Hyperbolic Random Graphs

• Hyperbolic random graphs are network models set in negatively curved spaces that generate scale-free graphs with power-law degree distributions and high clustering.
• They employ both hard threshold and soft Fermi–Dirac edge rules using polar coordinates and hyperbolic distance, capturing real-world connectivity and clustering nuances.
• Advanced sampling and geometric decomposition methods enable efficient algorithms for tasks like maximum clique detection and dynamic updates on large-scale networks.

Hyperbolic random graphs (HRGs), also called random hyperbolic graphs or RHGs, are random geometric graph models in negatively curved latent spaces used to model large complex networks with heterogeneous degree distributions and strong clustering. In the standard two-dimensional formulation, one samples $n$ vertices independently in a hyperbolic disk of radius $R=2\ln n + C$, draws angular coordinates uniformly, draws radial coordinates from a density proportional to $\sinh(\alpha r)$, and then connects pairs either by a hard distance threshold or by a Fermi–Dirac-type probability depending on hyperbolic distance (Gugelmann et al., 2012). In the sparse scale-free regime, the model yields a power-law degree distribution with exponent $2\alpha+1$, nonvanishing clustering, and polylogarithmic diameter bounds, while also admitting efficient generation procedures and, for some problems, unexpectedly efficient exact algorithms (Gugelmann et al., 2012).

1. Canonical definition and geometric formulation

The standard threshold HRG is defined in the hyperbolic plane $\mathbb H^2$ of curvature $-1$, usually in native polar coordinates. A point $v$ is represented as $(r_v,\phi_v)$, where $r_v$ is hyperbolic distance from the origin and $\phi_v\in[0,2\pi)$ is angular position. Hyperbolic distance between $R=2\ln n + C$0 and $R=2\ln n + C$1 satisfies

$R=2\ln n + C$2

with angular separation defined modulo $R=2\ln n + C$3 (Oh et al., 2023).

For the classical sparse model one fixes $R=2\ln n + C$4, $R=2\ln n + C$5, and

$R=2\ln n + C$6

Then one samples $R=2\ln n + C$7 independent points in the disk $R=2\ln n + C$8 with angular coordinate uniform in $R=2\ln n + C$9 and radial density

$\sinh(\alpha r)$0

In the threshold version, two vertices are adjacent if and only if $\sinh(\alpha r)$1 (Oh et al., 2023).

A widely used soft variant replaces the threshold rule by an independent edge probability

$\sinh(\alpha r)$2

where $\sinh(\alpha r)$3 is a temperature parameter. In the limit $\sinh(\alpha r)$4, this reduces to the threshold model $\sinh(\alpha r)$5 if $\sinh(\alpha r)$6, and $\sinh(\alpha r)$7 otherwise (Bläsius et al., 2019). The temperature controls clustering strength: smaller $\sinh(\alpha r)$8 sharpens the geometric cutoff, whereas larger $\sinh(\alpha r)$9 smooths it (Aldecoa et al., 2015).

This formulation combines a popularity coordinate, encoded by radius, with a similarity coordinate, encoded by angle. The model therefore functions both as a probabilistic graph ensemble and as a latent-space representation of network structure.

2. Degree laws, clustering, and parameter regimes

In the scale-free regime $2\alpha+1$0, the degree distribution follows a power law with exponent

$2\alpha+1$1

and this relation is rigorously established for the threshold model (Gugelmann et al., 2012). Gugelmann, Panagiotou, and Peter proved exact asymptotic expressions for the number $2\alpha+1$2 of vertices of degree $2\alpha+1$3, concentration of $2\alpha+1$4 around its expectation, a maximum degree of order $2\alpha+1$5, and a global clustering coefficient bounded away from zero (Gugelmann et al., 2012). They also showed that the average degree remains bounded in $2\alpha+1$6 under the scaling $2\alpha+1$7, with $2\alpha+1$8 tuning the average degree through a factor $2\alpha+1$9 (Gugelmann et al., 2012).

For soft HRGs, the parameter $\mathbb H^2$0 controls a regime change. In the inverse-temperature notation of Fountoulakis, a phase transition occurs around $\mathbb H^2$1: for $\mathbb H^2$2, the degree of a typical vertex is bounded in probability and has a power-law tail whose exponent depends only on the curvature; for $\mathbb H^2$3, typical degrees grow logarithmically; and for $\mathbb H^2$4, expected degree grows polynomially in $\mathbb H^2$5 (Fountoulakis, 2012). The same work shows that, conditional on radial types, the cold regime $\mathbb H^2$6 is asymptotically equivalent up to a constant factor to a Chung–Lu kernel $\mathbb H^2$7, with weights $\mathbb H^2$8 (Fountoulakis, 2012).

