Navigability of Random Geometric Graphs in the Universe and Other Spacetimes

Cunningham, William J.; Zuev, Konstantin M.; Krioukov, Dmitri

doi:10.1038/s41598-017-08872-4

Cited by 6 publications

(6 citation statements)

References 65 publications

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“…However, we always call it the hyperbolic random geometric graph. In a broader sense, our work is an addition to the developing literature on random structures of hyperbolic spaces (see [12,6,5,34,31,40,17,21]).…”

Section: Introductionmentioning

confidence: 99%

Sub-tree counts on hyperbolic random geometric graphs

Owada

Yogeshwaran

2022

Adv. Appl. Probab.

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The hyperbolic random geometric graph was introduced by Krioukov et al. (Phys. Rev. E82, 2010). Among many equivalent models for the hyperbolic space, we study the d-dimensional Poincaré ball ( $d\ge 2$ ), with a general connectivity radius. While many phase transitions are known for the expectation asymptotics of certain subgraph counts, very little is known about the second-order results. Two of the distinguishing characteristics of geometric graphs on the hyperbolic space are the presence of tree-like hierarchical structures and the power-law behaviour of the degree distribution. We aim to reveal such characteristics in detail by investigating the behaviour of sub-tree counts. We show multiple phase transitions for expectation and variance in the resulting hyperbolic geometric graph. In particular, the expectation and variance of the sub-tree counts exhibit an intricate dependence on the degree sequence of the tree under consideration. Additionally, unlike the thermodynamic regime of the Euclidean random geometric graph, the expectation and variance may exhibit different growth rates, which is indicative of power-law behaviour. Finally, we also prove a normal approximation for sub-tree counts using the Malliavin–Stein method of Last et al. (Prob. Theory Relat. Fields165, 2016), along with the Palm calculus for Poisson point processes.

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Section: Introductionmentioning

confidence: 99%

Sub-tree counts on hyperbolic random geometric graphs

Owada

Yogeshwaran

2022

Adv. Appl. Probab.

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“…Perhaps the simplest example of a random spatial network [1][2][3][4][5][6][7][8][9] is the 1d random geometric graph (1d RGG) [10][11][12][13][14][15][16]. 1d problems are widely studied in order to first understand a simpler case, such as the Ising and Heisenberg models of magnetism studied by Ising and Hans Bethe in 1924 and 1931 respectively, in the later case resulting in the famous Bethe ansatz, as well as more modern examples of e.g.…”

Section: Introductionmentioning

confidence: 99%

Connectivity of 1d random geometric graphs

Kartun-Giles¹,

Koufos²,

Privault³

2021

Preprint

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A 1d random geometric graph (1d RGG) is built by joining a random sample of n points from an interval of the real line with probability p. We count the number of k-hop paths between two vertices of the graph in the case where the space is the 1d interval [0, 1]. We show how the k-hop path count between two vertices at Euclidean distance |x − y| is in bijection with the volume enclosed by a uniformly random d-dimensional lattice path joining the corners of a (k − 1)-dimensional hyperrectangular lattice. We are able to provide the probability generating function and distribution of this k-hop path count as a sum over lattice paths, incorporating the idea of restricted integer partitions with limited number of parts. We therefore demonstrate and describe an important link between spatial random graphs, and lattice path combinatorics, where the d-dimensional lattice paths correspond to spatial permutations of the geometric points on the line.

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“…Given that the class of "nice" d-dimensional manifolds as considered above includes d-dimensional spheres (having constant positive curvature) and d-dimensional Euclidean spaces (having zero curvature), it is natural to ask about random geometric graphs on negatively curved spaces. Such an investigation was initiated recently in [28] on hyperbolic spaces and even more recently in [16] on more general spaces such as Lorentzian manifolds. However, since the d-dimensional Poincaré ball is one of the canonical and well-understood models of non-Euclidean and non-compact spaces, we shall restrict our attention to the same.…”

Section: Introductionmentioning

confidence: 99%

Sub-tree counts on hyperbolic random geometric graphs

Owada¹,

Yogeshwaran²

2018

Preprint

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In this article, we study the hyperbolic random geometric graph introduced recently in [28]. For a sequence Rn → ∞, we define these graphs to have the vertex set as Poisson points distributed uniformly in balls B(0, Rn) ⊂ B (α) d , the d-dimensional Poincaré ball (i.e., the unit ball on R d with the Poincaré metric dα corresponding to negative curvature −α 2 , α > 0) by connecting any two points within a distance Rn according to the metric d ζ , ζ > 0. Denoting these graphs by HGn(Rn; α, ζ), we study asymptotic counts of copies of a fixed tree Γ k (with the ordered degree sequence d (1) ≤ . . . ≤ d (k) ) in HGn(Rn; α, ζ). Unlike earlier works, we count more involved structures, allowing for d > 2, and in many places, more general choices of Rn rather than Rn = 2[ζ(d − 1)] −1 log(n/ν), ν ∈ (0, ∞). The latter choice of Rn for α/ζ > 1/2 corresponds to the thermodynamic regime in which the expected average degree is asymptotically constant. We show multiple phase transitions in HGn(Rn; α, ζ) as α/ζ increases, i.e., the space B (α) dbecomes more hyperbolic. In particular, our analyses reveal that the sub-tree counts exhibit an intricate dependence on the degree sequence d (1) , . . . , d (k) of Γ k as well as the ratio α/ζ. Under a more general radius regime Rn than that described above, we investigate the asymptotics of the expectation and variance of sub-tree counts. Moreover, we prove the corresponding central limit theorem as well. Our proofs rely crucially on a careful analysis of the sub-tree counts near the boundary using Palm calculus for Poisson point processes along with estimates for the hyperbolic metric and measure. For the central limit theorem, we use the abstract normal approximation result from [31] derived using the Malliavin-Stein method.

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Navigability of Random Geometric Graphs in the Universe and Other Spacetimes

Cited by 6 publications

References 65 publications

Sub-tree counts on hyperbolic random geometric graphs

Sub-tree counts on hyperbolic random geometric graphs

Connectivity of 1d random geometric graphs

Sub-tree counts on hyperbolic random geometric graphs

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