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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 global resistances than PoPPI networks, indicating that edges with moderately negative curvatures need not create severe bottlenecks to transport. 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. These results collectively demonstrate potential advantages of hyperuniformity in network design and motivate further theoretical and experimental exploration of HuPPI networks. 

Formulating order metrics that sensitively quantify the degree of order/disorder in many-particle systems in $d$-dimensional Euclidean space ${\mathbb{R}}^d$across length scales is an outstanding challenge in physics, chemistry, and materials science. Since an infinite set of $n$-particle correlation functions is required to fully characterize a system, one must settle for a reduced set of structural information, in practice. We initiate a program to use the local number variance ${\sigma }_N^2\left(R\right)$associated with a spherical sampling window of radius $R$(which encodes pair correlations) and an integral measure derived from it ${\mathrm{\Sigma }}_N(R_i,R_j)$that depends on two specified radial distances $R_i$and $R_j$. Across the first three space dimensions ( $d=1,2,3$), we find these metrics can sensitively describe and categorize the degree of order/disorder of 41 different models of antihyperuniform, nonhyperuniform, disordered hyperuniform, and ordered hyperuniform many-particle systems at a specified length scale $R$. Using our local variance metrics, we demonstrate the importance of assessing order/disorder with respect to a specific value of $R$. These local order metrics could also aid in the inverse design of structures with prescribed length-scale-specific degrees of order/disorder that yield desired physical properties. In future work, it would be fruitful to explore the use of higher-order moments of the number of points within a spherical window of radius $R$[S. Torquato , ] to devise even more sensitive order metrics. Published by the American Physical Society2024