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 {dij}, 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 ∈ [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 = {rl, θ1, ..., θd}: 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 √ 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 ∈ [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 √ 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 {r, r , R , m} = {r, r ,R , µ}, 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. (15) reads 2d πτ hki = (n − 1) he−ri2em, dI sin (πτ) d d,1 R π (27) 2d Z H Z ( 2 ) (n − 1)ρ(r)drdφ hki r hk(rl)i = , (19) − r+rl−m 1 hk(r)i = −r e , dId,1 0 0 1 + e τ φ τ he i where −r R RH −r where he i ≡ 0 drρ(r)e . The explicit expression −r ρ(r) ≡ aea(r−RH ). (20) for he i follows from (20): a By taking the inner integral in Eq. (19) we obtain he−ri = e−RH − e−aRH . (28) a − 1 R d Z H 1 (n − 1)π τ hk(rl)i = drρ(r) 2F1 1, τ, 1 + τ, −umax , In the case a > 1 we neglect the second term in (28) to dId,1 0 obtain π d rl+r−m m−r −R umax = e , hk(r)(n)i ∼ ne l H , 2 (29) (21) hk(n)i ∼ nem−2RH . where 2F1 is the Gauss hypergeometric function. The expected average degree of the graph is given by f(x) Henceforth, we write f(x) ∼ g(x) when limx→∞ g(x) = Z RH K, where K > 0 is a constant. hki = drρ(r)hk(r)i, (22) We note that hk(r)i decreases exponentially as a func- 0 tion of r with largest (smallest) expected degree corre- and degree distribution of the RHG can be expressed as sponding to r = 0 (r = RH ). By demanding that the largest and smallest expected degrees scale as Z RH P (k) = drρ(r)P (k|r), (23) hkmaxi = hk(0)i ∼ n, (30) 0 hkmini = hk (RH (n))i ∼ 1, (31) where P (k|r) is a conditional probability that node with radial coordinate r has exactly k connections. we obtain RH ∼ ln n and m = RH + λ, where λ is an In the case of sparse graphs P (k|r) is closely approxi- arbitrary constant. mated by the Poisson distribution: First, we note that the scaling for RH is consistent with our initial assumption of R 1 for large graphs. 1 H P (k|r) = e−hk(r)i [hk(r)i]k , (24) We also note that the exact value of the parameter λ is k! not important as long as it is independent of n. To be see Ref. [25], and the resulting degree distribution P (k) consistent with the original H2 formulation we set λ = 0, is a mixed Poisson distribution: obtaining 1 Z RH m = R = ln (n/ν) , (32) P (k) = e−hk(r)i [hk(r)i]k ρ(r)dr (25) H k! 0 where ν is the parameter controlling the expected degree with mixing parameter hk(r)i. of the RHG. 4

Applying scaling relationships (32) to (27) we obtain indicating that in the critical regime, hk(n)i ∼ ln(n) and hk(r)(n)i ∼ nre−r. 2d a 2 πτ hki = ν ∼ O(1), If a = 1, (21) and (22) result in dId,1 a − 1 sin(πτ) (33) ν2d π n a − 1 −r hki = ln R2 , hk(r)i = hkie . I 2 H ν a d,1 (39) d 2 Finally, using (24) and (25) we obtain 2 −r π RH hk(r)i = n e RH r + d ln − , dId,1 2 2 γ−1 Γ[k − a] −γ P (k) = (γ − 1)κ0 ∼ k , (34) Γ[k + 1] indicating that hk(n)i ∼ (ln(n))2. γ−2 where γ = a + 1, and κ0 ≡ γ−1 hki. Hence, the cold regime corresponds to sparse scale-free 3. Hot regime, τ > 1 graphs with γ ∈ (2, ∞). We note that degree distribution is called scale-free if it takes the form of P (k) = `(k)k−γ , where `(k) is a slowly varying function, i.e., a function In the case τ > 1, (21) and (22) can be approximated that varies slowly at inﬁnity, see Ref. [26]. Any function as 1 d converging to a constant is slowly varying. In the case of π 1− τ 2 τ h i2 γ−1 hki = (n − 1) he−r/τ i em/τ , Eq. (34), `(k) → (γ − 1)κ0 as k → ∞. 2 dI τ − 1 The special case of a = 1 is also well deﬁned. In this d,1 hki case the expressions for hki and hk(r)i given by Eq. (29) hk(r)i = e−r/τ , remain valid but he−ri is now given by he−r/τ i (40) −r −RH he i = RH e . (35) where