Recent results refine the structure of large degrees. In the scale-free regime $\mathbb H^2$9, the ranking of vertices by decreasing degree coincides with the ranking by increasing distance to the center up to rank $-1$0 with high probability (Gassmann, 2024). The same work proves convergence of the normalized top-degree process to a Poisson point process and identifies a phase transition at $-1$1: for $-1$2, the maximum degree is of order $-1$3, whereas for $-1$4, it is of order $-1$5 (Gassmann, 2024).

These results make the geometric meaning of hub formation unusually explicit. Small radius is not merely correlated with large degree; over a substantial initial rank range, it determines the high-degree ordering itself.

3. Components, distances, and spectral structure

For $-1$6, the standard threshold HRG has with high probability a connected small-diameter highly clustered structure in the sparse regime, and in the range $-1$7 the graph contains a unique giant component of linear size (Oh et al., 2023, Kiwi et al., 2016). The ball $-1$8 forms a central clique with expected size $-1$9, and this core plays a central role in proofs of distance and connectivity bounds (Kiwi et al., 2014).

Kiwi and Mitsche proved that when $v$0, any two vertices in the same component are at graph distance

$v$1

and obtained as a corollary that the second largest component has size $v$2 (Kiwi et al., 2014). They also showed that isolated components may form a path of length $v$3, providing a lower bound on the size of the second largest component (Kiwi et al., 2014).

Friedrich and Krohmer later gave simpler proofs of an improved upper bound for the diameter of the giant component,

$v$4

together with a lower bound $v$5 (Friedrich et al., 2015). Their argument separates the disk into an inner band, whose vertices reach a central clique in $v$6 hops, and an outer band, whose vertices find a path into the inner band in polylogarithmic length (Friedrich et al., 2015).

The spectral picture is correspondingly anisotropic. For the normalized Laplacian of the giant component $v$7, the second eigenvalue satisfies

$v$8

and also

$v$9

where $(r_v,\phi_v)$0 is the diameter of $(r_v,\phi_v)$1 (Kiwi et al., 2016). The conductance upper bound obtained via Cheeger’s inequality is essentially tight, and the bottleneck is attained by a linear-size subset, while subsets of volume $(r_v,\phi_v)$2 have much larger conductance (Kiwi et al., 2016). This implies that HRGs combine short metric distances with weak global expansion.

4. Sampling, generation, and dynamic updates

A direct generator samples coordinates and then tests all $(r_v,\phi_v)$3 pairs, yielding $(r_v,\phi_v)$4 distance-and-Bernoulli steps (Aldecoa et al., 2015). Aldecoa and Krioukov’s “Hyperbolic Graph Generator” implements this approach in C++, uses GSL for Monte Carlo integration and special functions, and supports several limiting regimes, including the hard-threshold hyperbolic random geometric graph, the soft configuration model, spherical random geometric graphs, and Erdős–Rényi graphs (Aldecoa et al., 2015).

Efficient large-scale generation relies on geometric decomposition. Bringmann, Keusch, and Lengler gave an expected $(r_v,\phi_v)$5-time generator for geometric inhomogeneous random graphs and adapted it to HRGs by flattening the angular coordinate to the torus, assigning weights

$(r_v,\phi_v)$6

and reusing GIRG machinery based on weight buckets, hierarchical cells, Morton codes, and geometric-jump sampling (Bläsius et al., 2019). Their implementation supports non-zero temperatures, higher-dimensional underlying geometries on the GIRG side, and produces graphs with ten million edges in under a second on commodity hardware; they also emphasize that the generators draw from the correct probability distribution and involve no approximation (Bläsius et al., 2019).

The same paper clarifies the relation to GIRGs. HRGs embed into GIRGs via $(r_v,\phi_v)$7 and $(r_v,\phi_v)$8, and the edge probability takes the GIRG-binomial form up to constant factors (Bläsius et al., 2019). At the same time, a straight-forward inclusion does not hold in practice, although the difference is negligible for most use cases (Bläsius et al., 2019). This qualification is important because it limits any literal identification of the two models.

Dynamic HRGs can also be updated sublinearly. Von Looz, Meyerhenke, and Prutkin introduced probabilistic neighborhood queries and a polar quadtree data structure that supports node movement while preserving the stationary distribution of point positions (Looz et al., 2018). For suitable planar point distributions, the query time is

$(r_v,\phi_v)$9

with high probability, yielding about one order of magnitude practical speedup over the fastest previous dynamic approach (Looz et al., 2018).

