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Random Hyperbolic Graphs in D + 1 Dimensions

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Random Hyperbolic Graphs in D + 1 Dimensions Random hyperbolic graphs in d + 1 dimensions Maksim Kitsak,1, 2, 3 Rodrigo Aldecoa,2, 3 Konstantin Zuev,4 and Dmitri Krioukov5, 3 1Faculty of Electrical Engineering, Mathematics and Computer Science, Delft University of Technology, 2628 CD, Delft, Netherlands 2Department of Physics, Northeastern University, 110 Forsyth Street, 111 Dana Research Center, Boston, MA 02115, USA. 3Network Science Institute, Northeastern University, 177 Huntington avenue, Boston, MA, 022115 4Department of Computing and Mathematical Sciences, California Institute of Technology, 1200 E. California Blvd. Pasadena, CA, 91125, USA 5Department of Physics, Department of Mathematics, Department of Electrical&Computer Engineering, Northeastern University, 110 Forsyth Street, 111 Dana Research Center, Boston, MA 02115, USA. (Dated: October 23, 2020) We generalize random hyperbolic graphs to arbitrary dimensionality. We find the rescaling of network parameters that allows to reduce random hyperbolic graphs of arbitrary dimensionality to a single mathematical framework. Our results indicate that RHGs exhibit similar topological properties, regardless of the dimensionality of their latent hyperbolic spaces. I. INTRODUCTION The spherical coordinate system on the hyperboloid (r; θ1; :::; θd) is defined by Random hyperbolic graphs (RHGs) have been intro- 1 duced as latent space models, where nodes correspond to x0 = cosh ζr; 2 ζ points in the 2-dimensional hyperboloid H and connec- 1 tions between the nodes are established with distance- x = sinh ζr cos θ ; dependent probabilities [1,2]. RHGs have been shown 1 ζ 1 to accurately model structural properties of real net- 1 x = sinh ζr sin θ cos θ ; (3) works including sparsity, self-similarity, scale-free de- 2 ζ 1 2 gree distribution, strong clustering, the small-world prop- :: erty, and community structure [2{6]. Using the RHG as a null model, one can map real networks to hyper- 1 xd = sinh ζr sin θ1::: sin θd−1 cos θd; bolic spaces [7{9]. Practical interest in the RHGs is ζ driven by promising applications in routing and naviga- 1 x = sinh ζr sin θ ::: sin θ sin θ ; tion [7, 10{13], link prediction [8, 14{20], network scal- d+1 ζ 1 d−1 d ing [3,6, 12, 21], and semantic analysis [22]. Of relevance to this work and also of potential interest where r > 0 is the radial coordinate and (θ1; :::; θd) to the reader is the hyperbolic graph generator [23], al- are the standard angular coordinates on the unit d- 2 lowing to generate RHGs in H , as well as the analysis of dimensional sphere Sd. d small-world and clustering in related S models, defined The coordinate transformation in (3) yields the Hd+1 in Euclidean spaces [24]. metric In this work we extend the RHG to the arbitrary di- 1 mensionality, compute its basic structural properties, and ds2 =dr2 + sinh2 (ζr) dΩ2; (4) discuss its limiting regimes. ζ2 d 2 2 2 2 dΩd = dθ1 + sin (θ1)dθ2 + :: 2 2 2 +sin (θ1):::sin (θd−1)dθd; (5) II. HYPERBOLIC RANDOM GRAPH MODEL IN d + 1 DIMENSIONS resulting in the volume element in Hd+1: Consider the upper sheet of the d + 1-dimensional hy- d d 1 Y perboloid of curvature K = −ζ2 dV = sinh ζr dr sind−k(θ )dθ : (6) ζ k k k=1 2 2 2 1 x0 − x1 − ::: − xd+1 = ; x0 > 0 (1) ζ2 The distance between two points i and j in Hd+1 is given by the hyperbolic law of cosines: in the d + 2-dimensional Minkowski space with metric cosh ζdij = cosh ζri cosh ζrj − sinh ζri sinh ζrj cos ∆Θij; 2 2 2 2 ds = −dx0 + dx1 + ::: + dxd+1: (2) (7) 2 d+1 where ∆Θij is the angle between i and j: Taken together, RHGs in B are fully defined by 6 parameters: properties of the hyperbolic ball, RH and ζ; cos(∆Θij) = cos θi;1 cos θj;1 number of nodes n; radial component of node distribution + sin θi;1 sin θj;1 cos θi;2 cos θj;2 + ::: α; chemical potential µ and temperature T . Only four parameters, however, (n; α; T; RH ) are inde- + sin θi;1 sin θj;1::: sin θi;d−1 sin θj;d−1 cos θi;d cos θj;d pendent. It follows from (7) that ζ is merely a rescaling + sin θi;1 sin θj;1::: sin θi;d−1 sin θj;d−1 sin θi;d sin θj;d; (8) parameter for distances fdijg, and can be absorbed into r coordinates by the appropriate rescaling. Chemical po- (θ ; :::; θ ) and (θ ; :::; θ ) are the coordinates of i;1 i;d j;1 j;d tential µ controls the expected number of links and the points i and j on d. S sparsity of resulting network models. We demonstrate For sufficiently large ζr and ζr values, the hyperbolic i j below that the sparsity requirement uniquely determines law of cosines in Eq. (7) is closely approximated by µ in terms of other RHGs parameters. 