Metric Embedding, Hyperbolic Space, and Social Networks∗
Kevin Verbeek† Subhash Suri†
ABSTRACT matical content, these embeddings have also become a ma- jor tool in the analysis of high-dimensional data: the general We consider the problem of embedding an undirected graph idea is to embed a high-dimensional metric space into a much into hyperbolic space with minimum distortion. A funda- smaller-dimensional space without creating large distortion mental problem in its own right, it has also drawn a great in pair-wise distances. A celebrated result in this area is the deal of interest from applied communities interested in em- Johnson-Lindenstrauss Lemma [7], showing that metrics in- pirical analysis of large-scale graphs. In this paper, we es- duced by n points in an Euclidean space can be embedded tablish a connection between distortion and quasi-cyclicity 2 with (1 + ε) distortion into O(ε− log n)-dimensional space. of graphs, and use it to derive lower and upper bounds The reader may consult the book by Matouˇsek [13] for an on metric distortion. Two particularly simple and natu- excellent survey and exposition to many elegant results on ral graphs with large quasi-cyclicity are n-node cycles and metric embeddings. n n square lattices, and our lower bound shows that any In a different context, researchers in the networking com- hyperbolic-space× embedding of these graphs incurs a multi- munities have also explored geometric embeddings for net- plicative distortion of at least Ω(n/ log n). This is in sharp work measurements and geographic routing [8, 17, 18, 20]. contrast to Euclidean space, where both of these graphs can A number of network services use shortest path distances to be embedded with only constant multiplicative distortion. implement application level multicast trees or routes [8, 14, We also establish a relation between quasi-cyclicity and δ- 17]. While maintaining the full matrix of pairwise distances hyperbolicity of a graph as a way to prove upper bounds is infeasible in Internet scale networks, a geometric embed- on the distortion. Using this relation, we show that graphs ding gives a highly compact representation of those dis- with small quasi-cyclicity can be embedded into hyperbolic tances. Similarly, virtual coordinates and geographic rout- space with only constant additive distortion. Finally, we also ing are an elegant way to avoid the need for full network present an efficient (linear-time) randomized algorithm for topology at each network router. There is also a great deal of embedding a graph with small quasi-cyclicity into hyperbolic interest in estimating pair-wise distances in social networks, space, so that with high probability at least a (1 ε) frac- as a means to measure influence, to recommend friends, and tion of the node-pairs has only constant additive distortion.− to show selective profiles, but running all-pair shortest path Our results also give a plausible theoretical explanation for algorithms is infeasible at their scale. Fortunately, a number why social networks have been observed to embed well into of empirical studies have shown that simple geometric em- hyperbolic space: they tend to have small quasi-cyclicity. beddings are good enough for estimating (most) pair-wise network distances in large graphs [19, 20]. The embedding 1. INTRODUCTION dimension is typically small (e.g. 10), and therefore the dis- Metric embeddings have been a topic of intense research tance between any two nodes can be calculated in constant in many areas of computer science, including geometry, the- time. Interestingly, these experimental results also show ory, databases and machine learning, during the past few that hyperbolic-space embeddings lead to higher