Ordering and Convergence of Large Degrees in Random Hyperbolic Graphs

TL;DR

This work provides a comprehensive, regime-aware analysis of the large-degree structure in random hyperbolic graphs. By linking node degrees to their radii in hyperbolic space, the authors establish that the top-degree ordering matches the center-distance ordering up to a fixed rank and, for $α>1/2$, up to rank $k_n=n^{β}$ with $β=1/(1+8α)$, while exceeding this bound disrupts the ordering. They prove that the normalized degree process converges to a Poisson point process across all $α>0$, yielding explicit extreme-value limits: Weibull for $α<1/2$ and Fréchet for $α>1/2$ (with a special coordinated description at $α=1/2$). The radii themselves converge as point processes with regime-dependent scalings, and the measures of the connection balls are carefully approximated to drive the analysis. Collectively, these results refine prior maximum-degree estimates and extend large-degree analysis from the sparse, scale-free regime to the dense regime $α\le 1/2$, providing a unified view of hub formation, connectivity, and extreme degrees in RHGs.

Abstract

We describe the asymptotic behaviour of large degrees in random hyperbolic graphs, for all values of the curvature parameter $ α$. We prove that, with high probability, the node degrees satisfy the following ordering property: the ranking of the nodes by decreasing degree coincides with the ranking of the nodes by increasing distance to the centre, at least up to any constant rank. In the scale-free regime $ α>1/2$, the rank at which these two rankings cease to coincide is $n^{1/(1+8 α)+o(1)}$. We also provide a quantitative description of the large degrees by proving the convergence in distribution of the normalised degree process towards a Poisson point process. In particular, this establishes the convergence in distribution of the normalised maximum degree of the graph. A transition occurs at $ α= 1/2$, which corresponds to the connectivity threshold of the model. For $ α< 1/2$, the maximum degree is of order $n - O(n^{ α+ 1/2})$, whereas for $ α\geq 1/2$, the maximum degree is of order $n^{1/(2 α)}$. In the cases $ α< 1/2$ and $ α> 1/2$, the limit distribution of the maximum degree belongs to the class of extreme value distributions (Weibull for $ α< 1/2$ and Fréchet for $ α> 1/2$). This refines previous estimates on the maximum degree for $ α> 1/2$ and extends the study of large degrees to the dense regime $ α\leq 1/2$.

Figures (4)

• Figure 1: Simulations of random hyperbolic graphs (native representation) with $n = 500$, $\nu = 1$, $\alpha = 0.45$ (left), $\alpha = 0.50$ (middle) and $\alpha = 0.55$ (right). The boundary of $\mathcal{B}_{0}(R_n)$ is represented by a black circle and its centre by a blue dot.
• Figure 2: Depiction of a ball $\mathcal{B}_{X}(R_n)$ (native representation)
• Figure 3: Representation of $\theta_r(y)$ (in a Euclidean setting)
• Figure 4: Depiction of the closeness event $C_n$. By condition \ref{['cond:small_gap']}, for all $i$, the radius gap $r(v_i') - r(v_i)$ is small, ensuring that $\deg_{\varepsilon}(v_i) \leq \deg_{\varepsilon}(v_i')$ holds with a probability bounded away from $0$. By condition \ref{['cond:disjoint_ep_balls']}, the portions of the balls $\mathcal{B}_{v_i}(R_n)$ and $\mathcal{B}_{v_i'}(R_n)$ that lie beyond the circle of radius $R_n^{\varepsilon}$ are all disjoints, ensuring the independence of the corresponding $\varepsilon$-degrees.

Theorems & Definitions (23)

• Theorem 3.1
• Theorem 3.2
• Theorem 3.3
• Remark 3.4
• Proposition 4.1
• proof
• Lemma 5.1
• Lemma 5.2
• Lemma 5.3
• proof