Hyperbolic Random Graphs Research Articles

Limit theory for isolated and extreme points in hyperbolic random geometric graphs

Given $\alpha \in (0, \infty )$ and $r \in (0, \infty )$, let ${\mathcal {D}}_{r, \alpha }$ be the disc of radius $r$ in the hyperbolic plane having curvature $-\alpha ^{2}$. Consider the Poisson point process having uniform intensity density on ${\mathcal {D}}_{R, \alpha }$, with $R = 2 \log (n/ \nu )$, $n \in \mathbb {N}$, and $\nu < n$ a fixed constant. The points are projected onto ${\mathcal {D}}_{R, 1}$, preserving polar coordinates, yielding a Poisson point process ${\mathcal {P}}_{\alpha , n}$ on ${\mathcal {D}}_{R, 1}$. The hyperbolic geometric graph ${\mathcal {G}}_{\alpha , n}$ on ${\mathcal {P}}_{\alpha , n}$ puts an edge between pairs of points of ${\mathcal {P}}_{\alpha , n}$ which are distant at most $R$. This model has been used to express fundamental features of complex networks in terms of an underlying hyperbolic geometry. For $\alpha \in (1/2, \infty )$ we establish expectation and variance asymptotics as well as asymptotic normality for the number of isolated and extreme points in ${\mathcal {G}}_{\alpha , n}$ as $n \to \infty $. The limit theory and renormalization for the number of isolated points are highly sensitive on the curvature parameter. In particular, for $\alpha \in (1/2, 1)$, the variance is super-linear, for $\alpha = 1$ the variance is linear with a logarithmic correction, whereas for $\alpha \in (1, \infty )$ the variance is linear. The central limit theorem fails for $\alpha \in (1/2, 1)$ but it holds for $\alpha \in (1, \infty )$.

Explosion in weighted hyperbolic random graphs and geometric inhomogeneous random graphs

In this paper we study weighted distances in scale-free spatial network models: hyperbolic random graphs, geometric inhomogeneous random graphs and scale-free percolation. In hyperbolic random graphs, n=Θ(eR∕2) vertices are sampled independently from the hyperbolic disk with radius R and two vertices are connected either when they are within hyperbolic distance R, or independently with a probability depending on the hyperbolic distance. In geometric inhomogeneous random graphs, and in scale-free percolation, each vertex is given an independent weight and location from an underlying measured metric space and Zd, respectively, and two vertices are connected independently with a probability that is a function of their distance and their weights. We assign independent and identically distributed (i.i.d.) weights to the edges of the obtained random graphs, and investigate the weighted distance (the length of the shortest weighted path) between two uniformly chosen vertices, called typical weighted distance. In scale-free percolation, we study the weighted distance from the origin of vertex-sequences with norm tending to infinity.In particular, we study the case when the parameters are so that the degree distribution in the graph follows a power law with exponent τ∈(2,3) (infinite variance), and the edge-weight distribution is such that it produces an explosive age-dependent branching process with power-law offspring distribution, that is, the branching process produces infinitely many individuals in finite time. We show that in all three models, typical distances within the giant/infinite component converge in distribution, that is, no re-scaling is necessary. This solves an open question in Hofstad and Komjáthy (2017).The main tools of our proof are to develop elaborate couplings of the models to infinite versions, to follow the shortest paths to infinity and then to connect these paths by using weight-dependent percolation on the graphs, that is, we delete edges attached to vertices with higher weight with higher probability. We realise the percolation using the edge-weights: only very short edges connected to high weight vertices are allowed to stay, hence establishing arbitrarily short upper bounds for connections.

On the Second Largest Component of Random Hyperbolic Graphs

We show that in the random hyperbolic graph model as formalized by Gugelmann, Panagiotou, and Peter (2012) in the most interesting range of $\frac12 < \alpha < 1$ the size of the second largest com...

Updating Dynamic Random Hyperbolic Graphs in Sublinear Time

Generative network models play an important role in algorithm development, scaling studies, network analysis, and realistic system benchmarks for graph data sets. A complex network model gaining considerable popularity builds random hyperbolic graphs, generated by distributing points within a disk in the hyperbolic plane and then adding edges between points with a probability depending on their hyperbolic distance. We present a dynamic extension to model gradual network change, while preserving at each step the point position probabilities. To process the dynamic changes efficiently, we formalize the concept of a probabilistic neighborhood : Let P be a set of n points in Euclidean or hyperbolic space, q a query point, dist a distance metric, and f : R + → [0, 1] a monotonically decreasing function. Then, the probabilistic neighborhood N ( q , f ) of q with respect to f is a random subset of P and each point p ∈ P belongs to N ( q , f ) with probability f (dist( p , q )). We present a fast, sublinear-time query algorithm to sample probabilistic neighborhoods from planar point sets. For certain distributions of planar P , we prove that our algorithm answers a query in O ((| N ( q , f )| + √ n ) log n ) time with high probability. This enables us to process a node movement in random hyperbolic graphs in sublinear time, resulting in a speedup of about one order of magnitude in practice compared to the fastest previous approach. Apart from that, our query algorithm is also applicable to Euclidean geometry, making it of independent interest for other sampling or probabilistic spreading scenarios.