accuracy decades. Besides their fundamental nature and rich mathe- (lower distortion) than Euclidean space [18, 20]. Kleinberg’s well-known result [8] that every connected finite graph has ∗This research was partially supported by the National a greedy embedding into the hyperbolic plane, while many Science Foundation Grant CNS-1035917 and by DARPA GRAPHS (BAA-12-01). graphs do not admit such an embedding in the Euclidean plane, also underscores the relative advantage of hyperbolic †Department of Computer Science, University of California Santa Barbara [kverbeek|suri]@cs.ucsb.edu space. Some researchers have even argued that the “hidden space” of the Internet and social networks is naturally the negatively curved hyperbolic space [12]. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not Our work is a theoretical counterpart to this line of em- made or distributed for profit or commercial advantage and that copies bear pirical research. We are interested both in establishing the this notice and the full citation on the first page. Copyrights for components worst-case distortion bounds for hyperbolic space embed- of this work owned by others than ACM must be honored. Abstracting with ding in general, and in discovering useful characterizations credit is permitted. To copy otherwise, or republish, to post on servers or to of graphs that may explain the observed low distortion in redistribute to lists, requires prior specific permission and/or a fee. Request hyperbolic embedding of large-scale social networks. We permissions from [email protected]. SoCG’14, June 8–11, 2014, Kyoto, Japan. begin with some formal definitions necessary to formulate Copyright 2014 ACM 978-1-4503-2594-3/14/06 ...$15.00. the problem and state our results.
1 1.1 Metric Spaces and Embeddings Using this relation, we show that graphs with small quasi- A metric space is a tuple ( ,d) where is the underlying cyclicity can be embedded into hyperbolic space with only X X space and d: R is the metric (or distance func- constant additive distortion. Finally, we present an efficient tion) measuringX× theX distance→ between two elements of . (linear-time) randomized algorithm for embedding a graph X A metric embedding (or simply embedding)of( 1,d1)into with small quasi-cyclicity into hyperbolic space, so that at X ( 2,d2)isamapf : 1 2. In this paper, we are primar- least a (1 ε) fraction of the node-pairs have constant ad- X X →X − ily concerned with embedding the graph metric (V,dG)of ditive distortion, with high probability. an undirected graph G =(V,E)intok-dimensional hyper- k 1.3 Related Work bolic space (H ,dH), where dG measures the length of the The body of research on hyperbolic space and its connec- shortest path between two nodes in G, and dH is the stan- k dard metric of H . In all our embeddings, the dimension k tions to various network properties and dynamics is large. of the target space is a constant, independent of the size of Due to lack of space, we just mention a few papers that the input graph. When the metric is clear from the context, are algorithmic in nature and explore questions similar to we simply refer to it as or G. the ones discussed in our paper. In [11], Krauthgamer and The quality of an embeddingX is measured by the worst- Lee present algorithms for approximate nearest-neighbors case distortion over all the distances in the metric, either and low-stretch routing, as well as a PTAS for the Travel- in relative or absolute terms. In particular, let f be an ing Salesman Problem in hyperbolic space. Their results embedding of the metric space ( 1,d1) into the metric space also apply to δ-hyperbolic spaces. In [3], Chepoi et al. X ( 2,d2). We say that f has multiplicative distortion