5. Algorithmic consequences on HRGs

One reason HRGs attract algorithmic interest is that latent geometry can make hard graph problems markedly easier on this distribution. Oh and Oh studied the Maximum Clique problem on hyperbolic random graphs and proposed a two-phase exact algorithm: a greedy degree-ordered clique initialization followed by peeling of low-degree vertices, leaving a kernel of size

$r_v$0

(Oh et al., 2023). With a geometric embedding, they then apply the BFK18 procedure to the kernel and obtain expected running time

$r_v$1

Without the embedding, they use a Co-Bipartite Neighborhood Edge Elimination Ordering (CNEEO) and obtain

$r_v$2

expected time (Oh et al., 2023).

The empirical results are equally explicit. On synthetic HRGs, total running time at $r_v$3 is under $r_v$4 s, described as a $r_v$5 speedup over BFK18; on several real-world collaboration and web graphs, the robust algorithm either finds the exact clique or a near-optimum lower bound within seconds, and on the 4M-vertex LiveJournal network the entire computation completes in minutes (Oh et al., 2023). The same paper situates this within a broader pattern: preprocessing-plus-kernelization also yields efficient algorithms for shortest paths, matchings, and separators in hyperbolic-network settings (Oh et al., 2023).

Subsequent work has sharpened the clique-theoretic picture. For threshold HRGs, degeneracy-based greedy colouring admits an approximation ratio ranging from $r_v$6 to $r_v$7, depending on the power-law exponent (Baguley et al., 2024). The same study shows that degeneracy and clique number are substantially different, derives an improved upper bound on the clique number, and proves that the core of HRGs does not constitute the largest clique (Baguley et al., 2024). A plausible implication is that core-based intuition, while often effective algorithmically, does not by itself characterize extremal clique structure.

6. Generalizations, discrete analogues, and mesoscale extensions

Several extensions modify either the ambient geometry or the constraints imposed on the graph ensemble. One line of work replaces continuous hyperbolic space by a regular tessellation. In the discrete hyperbolic random graph model (DHRG), nodes are mapped to vertices of a triangulation, edge probabilities depend on graph distance in the tessellation, and the main computations become combinatorial rather than floating-point geometric (Celińska-Kopczyńska et al., 2021). DHRG preserves the power-law exponent $r_v$8, retains high clustering at low temperature, and supports log-likelihood computation and local search in $r_v$9 time (Celińska-Kopczyńska et al., 2021). Related work on hyperbolic triangulations reports that discrete embeddings can match or slightly improve continuous maximum-likelihood embeddings on real networks at comparable wall-clock time (Kopczyński et al., 2017).

A second line generalizes the model to dimension $\phi_v\in[0,2\pi)$0. Budel et al. give a rescaling that leaves the degree distribution invariant across dimensions, with degree exponent $\phi_v\in[0,2\pi)$1 in the cold regime and dimension-dependent clustering that decreases to $\phi_v\in[0,2\pi)$2 as $\phi_v\in[0,2\pi)$3 (Budel et al., 2020). They also analyze limiting regimes connecting RHGs to spherical random geometric graphs, soft configuration models, and Erdős–Rényi graphs (Budel et al., 2020).

A third line addresses a structural limitation of purely geometric HRGs: mesoscale mixing patterns are constrained by the geometry. The Random Hyperbolic Block Model (RHBM) extends the $\phi_v\in[0,2\pi)$4 framework by adding a symmetric block-mixing matrix $\phi_v\in[0,2\pi)$5 and latent block-specific forces $\phi_v\in[0,2\pi)$6 within a maximum-entropy construction (Guarino et al., 3 Jun 2025). The resulting model preserves heavy-tailed degrees, high clustering, and small-world structure while enforcing arbitrary expected block-to-block edge counts (Guarino et al., 3 Jun 2025). This directly addresses the fact that purely geometric RHGs cannot reproduce certain non-geometric community patterns because angular similarity obeys triangle-inequality constraints (Guarino et al., 3 Jun 2025).

Taken together, these variants indicate both the reach and the limitations of the classical HRG. The original model captures sparsity, heavy tails, clustering, and navigable geometry with unusual analytical tractability; however, higher-dimensional, discrete, and block-structured extensions show that some empirically important mesoscale phenomena require additional structure beyond pure hyperbolic proximity.