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

TL;DR

The paper investigates how hyperuniform point patterns influence the transport and robustness of spatial networks by constructing HuPPI and PoPPI networks via four tessellations (Delaunay, Gabriel, Voronoi, Delaunay-centroidal) and analyzing ORC, TER, mixing times, and robustness. It finds that HuPPI networks typically exhibit smaller TER and faster mixing than PoPPI networks, and improved robustness, but these advantages arise from local edge-geometry biases rather than large-scale hyperuniformity, with network-generation rules playing a major role. The results show HuPPI networks have narrower ORC distributions and fewer extreme bottleneck edges, explaining enhanced transport despite more negative average curvatures, and highlight the decoupling of hyperuniformity from network structure at large scales. These insights inform design principles for spatially embedded transport/metamaterial networks by emphasizing local geometry and tessellation choices alongside underlying point-pattern design.

Abstract

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.

Figures (9)

• Figure 1: Visualizations of the types of networks that we construct from hyperuniform point patterns. In the top row, we show examples of Gabriel, Delaunay, Delaunay-centroidal, and Voronoi networks that we generate from a single point pattern with disorder strength $a = 1$. In the bottom row, we show Delaunay-centroidal networks that we generate using disorder strengths $a \in \{0.1, 0.5, 1.0, 2.0\}$, illustrating the transition from a near-lattice structure to a disordered structure. We generate and analyze networks with periodic boundary conditions, but we truncate edges at the boundaries in this figure for visualization purposes.
• Figure 2: Normalized total effective resistance (TER) $\mathcal{R}_{\mathrm{norm}}$ as a function of disorder strength $a$ for HuPPI networks (solid curves) and PoPPI networks (dashed curves) for Gabriel, Delaunay, Delaunay-centroidal, and Voronoi networks. We generate these networks from $15 \times 15$ lattices (which have 225 points). Smaller normalized TERs indicate the presence of more redundant paths and hence better connectivity.
• Figure 3: Normalized TER $\mathcal{R}_{\mathrm{norm}}$ versus the network size (i.e., number of nodes) $n$ for HuPPI networks (solid curves) and PoPPI networks (dashed curves) for Gabriel, Delaunay, Delaunay-centroidal, and Voronoi networks. The HuPPI networks, which we generate from hyperuniform point patterns, have a disorder strength of $a = 1$. As we increase $n$, it becomes easier to distinguish between the curves for the hyperuniform and random point patterns.
• Figure 4: Random-walk mixing time as a function of the disorder strength $a$ for HuPPI networks (solid curves) and PoPPI networks (dashed curves) for Gabriel, Delaunay, Delaunay-centroidal, and Voronoi networks. We generate networks from $15 \times 15$ lattices (which have 225 points). By definition, the PoPPI networks do not depend on $a$.
• Figure 5: Histograms of ORCs for HuPPI networks (blue) with disorder strength $a = 1$ and PoPPI networks (orange) for Gabriel, Delaunay, Delaunay-centroidal, and Voronoi networks. We generate these networks from $15 \times 15$ square lattices (which have 225 points). The histograms aggregate data from five independent realizations of each type of point pattern.