c, for present schemes for computing an additive approximation cX 1, if of the diameter, center, and radius of δ-hyperbolic spaces ≥ and graphs. They also show that several graph classes are d2(f(x),f(y)) d1(x, y) cd2(f(x),f(y)) x, y 1 (1) ≤ ≤ ∀ ∈X δ-hyperbolic and present a linear-time algorithm for approx- The embedding f has additive distortion c if imating trees of n-node δ-hyperbolic graphs with O(δ log n) additive distortion. In [16], Sarkar shows that every finite d1(x, y) d2(f(x),f(y)) c x, y 1 (2) 2 | − |≤ ∀ ∈X tree admits an embedding into H with only 1 + ε multi- All our lower bounds in Section 3 are shown using multi- plicative distortion, for any ε>0; for this result, however, plicative distortion, which implies similar (in fact, stronger) the input metric needs to be scaled by a factor that depends lower bounds on the additive distortion as well. Our up- on both ε and the maximum degree of the tree, which may per bounds and algorithms, however, produce results with be non-constant. Nonetheless, this suggests a natural ap- additive distortion. proach to embed graph metrics into hyperbolic space: first embed the graph metric into a tree metric, and then embed Remark. Some papers on embeddings allow the input the tree metric into 2 using the result of [16]. Some graph distances to be scaled before constructing the embedding, H metrics (e.g. the n-cycle), however, require Ω(n) distortion which may produce smaller distortion especially for the ad- when embedded into a tree metric, as shown in [15]. On the ditive measure. We also utilize such a scaling by a constant other hand, under a relaxed model where the target metric factor for our upper bounds and algorithms. We do not con- is not a single tree but rather a distribution over several tree sider scaling of distances by a non-constant factor because metrics, one can achieve an expected distortion O(log n), as our intention is to investigate the intrinsic difference between shown by Fakcharoenphol et al. [4]. In our paper, we only hyperbolic and Euclidean space: after all, hyperbolic space consider embedding the graph into a single target metric behaves like Euclidean space in the limit at small distances. (hyperbolic space), with the goal of minimizing the worst- For the sake of simplicity, we do not use scaling in our lower case distortion. bounds, although the results hold with any constant factor The hyperbolic-space embeddings have also proved use- scaling. Finally, we point out that, by definition, our em- ful for greedy routing: in [8], Kleinberg proves that every fi- beddings (for multiplicative distortion) are non-expanding: nite graph admits a greedy embedding into hyperbolic space, the distances in the target space are upper-bounded by the which is known not to be possible in Euclidean space. input space distances. This restriction does not influence our lower bounds but simplifies the presentation. 2. PRELIMINARIES 1.2 Results We begin by establishing several technical preliminaries We begin with a simple lower bound result: any embed- necessary for our proofs, including formal definitions of hy- ding of an n-node cycle in hyperbolic space has multiplica- perbolic space. Given two points x, y of a metric space tive distortion at least Ω(n/ log n). This may seem surpris- ( ,d), we use [x, y] to denote a shortest∈ pathX (geodesic) from ing since the cycle is trivially embedded in Euclidean space xXto y. A metric space is geodesic if, for every x, y ,all with distortion π/2. We extend this lower bound using the points on the shortest path from x to y are in , where∈X the concept of quasi-cyclicity: any graph with quasi-cyclicity n shortest path is seen as a continuous curve. AX graph met- has multiplicative