Geometric inhomogeneous random graphs

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 behavior 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) 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). (2) We establish that GIRGs have clustering coefficients in Ω(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.

Spectral gap of random hyperbolic graphs and related parameters

Random hyperbolic graphs have been suggested as a promising model of social networks. A few of their fundamental parameters have been studied. However, none of them concerns their spectra. We consider the random hyperbolic graph model, as formalized by [Automata, Languages, and Programming—39th International Colloquium—ICALP Part II. (2012) 573–585 Springer], and essentially determine the spectral gap of their normalized Laplacian. Specifically, we establish that with high probability the second smallest eigenvalue of the normalized Laplacian of the giant component of an $n$-vertex random hyperbolic graph is at least $\Omega(n^{-(2\alpha-1)}/D)$, where $\frac{1}{2}<\alpha<1$ is a model parameter and $D$ is the network diameter (which is known to be at most polylogarithmic in $n$). We also show a matching (up to a polylogarithmic factor) upper bound of $n^{-(2\alpha-1)}(\log n)^{1+o(1)}$. As a byproduct, we conclude that the conductance upper bound on the eigenvalue gap obtained via Cheeger’s inequality is essentially tight. We also provide a more detailed picture of the collection of vertices on which the bound on the conductance is attained, in particular showing that for all subsets whose volume is $O(n^{\varepsilon})$ for $0<\varepsilon<1$ the obtained conductance is with high probability $\Omega(n^{-(2\alpha-1)\varepsilon+o(1)})$. Finally, we also show consequences of our result for the minimum and maximum bisection of the giant component.

Navigability of Random Geometric Graphs in the Universe and Other Spacetimes

Random geometric graphs in hyperbolic spaces explain many common structural and dynamical properties of real networks, yet they fail to predict the correct values of the exponents of power-law degree distributions observed in real networks. In that respect, random geometric graphs in asymptotically de Sitter spacetimes, such as the Lorentzian spacetime of our accelerating universe, are more attractive as their predictions are more consistent with observations in real networks. Yet another important property of hyperbolic graphs is their navigability, and it remains unclear if de Sitter graphs are as navigable as hyperbolic ones. Here we study the navigability of random geometric graphs in three Lorentzian manifolds corresponding to universes filled only with dark energy (de Sitter spacetime), only with matter, and with a mixture of dark energy and matter. We find these graphs are navigable only in the manifolds with dark energy. This result implies that, in terms of navigability, random geometric graphs in asymptotically de Sitter spacetimes are as good as random hyperbolic graphs. It also establishes a connection between the presence of dark energy and navigability of the discretized causal structure of spacetime, which provides a basis for a different approach to the dark energy problem in cosmology.

Cliques in Hyperbolic Random Graphs

Most complex real world networks display scale-free features. This characteristic motivated the study of numerous random graph models with a power-law degree distribution. There is, however, no established and simple model which also has a high clustering of vertices as typically observed in real data. Hyperbolic random graphs bridge this gap. This natural model has recently been introduced by Krioukov et al. (in Phys Rev E 82(3):036106, 2010) and has shown theoretically and empirically to fulfill all typical properties of real world networks, including power-law degree distribution and high clustering. We study cliques in hyperbolic random graphs G and present new results on the expected number of k-cliques $$\mathbb {E}\left[ K_k\right] $$ and the size of the largest clique $$\omega (G)$$ . We observe that there is a phase transition at power-law exponent $$\beta = 3$$ . More precisely, for $$\beta \in (2,3)$$ we prove $$\mathbb {E}\left[ K_k\right] =n^{k (3-\beta )/2} \varTheta (k)^{-k}$$ and $$\omega (G)=\varTheta (n^{(3-\beta )/2})$$ , while for $$\beta \geqslant 3$$ we prove $$\mathbb {E}\left[ K_k\right] =n \, \varTheta (k)^{-k} $$ and $$\omega (G)=\varTheta (\log (n)/ \log \log n)$$ . Furthermore, we show that for $$\beta \geqslant 3$$ , cliques in hyperbolic random graphs can be computed in time $$\mathcal {O}(n)$$ . If the underlying geometry is known, cliques can be found with worst-case runtime $$\mathcal {O}(m \cdot n^{2.5})$$ for all values of $$\beta $$ .