Dynamical processes on graphs with hyperuniform spatial embeddings

The concept of hyperuniformity provides a general framework to describe systems with suppressed density fluctuations at large length scales; it includes lattices but also disordered systems. Current studies largely focus on hyperuniformity in Euclidean space. For applications outside of condensed matter systems, it is of great interest to generalize the concept of hyperuniformity to graphs and networks. In this study, we construct graphs from non-hyperuniform and hyperuniform point patterns, including one-component plasma and stealthy systems, and investigate their transport properties. We examine both Fickian and non-Fickian diffusion on Delaunay triangulated graphs and show that the graph diffusion process directly depends on the correlation of vertices of the graphs. In particular, graphs with hyperuniform spatial embedding exhibit a highly uniform diffusion pattern, with their diffusion distance approximately invariant under permutation. We demonstrate that dynamical processes capture crucial structural properties of the networks, which provides key insights into the generalization of hyperuniformity on graphs.