Hyperbolic random geometric graphs: Structural and spectral properties

In this paper we perform a thorough numerical study of structural and spectral properties of hyperbolic random geometric graphs (HRGs) G(n,ρ,α,ζ) by means of a random matrix theory (RMT) approach. HRGs are formed by distributing n nodes in a Poincaré disk of fixed radius ρ; the radial node distribution is characterized by the exponent α and ζ controls the curvature of the embedding space. Specifically, we report and analyze average structural properties [by means of the number of nonisolated vertices V_{x}(G), topological indices, and clustering coefficients] and average spectral properties [by means of standard RMT measures: the ratio between consecutive eigenvalue spacings r_{R}(G), the ratio between nearest- and next-to-nearest-neighbor eigenvalue distances r_{C}(G), and the inverse participation ratio and the Shannon entropy S(G) of the eigenvectors]. Even though HRGs are, in general, more elaborated than Euclidean random geometric graphs, we show that both types of random graphs share important average properties, namely: (i) 〈V_{x}(G)〉 is a simple function of the average degree 〈k〉, 〈V_{x}(G)〉≈n[1-exp(-γ〈k〉)], while (ii) properly normalized 〈r_{R}(G)〉, 〈r_{C}(G)〉 and 〈S(G)〉 scale with the parameter ξ∝〈k〉n^{δ}. Here, γ≡γ(α/ζ), δ≡δ(α/ζ), and 〈·〉 is the average over a graph ensemble.