Local Geometric and Transport Properties of Networks that are Generated from Hyperuniform Point Patterns

Hyperuniformity, which is a type of long-range order that is characterized by the suppression of long-range density fluctuations in comparison to the fluctuations in standard disordered systems, has emerged as a powerful concept to aid in the understanding of diverse natural and engineered phenomena. In the present paper, we harness hyperuniform point patterns to generate a class of disordered, spatially embedded networks that are distinct from both perfectly ordered lattices and uniformly random geometric graphs. We refer to these networks as \emph{hyperuniform-point-pattern-induced (HuPPI) networks}, and we compare them to their counterpart \emph{Poisson-point-pattern-induced (PoPPI) networks}. By computing the local geometric and transport properties of HuPPI networks, we demonstrate how hyperuniformity imparts advantages in both transport efficiency and robustness. Specifically, we show that HuPPI networks have systematically smaller total effective resistances, slightly faster random-walk mixing times, and fewer extreme-curvature edges than PoPPI networks. Counterintuitively, we also find that HuPPI networks simultaneously have more negative mean Ollivier--Ricci curvatures and smaller total effective resistances than PoPPI networks, indicating that edges with moderately negative curvatures need not create severe bottlenecks to transport. Moreover, HuPPI networks are consistently more robust under both random edge removals and curvature-based targeted edge removals, maintaining larger connected components for larger fractions of removed edges than their PoPPI counterparts. We also demonstrate that the network-generation method strongly influences these properties and in particular that it often overshadows differences that arise from underlying point patterns.