Economical representation of spatial networks

Network representation is crucial across various scientific, societal, technological, and artistic domains. The primary goal is to highlight patterns out of nodes interconnected by edges that are easy to understand, facilitate communication, and support decision-making. This is typically achieved by rearranging the nodes to minimize the edge crossings responsible of unintelligible and often unesthetic trends. But when the nodes cannot be moved, as in spatial and physical networks, this procedure is not viable. Here, we overcome this impasse by turning the edge crossing problem into a graph filtering optimization. By introducing the concept of progressive cost, we demonstrate that longer connections prompt the optimal solution to yield sparser networks, thereby limiting the number of intersections and getting more readable layouts. This theoretical result matches human behavior and provides an ecologically inspired criterion to visualize and model real-world interconnected systems.