Random Geometric Graphs

• Random Geometric Graphs are spatial network models where nodes are placed in a metric space and connected if their distance is below a threshold, providing a clear framework for studying spatial effects.
• They exhibit distinctive properties such as high clustering, positive assortativity, and unique spectral behavior driven by geometric constraints, which inform theoretical analysis and modeling.
• Applications extend to wireless communication, biological networks, and machine learning, making RGGs a versatile tool for analyzing and optimizing spatial interactions.

A random geometric graph (RGG) is a fundamentally spatial random graph model in which nodes are embedded in a metric space and edges are determined, probabilistically or deterministically, by inter-node distances. RGGs were initially formalized to model wireless communication networks, but now play a central role in areas such as biological networks, machine learning, statistical physics, combinatorial optimization, and theoretical network science. Their spatial constraints result in distinctive properties, including clustering, dependence structures, and phase transitions, that are absent in purely random (Erdős–Rényi) graphs.

1. Definition, Construction, and Generalizations

Classic RGGs are constructed by placing $N$ nodes independently and uniformly in a domain %%%%1%%%% (commonly the unit cube or torus), connecting existing pairs $(i, j)$ if and only if their metric distance $d(x_i, x_j)$ is less than or equal to a threshold $r$. This deterministic, hard-threshold rule is known as the Gilbert disk model. Extensions include soft RGGs, where the connection is random with probability $H(\|x_i-x_j\|)$ for a prescribed connection function $H$ that encodes spatial attenuation or channel fading in wireless systems (Dettmann et al., 2014). More generally, both the node distribution and the connection function can be non-uniform or data-driven, and RGGs have been adapted for arbitrary metric spaces, including manifolds (Huang et al., 14 Feb 2024), circles (Angel et al., 2019), and unbounded ambient spaces.

2. Topological and Degree Correlation Properties

The spatial embedding of RGGs leads to high clustering and non-trivial degree correlations. In two-dimensional RGGs, nodes of high degree tend to be connected to other high-degree nodes—a haLLMark of positive assortativity. The average degree of nearest neighbors for a node of degree $k$ is given by

$k_{nn}(k) = A_2 k + (1 - A_2)\bar{k},$

with $A_2 = 1 - \frac{3\sqrt{3}}{4\pi} \approx 0.5865$ as the asymptotic clustering coefficient (Antonioni et al., 2012). This relation generalizes to $d$ dimensions with $A_d$ as the ratio of average neighborhood overlap to neighborhood volume. The assortativity coefficient converges to $A_d$, reflecting a direct geometric control of higher-order network structure.

Moreover, topological indices such as the Randić index and harmonic index are nearly perfect predictors of global spectral and information-theoretic properties of RGGs, owing to their tight correlation to the number of non-isolated vertices (Aguilar-Sanchez et al., 2020).

3. Spectral Properties and Mesoscopic Structure

The Laplacian and adjacency spectra of RGGs reveal features fundamentally distinct from non-spatial graphs. The Laplacian spectrum of RGGs possesses both continuous and discrete components: notably, Dirac delta peaks at integer-valued eigenvalues reflecting the abundance of mesoscopic symmetric motifs (e.g., cliques and orbits) (Nyberg et al., 2014). These motifs, arising from the spatial arrangement, lead to localization of eigenvectors and can constitute a finite fraction of the spectrum even in the large size limit.

The regularized Laplacian spectrum in the thermodynamic regime displays a power-law tail near the smallest nonzero eigenvalue, implying that the spectral dimension $d_s$ coincides with the underlying spatial dimension $d$ (Avrachenkov et al., 2019). For adjacency matrices, the limiting eigenvalue distribution (LED) of RGGs in the connectivity regime converges to that of a corresponding deterministic geometric graph (DGG), effectively enabling analytical computation via multidimensional Fourier techniques (Hamidouche et al., 2019).

In the sparse regime, the first few nontrivial "edge" eigenvalues of the scaled Laplacian converge, with high probability, to those of a weighted differential operator (e.g., the Laplace–Beltrami operator); this convergence is robust for RGGs generated by samples with unbounded support and non-uniform densities (Ding et al., 9 Sep 2025).

4. High-Dimensional Phenomena: CLT, Entropy, and Limiting Behavior

In high dimensions ($d \gg 1$), the geometry of RGGs simplifies due to the concentration of measure. Pairwise distances become approximately independent and Gaussian-distributed via the multivariate central limit theorem (CLT) (Erba et al., 2020, Baker et al., 14 Mar 2025). For RGGs on the $d$-torus with uniform sampling, the ensemble distribution converges to the classical Erdős–Rényi (ER) ensemble, with factorization of edge probabilities and maximal Shannon entropy at connection probability $1/2$ (Baker et al., 14 Mar 2025).

In contrast, for RGGs embedded in the cube with non-negligible kurtosis in the coordinate distributions, unavoidable positive correlations between adjacencies of edges sharing a vertex result in systematically lower entropy than the ER maximum, even in the $d\to\infty$ limit. The scaling of the entropy correction is $O(d^{-1/2})$, as established by a third-order Edgeworth expansion around the CLT (Baker et al., 14 Mar 2025). For soft RGGs (with continuous connection functions), these dependencies are erased in the high-$d$ limit and convergence to ER is restored (Erba et al., 2020).

