Delayed threshold and spatial diffusion in k-core percolation induced by long-range connectivity

Abstract Most real-world complex networks are spatially embedded, with nodes positioned in physical space. In such systems, distance-based connectivity shapes not only the information transmission efficiency but also the network robustness and vulnerability behavior. Here, we systematically examined how spatial distance influences network robustness using k -core percolation across a pair of models of long-range connectivity. For structural connected components, we found that long-range connectivity can trigger explosive phase transitions, but with a delayed threshold. For spatial neighborhoods, two core phenomena emerge: spatial diffusion and clustering. Both synthetic model networks and the empirical Drosophila neuronal network, that follow geometric scaling laws with small exponents, exhibit gradual spatial spread. In contrast, strong geometric constraints lead to spatial clustering. Despite varying connectivity models, spatial neighborhoods consistently demonstrate self-organized criticality throughout the percolation process. Our findings reveal how geometric constraints influence structural and spatial robustness, which may in turn support functional robustness.