Emergence of loops in spatial networks with self-organized growth

Spatial networks play a central role in the analysis of real-world systems' structures. However, the modelling of spatial networks remains a challenge, primarily due to the elusive mechanisms governing the emergence of loops. Since loops are intimately related to network functionality, it is imperative to gain an understanding of their formation. Early models, often based on top-down planning, tend to overlook the evolving nature of spatial networks, resulting in exceedingly high computational complexity. To address these limitations, self-organized models have been introduced, though they typically yield tree structures or are designed for specific types of networks. In this paper, to facilitate the classification and reconstruction of spatial networks, we propose a class of economical-efficient models. They consider cost and efficiency as the primary driving forces behind network evolution. Different network patterns emerge as a consequence of the interplay between node heterogeneity and spatial constraints. Our model is grounded in the concept of self-organization and demonstrates the ability to replicate many macroscopic properties observed in real-world systems. The model can also serve as a tool for network reconstruction as we further show how to fit the parameter and apply it to domestic airline networks.