5. Continuum, Boundary, and Manifold Effects

Precise connectivity and percolation properties in finite domains are determined by both bulk and boundary effects. In general convex domains, the probability of full connectivity is expressible as a sum of contributions from the interior (bulk), edges/faces, corners/vertices, with prefactors determined by moments of the connection function (Dettmann et al., 2014). Boundary effects dominate disconnection probability in high-density regimes, rendering design optimization (e.g., smoothing out corners) particularly impactful.

For RGGs sampled from manifolds, even in the absence of explicit distance information, the intrinsic and extrinsic geometry can be reconstructed with high accuracy from the observed graph using local statistics—essentially, from counts of common neighbors and clustering patterns (Huang et al., 14 Feb 2024). This geometrical inference is algorithmically tractable (polynomial time) and directly relevant to unsupervised manifold learning methods such as Isomap and Laplacian eigenmaps.

In non-Euclidean settings, such as the circle, RGGs exhibit strong rigidity: the isomorphism class of the graph can encode geometric invariants like the circumference, recovering even the vertex location up to isometry for irrational lengths (Angel et al., 2019). This stands in sharp contrast to the Rado property of universal connectivity in certain other metric spaces.

6. Statistical Physics, Percolation, and Optimization

RGGs provide a well-suited spatial backbone for models in statistical mechanics and combinatorial optimization. First-passage percolation (FPP) models on RGGs exhibit limiting Euclidean ball-shaped growth, with quantitative control given by moderate deviations and shape theorems (Lima, 21 Aug 2024, Lima et al., 4 Nov 2024). These results offer lower bounds on percolation thresholds, linking critical connectivity directly to geometric and intensity parameters.

Minimum spanning tree (MST) weights in RGGs, crucial for infrastructure optimization, can be analyzed with location-dependent weights, yielding sharp deviation and variance bounds and $L^2$-convergence to deterministic scaling laws (Ganesan, 2021). These results are robust to various node density functions and extend naturally to higher dimensions.

In ecological models, RGGs model the spatial competition and coexistence of species: interacting particle systems coupling growth (first-passage/percolation) and invasion (voter dynamics) demonstrate strictly positive probability of coexistence in the supercritical (giant component) phase (Coletti et al., 18 Jan 2025).

7. Applications: Machine Learning, Network Science, Planning, and Wireless Systems

RGGs are central null models in high-dimensional machine learning, dimensionality reduction, and manifold learning, especially for benchmarking graph-based methods relying on nearest neighbor structures (Erba et al., 2020, Duchemin et al., 2022). They enter as proxies in sampling-based robotic motion planning, where properties such as probabilistic completeness and asymptotic optimality can be directly inferred from corresponding RGG properties using localization–tessellation frameworks (Solovey et al., 2016).

The transferability of graph neural network (GNN) policies in large-scale sparse networks (such as wireless communication systems) can be theoretically understood by relating RGGs to deterministic geometric graphs; performance bounds for transferring learned GNN policies from small to larger RGGs (with the same geometric properties) are quantitatively established via the closeness of the graph shift operator spectra (Camargo et al., 1 Oct 2025).

Soft RGGs (with probabilistic connection functions) are essential in modeling realistic communication networks, capturing wireless fading, percolation thresholds, and boundary effects not addressed by hard-threshold models (Dettmann et al., 2014).

Summary Table: Key Properties of RGG Ensembles in Different Regimes

Regime/Geometry	Limiting Distribution	Entropy Scaling	Correlations	Spectral Properties
Torus, hard RGG, $d\to\infty$	Erdős–Rényi ($G(n,p)$)	$O(1)$ at $p=1/2$	Vanishing in limit	Integer peak, continuous bulk (Nyberg et al., 2014)
Cube, hard RGG, $d\to\infty$	Correlated edges (not ER)	$H < H_{\mathrm{ER}}$	Positive for shared nodes	Deviations from ER in adjacency LED
Soft RGG ($\forall$ geom.)	Erdős–Rényi ($G(n,p)$)	$O(1)$	Negligible in limit	As in ER
Low dimensions ($d=1,2,3$)	Non-ER, geometry-dependent	Strong finite-size effects	Strong local dependencies	Mesoscopic structures prominent

• Degree correlations, clustering: (Antonioni et al., 2012)
• Connection functions and boundary phenomena: (Dettmann et al., 2014)
• Spectral/mesoscopic structure: (Nyberg et al., 2014, Avrachenkov et al., 2019, Hamidouche et al., 2019, Ding et al., 9 Sep 2025)
• High-dimensional limits and entropy: (Erba et al., 2020, Baker et al., 14 Mar 2025)
• Manifold geometry reconstruction: (Huang et al., 14 Feb 2024)
• Topological–spectral correlations: (Aguilar-Sanchez et al., 2020)
• Percolation, FPP, shape theorems: (Lima, 21 Aug 2024, Lima et al., 4 Nov 2024)
• Minimum spanning trees: (Ganesan, 2021)
• Competition models/ecology: (Coletti et al., 18 Jan 2025)
• Machine learning/planning: (Solovey et al., 2016, Duchemin et al., 2022)
• GNN transferability: (Camargo et al., 1 Oct 2025)
• Circle and non-Euclidean metric spaces: (Angel et al., 2019)
• Path enumeration and 1D combinatorics: (Kartun-Giles et al., 2021)
• Distribution and entropy of small RGGs: (Badiu et al., 2018)

This synthesis reflects the rigorously characterized geometry-driven structure of RGGs and their pervasive impact across mathematical, algorithmic, and applied network domains.