Self-consistent gravity model for inferring node mass in flow networks

The gravity model, inspired by Newton's law of universal gravitation, has long served as a primary tool for interpreting trade flows between countries, using a country's economic `mass' as a key determinant. Despite its wide application, the definition of `mass' within this model remains ambiguous. It is often approximated using indicators like GDP, which may not accurately reflect a country's true trade potential. Here, we introduce a data-driven, self-consistent numerical approach that redefines `mass' from a static proxy to a dynamic attribute inferred directly from flow data. We infer mass distribution and interaction nature through our method, mirroring Newton's approach to understanding gravity. Our methodology accurately identifies predefined embeddings and reconstructs system attributes when applied to synthetic flow data, demonstrating its strong predictive power and adaptability. Further application to real-world trade networks yields critical insights, revealing the spatial spectrum of trade flows and the economic mass of countries, two key features unexplored in depth by existing models. Our methodology not only enables accurate reconstruction of the original flow but also allows for a deep understanding of the unique capabilities of each node within the network. This study marks a significant shift in the understanding and application of the gravity model, providing a more comprehensive tool for analyzing complex systems and uncovering new insights into various fields, including global trade, traffic engineering, epidemic disease prevention, and infrastructure design.