2 d = r + r + ln (sin(∆Θ =2)) : (9) ij i j ζ ij III. DEGREE DISTRIBUTION IN THE RHG d+1 The hyperbolic ball B of radius RH > 0 is defined as the set of points with The structural properties of the RHG can be computed with the hidden variable formalism, Ref. [25], by treating r 2 [0;RH ]: (10) node coordinates as hidden variables. d+1 We begin by calculating the expected degree of node l Nodes of the HRG are points in B selected at ran- l l located at point xl = frl; θ1; :::; θdg: dom with density ρ(x) ≡ ρ(r)ρ(θ1)...ρ(θd), where Z d dxkρ(xk) ρ(r) = [sinh(αr)] =Cd; α ≥ ζ=2 hk(xl)i = (n − 1) (13) ζ(dlk−µ) Z RH 1 + e 2T d Cd ≡ [sinh(αr)] dr; 0 The symmetry in the angular distribution of points en- d−k ρk(θ) = [sin(θ)] =Id;k; sures that the expected degree of the node depends only Z π d−k+1 on its radial coordinate rl and not on its angular coordi- d−k p Γ[ 2 ] nates, hk(xl)i = hk(rl; 0; ::; 0)i ≡ hk(rl)i. This allows us Id;k ≡ [sin(θ)] dθ = π d−k k 2 [1; d − 1]; 0 Γ 1 + 2 to integrate out d angular coordinates in Eq. (13). Id;d ≡ 2π; We also note that the choice of radial coordinate dis- (11) tribution given by Eq. (11) with α ≥ ζ=2 results in most In other words, nodes are uniformly distributed on unit of the nodes having large radial coordinates, ri ≈ RH sphere Sd with respect to their angular coordinates. In 1. This fact allows us to approximate distances using the special case of α = ζ nodes are also uniformly dis- Eq. (9): tributed in Bd+1. Z RH Z π Pairs of nodes i and j are connected independently (n − 1)ρ(r)drρ1(θ)dθ hk(rl)i = 1 : (14) with connection probability 0 0 h 2i 2T ζ(r+rl−µ) θ 1 + e sin 2 1 pij = p (dij) = ; (12) 1 + exp (ζ (dij − µ) =2T ) Since the inner integral in Eq. (14) does not have a closed- form solution, to estimate hk(rl)i we need to employ sev- where µ > 0 and T > 0 are model parameters and dij d+1 eral approximations. We note that most nodes have large is the distance between points i and j in H , given by radial coordinates, e(r+rl−µ) 1, and the dominant con- Eq. (7). We refer to parameters T and µ as temperature tribution to the inner integral in (14) comes from small and chemical potential, respectively, using the analogy θ values. This allows us to estimate the integral by re- with the Fermi-Dirac statistics. We note that the factors placing sin(θ) and sin(θ=2) with the leading Taylor series of 2 and ζ in Eq. (11) are to agree with the 2-dimensional terms, as sin(x) = x + O x3: RHG [2] that corresponds to d = 1. Thus, RHG is formed in a two step network generation 1 Z RH Z π (n − 1)ρ(r)drθd−1dθ process: hk(rl)i = 1 ; (15) Id;1 0 0 h θ 2i 2T 1 + eζ(r+rl−µ) 1. Randomly select n points in Bd+1 with pdf ρ(x) in 2 Eq. (11). where 2. Connect i-j nodes pairs independently at random Z π with distance-dependent connection probabilities d−1 p d d + 1 Id;1 ≡ sin (θ)dθ = πΓ =Γ : (16) pij = p (dij), prescribed by Eq. (12). 0 2 2 3 The integral in Eq. (14) can be further simplified by IV. CONNECTIVITY REGIMES OF THE RHG the following change of RHG variables: dζ Depending on the value of the rescaled temperature fr; r ; R ; mg = fr; r ;R ; µg; τ = dT , there exist three distinct regimes of the RHG: (i) l H 2 l H cold regime (τ < 1), (ii) critical regime (τ = 1), and (iii) τ = dT; hot regime (τ > 1). We provide detailed analyses these (17) a = 2α/ζ; regimes below, and summarize our findings in TableI. θ d u ≡ e(r+rl−m) : 2 1. Cold regime, τ < 1 where the top line corresponds to 4 equations, each cor- responding to one variable in the brackets. In the τ < 1 regime the hypergeometric function in In terms of the rescaled variables the connection prob- (21) can be approximated as ability function takes the form of 1 πτ F 1; τ; 1 + τ; −u τ = u−1 ; (26) 1 2 1 max max sin (πτ) pij = d ; (18) ri+rj −m τ h ∆Θij i τ 1 + e sin 2 and hk(rl)i and hki are then given by: and Eq.
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