dimension, and correlation dimension several approaches have been proposed. We modify the commonly used definitions of the box dimension and information dimension to introduce a d-summable approach (a geometric notion that comes from geometric measure theory) of these dimensions. It is applied to calculate d-summable information dimension of several real complex networks. We offer empirical evidence to support the conjecture that d-summable information model worth carrying out than information model for several networks.