Deterministic Hyperbolic Graphs

• Deterministic hyperbolic graphs are explicitly constructed models that exhibit Gromov hyperbolicity using non-random, algorithmic procedures.
• They leverage combinatorial, algebraic, and dynamical systems frameworks to ensure properties like small-world behavior and hierarchical clustering.
• Their deterministic structure facilitates efficient computation of graph metrics and benchmarking in network science as well as fractal geometry.

A deterministic hyperbolic graph is a graph constructed through explicit, non-random procedures—often leveraging combinatorial, algebraic, or dynamical systems frameworks—so that its large-scale geometry is Gromov hyperbolic and the construction admits full a priori description. In the context of network science, deterministic hyperbolic graphs serve both as mathematical models for real-world network organization (exhibiting properties such as small-world phenomena and hierarchical clustering) and as canonical representatives within statistical ensembles, particularly as “most typical states” of grand canonical models. These graphs arise in various domains, including geometric group theory, fractal geometry, metric graph theory, and algorithmic network analysis. The theory provides tools for quantifying typicality, developing efficient algorithms, and connecting discrete structures to underlying continuous models.

1. Foundations of Deterministic Hyperbolic Graphs

A graph is Gromov hyperbolic (or $\delta$-hyperbolic) if, for a fixed $\delta \geq 0$, the metric space satisfies the $\delta$-slim triangle condition: for any four vertices $u, v, w, x$, the two largest of the three sums $d(u,v)+d(w,x)$, $d(u,w)+d(v,x)$, and $d(u,x)+d(v,w)$ differ by at most $2\delta$. This property captures large-scale tree-like geometry and underpins metric hyperbolicity.

Deterministic hyperbolic graphs differ from random hyperbolic graphs (where vertices are usually distributed according to probabilistic rules in hyperbolic space). Instead, their construction involves explicit deterministic algorithms that systematically generate hyperbolic structure. An important instance is the “deterministic construction of a state that converges to the most typical state in the thermodynamic limit,” implemented through rounds of derandomization that include combinatorial and point process derandomization (Sabhahit et al., 1 Dec 2025).

Several established deterministic frameworks yield hyperbolic graphs:

• Helly graphs and their injective hulls, central in metric graph theory (Dragan et al., 2021).
• Augmented trees associated with symbolic dynamics of contractive Iterated Function Systems (IFS), inducing canonical hyperbolic graphs (Lau et al., 2016, Kong et al., 2020).
• Generalizations to “expansive hyperbolic graphs,” which abstract the core combinatorics of IFS-induced hyperbolic graphs to a broader deterministic context (Kong et al., 2020).

2. Model Constructions and Typicality Principles

Construction methodologies for deterministic hyperbolic graphs are tailored to typify extremal or representative configurations within broad ensembles:

• Derandomization procedures: As introduced in “Deterministic construction of typical networks in network models” (Sabhahit et al., 1 Dec 2025), a hierarchy of deterministic mappings is constructed, which—under suitable conditions—identifies a unique “most typical” network for a given statistical ensemble, particularly effective for grand canonical models. This construction utilizes explicit derandomization steps for both discrete structures and underlying point processes. In the case of deterministic hyperbolic graphs, these procedures ensure convergence to a limit object in the thermodynamic regime.
• Symbolic dynamics and IFS: For a contractive IFS (composed of mappings $S_i:\mathbb{R}^d\to\mathbb{R}^d$ with $0<r_i\leq \mathrm{Lip}(S_i) < 1$), a deterministic symbolic coding is used to build a rooted tree structure. Adding specific horizontal (shortcut) edges according to geometric criteria—such as the closeness of image sets $K_x$ and $K_y$—results in the “augmented tree” $(X,\mathcal E)$, which is Gromov hyperbolic under explicit combinatorial chain-length criteria (Lau et al., 2016).

In both settings, the construction imparts fine control over the graph’s structure, ensures hyperbolicity, and allows for deterministic uniqueness of objects representing extreme values or typical points in statistical ensembles.

3. Structural Properties and Characterizations

Deterministic hyperbolic graphs exhibit rigorous structural properties:

• Bounded horizontal chain length: Gromov hyperbolicity in augmented trees is equivalent to the existence of a finite upper bound on the length of horizontal geodesics at any fixed level, ensuring the absence of arbitrarily large “flat” regions in the combinatorial geometry [(Lau et al., 2016), Theorem 3.2].
• Expansiveness and Departing Properties: Expansive hyperbolic graphs generalize the construction with conditions on vertical/horizontal edges, ensuring that horizontal connectivity does not collapse under taking children or successors (“expansive”) and that layer separation propagates at an explicit rate (“departing”) [(Kong et al., 2020), Defs. 2.1, 2.5].
• Hierarchical and isometric features: For $\delta$-hyperbolic Helly graphs, key properties include unimodality of the eccentricity function, isometricity and Helly property of the center, and succinct parameter equivalences between hyperbolicity, grid sizes, and center diameter [(Dragan et al., 2021), Theorem 3.2].

