Sampling Geometric Inhomogeneous Random Graphs in Linear Time

Real-world networks, like social networks or the internet infrastructure, have structural properties such as large clustering coefficients that can best be described in terms of an underlying geometry. This is why the focus of the literature on theoretical models for real-world networks shifted from classic models without geometry, such as Chung-Lu random graphs, to modern geometry-based models, such as hyperbolic random graphs. With this paper we contribute to the theoretical analysis of these modern, more realistic random graph models. Instead of studying directly hyperbolic random graphs, we use a generalization that we call geometric inhomogeneous random graphs (GIRGs). Since we ignore constant factors in the edge probabilities, GIRGs are technically simpler (specifically, we avoid hyperbolic cosines), while preserving the qualitative behaviour of hyperbolic random graphs, and we suggest to replace hyperbolic random graphs by this new model in future theoretical studies. We prove the following fundamental structural and algorithmic results on GIRGs. (1) As our main contribution we provide a sampling algorithm that generates a random graph from our model in expected linear time, improving the best-known sampling algorithm for hyperbolic random graphs by a substantial factor O(n^0.5). (2) We establish that GIRGs have clustering coefficients in {\Omega}(1), (3) we prove that GIRGs have small separators, i.e., it suffices to delete a sublinear number of edges to break the giant component into two large pieces, and (4) we show how to compress GIRGs using an expected linear number of bits.

💡 Research Summary

The paper introduces Geometric Inhomogeneous Random Graphs (GIRGs), a versatile random graph model that captures key properties of real-world networks—scale‑free degree distributions, high clustering, and small‑world distances—by embedding vertices in a geometric space. Each vertex v receives a weight wᵥ drawn from a power‑law distribution with exponent β>2 and a uniformly random position xᵥ on a d‑dimensional torus Tᵈ. An edge between distinct vertices u and v is created independently with probability

p_uv = Θ( min{ 1, (w_u w_v / W)^α / ‖x_u−x_v‖_∞^{α d} } ),

where W = Σ_wᵥ, α>1 controls how quickly connection probability decays with distance, and the ∞‑norm defines distance on the torus. By ignoring constant factors in the edge probabilities, the model becomes analytically simpler while remaining qualitatively equivalent to hyperbolic random graphs; indeed, hyperbolic graphs appear as a special case of GIRGs.

The authors first establish that GIRGs inherit the structural hallmarks of realistic networks. Using results from a broader class of augmented Chung‑Lu graphs, they show that the degree sequence follows a power law with exponent β, the average degree stays Θ(1), and for 2<β<3 the giant component occupies linear size, has logarithmic diameter, and average distance (2±o(1))·log log n / |log(β−2)|. These properties confirm that GIRGs exhibit the “small‑world” phenomenon.

The central technical contribution is an expected‑linear‑time sampling algorithm. The algorithm partitions the torus into a regular grid of cells whose side length is Θ(n^{−1/d}). Vertices are placed into cells according to their positions. For a given vertex, only vertices in its own cell and a constant‑radius neighbourhood of cells can be potential neighbors because the distance‑dependent factor makes the edge probability negligible for farther cells. By precomputing for each cell a list of candidate cells and using the distance bound to limit the number of Bernoulli trials, the algorithm generates all edges with O(1) work per vertex on average. Consequently, the total expected runtime is O(n), improving upon the naïve O(n²) approach and the best known hyperbolic‑graph sampler, which runs in O(n^{3/2}) time, by a factor of Θ(√n).

Next, the paper proves that GIRGs have a constant clustering coefficient. The geometric construction ensures that two neighbors of a vertex are likely to lie in the same or adjacent cells, making the distance between them small and the edge probability between them bounded away from zero. A careful probabilistic analysis shows that the expected fraction of neighbor pairs that are connected is Θ(1), in stark contrast to Chung‑Lu graphs where clustering decays as n^{−Ω(1)}.

The authors also demonstrate the existence of small separators. By analyzing the volume growth of balls in the torus and the distance‑dependent edge probabilities, they show that removing only O(n^{1−Ω(1)}) edges suffices to split the giant component into two large pieces. This separator size matches empirical observations on real networks and implies low tree‑width.

Leveraging the small separators, the paper presents a compression scheme that stores a GIRG using O(n) bits in expectation. The graph is recursively partitioned along the separators, and each subgraph is encoded by its adjacency to the separator and the positions/weights of its vertices. Because the separator size is sublinear, the total encoding cost remains linear, offering an efficient representation comparable to, but simpler than, previous hyperbolic‑graph compression methods.

Finally, the authors verify that hyperbolic random graphs are a special case of GIRGs (by choosing appropriate α, d, and a specific distance function). Therefore, all structural and algorithmic results—linear‑time sampling, constant clustering, small separators, and linear‑size compression—apply directly to hyperbolic graphs as well.

In summary, the paper provides a comprehensive theoretical framework for GIRGs, delivering both deep structural insights and practical algorithms. The linear‑time sampler makes it feasible to generate massive synthetic networks for experiments, while the clustering, separator, and compression results deepen our understanding of why geometry‑based models faithfully reproduce real‑world network phenomena.