As a result, Z RH aτ he−r/τ i ≡ ρ(r)e−r/τ dr ≈ e−RH /τ . (41) n2 2d πτ aτ − 1 hki = νln , 0 ν dI sin(πτ) d,1 (36) Note that the expression for he−r/τ i given by Eq. (41) is n 2d πτ hk(r)i = nln e−r. valid for all values of a and τ since aτ > 1. ν dId,1 sin(πτ) Similar to the τ < 1 regime, we demand hkmax(n)i ∼ n 2 and hkmin(n)i ∼ 1 to obtain the scaling relationships for As seen from Eq. (36), hki ∼ (ln(n)) , indicating that m and R : RHGs in this case are no longer sparse. H

m = RH = τ ln (n/ν) , (42) 2. Critical regime, τ = 1 which in combination with (40) leads to

d 1− 1 2 3 In the τ = 1, a > 1 regime (21) and (22) up to the ν2 π τ a τ hki = 2 , leading term can be approximated as: dId,1 2 (τ − 1) (aτ − 1) n aτ − 1 hk(r)i = hkie−r/τ , (43) 2d a 2 ν aτ hki = (n − 1) × dId,1 a − 1 aτ Γ[k − aτ] −γ P (k) =aτ [hk(RH )i] ∼ k , π d Γ[k + 1] × em−2RH ln 1 + e2RH −m , 2 (37) where γ = aτ + 1. 2d a hk(r)i = (n − 1) × dId,1 a − 1 V. LIMITING CASES OF THE RHG MODEL π d × em−r−RH ln 1 + er+RH −m , 2 In this section, we analyze several important parameter which in the case of scaling deﬁned by (32) further sim- limits of the RHG and show that they correspond to well- plify to deﬁned graph ensembles.

ν2d a 2 π d hki = ln 1 + eRH , A. τ → 0 limit in the cold regime dI a − 1 2 d,1 (38) 2d a π d The case of τ = 0 is well-deﬁned as the limit of τ → 0 of hk(r)i = n e−r ln 1 + er , dId,1 a − 1 2 the cold regime. In this limit the connection probability 5

a a a τ aa 1 (1, ∞) ∞ aa 0 m = RH = ln [n/ν] m = RH = ln [n/ν] Spherical Random hki = O (ln [n])2 hki = O(1) Geometric Graph, γ = 2 γ = a + 1 pij = Θ(θc − ∆θ) (0, 1) Spherical Soft 1 m = RH = ln [n/ν] Random Geometric hki = O(ln [n]) Graph, γ = a + 1 pij = f(∆θ) (1, ∞) m = RH = τ ln [n/ν] hki = O(1) γ = aτ + 1 2α τ → ∞, limτ→∞ ζ/τ = λ Hyper Soft Conﬁgurational Model, γ = dλ + 1 1 τ → ∞, limτ→∞ m/τ = λ G(n, p), p = 1+e−λ a aa α ζ ζ T a 2 2 , ∞ ∞ aa 0 µ = RH , RH ∼ ln [n] µ = RH , RH ∼ ln [n] Spherical Random hki = O (ln [n])2 hki = O(1) Geometric Graph, α γ = 2 γ = 2 ζ + 1 pij = Θ(θc − ∆θ) 1 0, d Spherical Soft 1 d µ = RH , RH ∼ ln [n] Random Geometric hki = O(ln [n]) Graph, α γ = 2 ζ + 1 pij = f(∆θ) 1 d , ∞ µ = RH , RH ∼ T ln [n] hki = O(1) α γ = 2 ζ dT + 1 ζ 2α T → ∞, limT →∞ T = dλ Hyper Soft Conﬁgurational Model γ = dλ + 1 ζµ 1 T → ∞, limT →∞ 2T = λ G(n, p), p = 1+e−λ

TABLE I: RHG regimes in (top) rescaled and (bottom) original variables. 6 function in Eg. (12) becomes the step function, and con- Hence, connection probabilities {pij} are fully deter- nections are established deterministically between node mined by angles ∆θij: pairs separated by distances smaller than µ. πτ 1 In this case sin(πτ) → 1 leading to pij = pij (∆θij) = ˜ , (51) 1 + exp dij (∆θij )−m 2d a 2 τ hki = ν , (44) dId,1 a − 1 ˜ dζ where dij (∆θij) ≡ dij (∆θij). n a − 1 2 hk(r)i = hkie−r. (45) Eﬀectively, in the a → ∞ regime nodes are placed ν a at the surface of the unit sphere Sd and connections are made with distance-dependent probabilities on the The resulting graphs are sparse and are characterized by sphere. Hence, RHG in the a → ∞ limit can be viewed the scale-free degree distribution P (k) ∼ k−γ , γ = a + 1. as the soft RGG on Sd.