distortion Ω(n/ log n) in hyperbolic space. ric is not geodesic, but can be made geodesic by adding all An example graph with large quasi-cyclicity but no large in- edges as continuous curves to the metric space. Hence, for a duced cycle is the √n √n regular lattice. This graph graph metric, [x, y] actually denotes a shortest path in the also embeds with constant× distortion in Euclidean space, geodesic version of the graph metric. but is shown to require Ω(√n/ log n) distortion in hyper- The distance between two sets A, B is written as ⊆X bolic space. d(A, B), that is, d(A, B)=minx A,y B d(x, y). ∈ ∈ Next, we establish a relation between quasi-cyclicity and The hyperbolic space is a type of non-Euclidean geometry δ-hyperbolicity (see Section 2 for a formal definition) of a with negative curvature. Formally speaking, real hyperbolic k graph as a way to prove upper bounds on the distortion. k-space H is a k-dimensional Riemannian manifold with
2 y metric space is δ-hyperbolic, then we assume that δ is a constant. The definitions of δ-hyperbolic metric spaces give only an upper bound on δ. When proving lower bounds, we say a metric space or graph is strictly δ-hyperbolic if it is δ-hyperbolic and not δ-hyperbolic for δ <δ. In this defini- tion we generally allow δ to depend on the input size (e.g., x the number of vertices of the graph metric). δ z 3. LOWER BOUNDS ON DISTORTION In this section, we prove that some simple graphs require Figure 1: A δ-thin triangle. large distortion in any hyperbolic space embedding. In the process we introduce the notion of quasi-cyclicity, which turns out to be closely related to the distortion. Let Cn constant sectional curvature 1. There are several models be the cycle graph with n nodes v0,...,vn 1, with edges − − of hyperbolic space, including the hyperboloid model, Klein of the form (vi,vi+1). We refer to the graph metric of Cn model, and Poincar´edisc/half-plane models. Our results as the cycle metric of size n, denoted by (Cn,dC ). Our are model-independent and rely only on general properties proof uses the following result of Knaster, Kuratowski and k of hyperbolic space. Because H is a manifold, angles locally Mazurkiewicz [10]. behave as in Euclidean space but, in contrast to Euclidean space, the angles of a triangle satisfy (α+β+γ) <πin hyper- Lemma 3.1 ([10]). Consider an n-dimensional simplex bolic space. The difference π (α+β+γ) is called the defect ∆n with vertices V = v0,...,vn ,andaclosedsetSv for of a triangle, and it equals the− area of the triangle in k. { } H each v V .If,foreachsubsetA V ,theunion v A Sv ∈ ⊆ ∈ One can define a more general notion of hyperbolic spaces covers the face of ∆n spanned by the vertices in A,then called δ-hyperbolic spaces. The following definition is due v V Sv = . to Gromov [6]. For a metric space ( ,d) and p, x, y , ∈ ∅ X ∈X k k the Gromov product of x and y at p, denoted by (x y)p is Lemma 3.2. Let p H and let P :[0, 1] H be a defined as: | ∈ → curve such that ∠P (0)pP (1) = α and dH(p, P (t)) R for 1 0 t 1.ThenthelengthofP is at least α sinh(R≥). (x y)p = (d(p, x)+d(p, y) d(x, y)) (3) ≤ ≤ | 2 − Proof. Our proof, which is straightforward but techni- A metric space ( ,d)isδ-hyperbolic for some δ 0 if the cal, is presented in Appendix A, to maintain the conceptual following holds forX all p, x, y, z : ≥ ∈X flow of the discussion. (x z)p min((x y)p, (y z)p) δ (4) | ≥ | | − The following lemma presents our key lower bound, namely, Another definition for geodesic metric spaces is due to Rips. that an n-cycle suffers a large distortion in any hyperbolic For three points x, y, z ,ageodesic triangle consists space embedding. of [x, y], [y, z], and [x, z].