These properties underlie efficient algorithmic techniques and cement the deterministic nature of the constructions.

4. Algorithmic Frameworks and Computational Implications

The deterministic structure of hyperbolic graphs enables strong algorithmic results:

• Eccentricity, radius, and diameter algorithms: For an $m$-edge $\delta$-hyperbolic Helly graph, both the radius and a central vertex can be determined in $\mathcal O(\delta m)$ time; all vertex eccentricities can be determined in $\mathcal O(\delta^2 m)$ time, outperforming the best-known general graph algorithms under the Strong Exponential Time Hypothesis [(Dragan et al., 2021), Theorems A, B].
• Center extraction: Extraction of the graph center relies on Helly property and local metric slice decomposition, ensuring deterministic and subquadratic complexity with explicit dependence on hyperbolicity $\delta$.
• Construction of the most typical network: For deterministic hyperbolic graphs as typical representatives, the deterministic procedure—consisting of rounds of derandomization with provable convergence—enables verifiable, reproducible assessment of how closely a real network matches the limit typical state from its ensemble (Sabhahit et al., 1 Dec 2025).

The deterministic algorithmic pipeline thus has both theoretical and practical value, especially in large-scale network analysis.

5. Connections with Fractal Geometry and Boundary Theory

Deterministic hyperbolic graphs built from symbolic dynamics of IFS yield hyperbolic boundaries that are Hölder equivalent to the underlying attractor of the IFS. Specifically, the augmented tree $(X,\mathcal E)$ associated to an IFS yields a boundary $\partial X$ equipped with a visual or Gromov metric, and a canonical bijection to the fractal attractor $K$ satisfying

$$C^{-1}\|\iota(\xi)-\iota(\eta)\| \leq [\rho_a(\xi,\eta)]^\beta \leq C\|\iota(\xi)-\iota(\eta)\|,$$

for all points $\xi, \eta \in \partial X$ and constants $C, \beta$ depending on the IFS and metric parameters [(Lau et al., 2016), Theorem 4.1; (Kong et al., 2020), Theorem 4.5].

These connections enable the paper of random walks, Martin boundaries, and the Dirichlet forms on fractals, drawing a bridge between discrete deterministic graph models and analysis on self-similar sets.

6. Bounded Degree, Separation Conditions, and Real-World Relevance

Deterministic hyperbolic graphs faciliate precise combinatorial and measure-theoretic control:

• Bounded degree: For IFS-based graphs, the augmented tree has bounded degree if and only if the IFS satisfies the open set condition (OSC), directly impacting the applicability to random walks, metric doubling, and harmonic analysis [(Lau et al., 2016), Theorem 5.2].
• Weak separation condition (WSC): When OSC fails, passage to the appropriate quotient and verification of WSC guarantees bounded degree in the quotient graph [(Lau et al., 2016), Theorem 5.4].
• Applicability to real networks: Many empirical networks are demonstrably close, in suitable metric spaces, to the most typical deterministic hyperbolic graph in their statistical ensemble, making these constructions particularly relevant for practical network science (Sabhahit et al., 1 Dec 2025).

A plausible implication is that deterministic hyperbolic graphs, due to their explicit structure, enable benchmarking, typicality analysis, and algorithmic acceleration in network modeling and data analysis.

7. Generalizations and Extensions

The core framework for deterministic hyperbolic graphs admits powerful generalizations:

• Expansive hyperbolic graphs: Abstract essential properties of IFS-based graphs to general combinatorial settings, including “weighted IFS” graphs and their asymptotic analysis (Kong et al., 2020).
• Connections to self-similar energy forms: Via boundary theory and effective resistance metrics, deterministic hyperbolic graphs provide deterministic models for analysis of self-similar Dirichlet forms and p.c.f. fractals [(Kong et al., 2020), Theorem 6.7].
• Dynamical and statistical derandomization: The deterministic most-typical-state construction extends to grand canonical ensembles and their mixtures satisfying suitable conditions, broadly applicable across network models and statistical physics (Sabhahit et al., 1 Dec 2025).

These directions integrate deterministic hyperbolic graph models with current research at the interface of combinatorics, statistical mechanics, and fractal geometry.