B. τ → ∞ limit: Erd˝osRényi model D. a → ∞, τ → 0 limit: Random Geometric Graphs The case of τ → ∞, on the other hand, is not the special case of the hot regime. Indeed, it follows from If a → ∞ and τ → 0 connection probabilities in r+rl−m τ 1 Eq. (12) become Eq. (42) that e ∼ n 1 and the small angle approximation of sin(x) ≈ x in Eq. (15) no longer holds. Therefore, to infer the proper RHG behavior in the pij = Θ(θc − ∆θij), (52) τ → ∞ limit, we retreat back to Eq. (12) and set T → ∞. ˜ We observe that in this case the RHG degenerates to where θc is the solution to the equation dij(θc) = m. the Erd˝osRényi (ER) model with connection probability Thus, the RHG in the T → 0 limit becomes the sharp random geometric graph (RGG) on Sd. 1 d p → p = , (46) Since nodes are uniformly placed on S , resulting de- ij 1 + e−λ gree distribution of the RGG is binomial where n − 1 P (k) = p˜k [1 − p˜]n−k−1 . (53) λ ≡ lim ζµ/2T = m/τ, (47) k T →∞ Degree distribution of the HRG in this limit is binomial Herep ˜ is the probability for a randomly chosen node to end up within the angle of θc of the θ1, ..., θd point: n − 1 k n−k−1 P (k) = p [1 − p] , (48) R θc k−1 k [sin (θ)] dθ p˜ = 0 . (54) R π [sin (θ)]k−1 dθ and 0 n − 1 hki = (n − 1)p = , (49) Since binomial distribution at large k values decays 1 + e−λ faster than any power-law function k−γ with finite posi- tive γ we refer to this regime as γ = ∞ case in TableI. Since binomial distribution at large k values decays faster than any power-law function k−γ with finite posi- tive γ we refer to this regime as γ = ∞ case in TableI. E. ζ → ∞, τ → ∞ limit: Hyper Soft Configuration Model C. a → ∞ limit: Soft Random Geometric Graphs In the ζ → ∞ limit hyperbolic distances degenerate to In this limit radial node distribution degenerates to dij = ri + rj. (55) ρ(r) → δ (r − RH ) . (50) Further, if lim ζ = λ > 0, where λ is a constant, As a result, all nodes are placed at the boundary of the ζ→∞ τ d+1 connection probability in Eq. (12) simplifies to hyperbolic ball B with ri = RH . Even though the distances between nodes are still hyperbolic, they are fully d 1 determined by the angles on : pij = , (56) S 1 + eωi eωj " # 2R 2 2R 2 ζd = cosh−1 cosh H − sinh H sin (∆θ ) . resulting in the Hyper Soft Configuration Model ij d d ij dλ µ (HSCM). Here ωi = 2 ri − 2 are Lagrange multipliers 7 controlling expected node degrees. Lagrange multipliers single mathematical framework. Summarized in TableI, are drawn from effective pdf our results indicate that RHGs exhibit similar topological properties, regardless of the dimensionality of their 2α α µ −R 2α ω ρ(ω) = e ( 2 H )e dλ , (57) latent hyperbolic spaces. dλ While the dimensionality of the hyperbolic space does dλµ dλ µ ω ∈ − , R − . (58) not seem to significantly degree distributions of the RHG 4 2 2 models, we conjecture that higher-dimensional RHGs may be instrumental in mapping of real networks. One Expected degrees in the HSCM are approximated by of the standard mapping approaches is the Maximum Likelihood Estimation (MLE), finding node coordinates hk(ω )i = (n − 1)he−ωie−ωi , i of the network of interest by maximizing the likelihood −ω 2 (59) hki = (n − 1)he i that the network was generated as the RHG in the latent space. The likelihood function in the case of the B2 has By demanding that hk (ω(r = 0))i ∼ n and been shown to be extremely non-convex with respect to 2 hk (ω(r = RH ))i ∼ 1 we obtain RH = dλ ln(n), while node coordinates [15], making the standard learning tools 2α 2α µ = RH in the case of dλ > 1, and µ = dλ RH in the case like the stochastic gradient descent inefficient. Raising 2α d+1 of dλ < 1. the dimensionality of the H may lift some of the local In both cases hk(ω)i ∼ e−ω and graphs are character- maxima of the likelihood function, resulting in faster and ized by the scale-free degree distribution: more accurate network mappings.

P (k) ∼ k−γ , 2α (60) γ = + 1. dλ VII. ACKNOWLEDGEMENTS

VI. SUMMARY We thank F. Papadopoulos, M. A.´ Serrano, M. Bogu˜na, and P. van der Hoorn for useful discussions In our work we have generalized the RHG to arbitrary and suggestions. This work was supported by ARO dimensionality. In doing so, we have found the rescal- Grant No. W911NF-17-1-0491 and NSF Grant No. IIS- ing of network parameters, given by Eq. (17), that al- 1741355. M. Kitsak was additionally supported by the lows to reduce RHGs of arbitrary dimensionality to a NExTWORKx project.

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