The diverse and fair structures of growth with scale-free saturations

Real-world growth processes and scalings have been broadly categorized into three growth regimes with distinctly different properties and driving forces. The first two are characterized by a positive and constant feedback between growth and growth rates which in the context of networks lead to scale-free or single-scale networks. The third, sublinear, regime is characteristic of biological scaling processes and those that that are driven by optimization and efficiency. These systems are characterized by a negative feedback in growth rates and as such naturally exhibit saturations, i.e., areas where growth ceases from a lack of resources. Motivated by this observation, we propose and analyze a simple network growth process that is analogous to this sublinear regime and characterize how its scale-free saturations impact the diversity and fairness of its structural properties and give rise to scaling relations observed throughout complex systems and science.