∈X A geodesic triangle is δ-thin if k for every p [x, y], we have d(p, [y, z] [x, z]) δ (and Lemma 3.3. Any embedding of Cn into H has multi- symmetrically∈ for p [x, z] and p [y, z∪]). See Fig.≤ 1. A n plicative distortion Ω( log n ). metric space is δ-hyperbolic∈ if all geodesic∈ triangles are δ- thin. These two definitions are equivalent upto a constant Proof. Let us assume n is a multiple of 3, and consider the nodes u1 = v0, u2 = v n , and u3 = v 2n . We have factor of δ, do not change the asymptotic bounds of our 3 3 n theorems, and therefore we use them interchangeably. dC (ui,uj )= 3 , for 1 i p p u1 1 1 p P α1 P2 3 Cn T β2 β3 α β1 u3 u2 α3 2 p3 p2 p3 p2 P P1 (a) (b) k Figure 2: (a) A metric embedding of Cn into H .(b)SubdividingT into three triangles. 3 2π First, suppose one of the angles, say α1,isatleast 3 . L βn k ≥ Consider the polygonal curve P :[0, 1] H that is the im- → k age of the geodesic segment [u2,u3]inH under f, where P (0) = p2 and P (1) = p3. Since distances do not ex- n pand under f, the length of P is at most 3 . On the other αL hand, by Lemma 3.2, the length of P is at least α1 sinh(R), ≥ where R is the minimum distance between p1 and P . Since x x sinh(x)=(e e− )/2, we get that if α1 sinh(R)=O(n), then R = O(log− n). Let p P be the point such that ∈ dH(p1,p)=R. Although p may not be the image of one of the nodes of Cn, there must exist a node u [u2,u3] such that d (p, f(u)) 1 (distances do not expand∈ under f), and H ≤ so dH(p1,f(u)) = O(log n) by the triangle inequality. This Figure 3: An (α,β)-quasi-cycle. leads to the desired lower bound: the node-pair u, u1 has n graph distance dC (u, u1) , but hyperbolic-space distance ≥ 3 O(log n), and therefore suffers a multiplicative distortion of such that the following inequality holds: n at least Ω( log n ). αdC (u, v) d(u, v) dC (u, v) u, v Cn, (5) The above argument also works if α1 =Ω(1). If none of ≤ ≤ ∀ ∈ the angles of triangle T is at least of size Ω(1), then we need where α defines the quasiness of the quasi-cycle. For techni- a different argument (recall that the sum of the angles of cal reasons we also introduce the notion of weak quasi-cycles a hyperbolic triangle is less than π, and possibly o(1)). In where the bound of Inequality (5) holds only if dC (u, v) this case, consider a point p in the interior of T , and the βn, for some constant 0 β 1 . See Figure 3 for illustra-≥ ≤ ≤ 2 three triangles formed by the geodesic segments [p, pi]. See tion. The constant β is the weakness of a quasi-cycle (for Fig. 2(b). Let βi be the angle at p opposite of pi. We define strong quasi-cycles, β = 0). We refer to strong quasi-cycles the closed set Si (1 i 3) as the set of points p such as α-quasi-cycles and weak quasi-cycles as (α,β)-quasi-cycles. 2π ≤ ≤ that βi 3 . It is easy to verify that the sets Si satisfy the conditions≤ of Lemma 3.1, and therefore there exist a Lemma 3.4. Any embedding of an (α,β)-quasi-cycle 2π 1 k point p such that βi for all 1 i 3. Since those (Cn,d),whereα>0 and β 3 ,intoH has multiplica- ≤ 3 ≤ ≤ n ≤ three angles sum to 2π, we must have β1 = β2 = β3 = tive distortion at least Ω( log n ). 2π k 3 .LetP1,P2,P3 :[0, 1] H be the polygonal curves Proof. The proof is analogous to that of Lemma 3.3. → k obtained by embedding [u2,u3], [u1,u3], and [u1,u2]into Given any embedding f of (Cn,d)intoH , there are two k n H , respectively. Using the same line of arguments as above, nodes u, v Cn whose distance in the cycle is dC (u, v) ∈ ≥ 3 there must be nodes v1 [u2,u3], v2 [u1,u3], and v3 but whose distance in hyperbolic space is d (f(u),f(v)) = ∈ ∈ ∈ H [u1,u2] such that d (p, f(vi)) = O(log n) for 1 i 3. By n H O(log n). By definition of quasi-cycles, we have d(u, v) α 3 ≤ ≤ 1 ≥ the triangle inequality, this implies that dH(f(vi),f(vj )) = if β 3 , and so the multiplicative distortion of node-pair O(log n) for 1 i 4 p3 r1 δ r2 ≥ s1 s2 δ 16 δ δ 4 4 q r1 r2 q q δ δ 1 3δ 2 8 q 8 ≥ 2 p1 q1 7δ q2 p2 ≥ 4 Case (i) Case (ii) Figure 4: The cases of Theorem 4.1. Theorem 4.1. If a geodesic metric space ( ,d) is strictly ist. Now the points q1,r1,s1,s2,r2, and q2 form a geodesic X1 7δ δ-hyperbolic, then ( ,d) must contain an (α, 3 )-quasi-cycle hexagon C with length at least 2 (see Fig. 4 right). Again of size Ω(δ) for someX constant α>0. we assume that the length of C is O(δ). We now show that if the distance along C between two points x, y C Proof. We prove the result using Rips’ definition of δ- δ δ ∈ is at least 2 , then d(x, y) 16 . This implies that C is hyperbolic metric spaces. By definition of strict δ-hyper- 1 ≥ 1 an (α, 3 )-quasi-cycle (actually, it is an (α, 7 )-quasi-cycle). bolicity, there must be three points p1,p2,p3 and a point δ ∈X It is easy to see that d([q1,q2], [r1,s1] [r2,s2]) and q [p1,p2] such that d(q, [p1,p3] [p2,p3]) = δ. Then, for 8 ∈ ∪ δ ∪ ≥ any given 0 x δ, there must exist points q1 [p1,q] and d([r1,s1], [r2,s2]) . The following cases remain (sym- ≤ ≤ ∈ ≥ 16 q2 [q, p2] such that d(q1, [p1,p3] [p2,p3]) = d(q2, [p1,p3] metric cases are ignored). ∈ ∪ ∪ [p2,p3]) = x. There are two cases to consider. (a) x [q1,q2] and y [q1,r1]. Because the distance along ∈ δ ∈ 3δ Case (i). There exists a point q1 [p1,q] such that C is at least 2 , we get that d(x, q1) 8 . Now, by the δ ∈ δ ≥ d(q1, [p2,p3]) = 4 . Note that we can switch p2 and p1 triangle inequality, d(x, y) 4 . without loss of generality, so this also covers the symmetric ≥ case. Let q [p ,q] be the point closest to q such that (b) x [q1,q2] and y [s1,s2]. By construction and the 1 1 ∈ ∈ δ ∈ δ triangle inequality we get that d(x, s1) d(x, y)+ d(q1, [p2,p3]) = . Define q2 in the same way, where q2 8 4 δ ≤ δ ≤ ∈δ d(y, s1) d(x, y)+ 16 . Thus d(x, y) 16 . [q, p2]. Let r1 [p2,p3] be the point such that d(q1,r1)= 4 ≤ ≥ ∈ δ and let r2 [p2,p3] be the point such that d(q2,r2)= (see ∈ 4 (c) x [q1,r1] and y [r1,s1]. Similar to Case (a). Fig. 4 left). The points q1,r1,r2, and q2 form a geodesic ∈ ∈ δ quadrilateral C. Since d(q, [p2,p3]) δ, the triangle in- (d) x [q1,r1] and y [s1,s2]. If d(x, y) 16 then 3δ ≥ 3δ ∈ ∈ δ ≤ equality implies that d(q, q1) 4 and d(q, q2) 4 , and d(q1,s2) d(q1,x)+d(x, y)+d(y, s1) 4 . That means 3δ ≥ ≥ ≤ ≤ δ hence d(q1,q2) 2 . This also means that d(r1,r2) δ, that we should be in Case (i), so d(x, y) 16 . ≥ 3δ ≥ ≥ since otherwise d(q1,q2) < 2 . Hence the length of C is at (e) x [q1,r1] and y [r2,s2]. Similar to Case (d). least 3δ. For now we assume that the length of C is also ∈ ∈ O(δ). We show that if the distance along C between two (f) x [q1,r1] and y [q2,r2]. By the triangle inequal- δ ∈ 7δ∈ points x, y C is at least δ, then d(x, y) 4 . This im- ity we get that d(q ,q ) d(q ,x)+d(x, y)+ ∈ ≥ 4 1 2 1 plies that C is an (α, 1 )-quasi-cycle. By construction we δ ≤ ≤ 3δ 3 d(y, q2) + d(x, y). Thus d(x, y) . δ ≤ 4 ≥ 2 directly get that d([q1,q2], [r1,r2]) 4 . By the triangle ≥ (g) x [r1,s1] and y [s1,s2]. Similar to Case (a). inequality we get that d([q1,r1], [q2,r2]) δ. Finally con- ∈ ∈ sider points x [q ,q ] and y [q ,r ]≥ such that the dis- 1 2 1 1 Thus, C is an (α, 1 )-quasi-cycle of size Ω(δ). tance between ∈x and y along C∈is at least δ. In that case 3 3δ Finally we consider the case that the length of C in Case (i) d(x, q1) 4 . Again, by the triangle inequality we get that 3δ ≥ δ or Case (ii) is ω(δ), or in particular, at least 37δ. This can d(x, q1) d(x, y)+d(y, q1) d(x, y)+ and hence 4 4 happen only if d(q1,q2), d(r1,s1), or d(r2,s2)isatleast12δ. ≤ δ ≤ ≤ d(x, y) 2 . The other cases are symmetric. Thus C is an Without loss of generality we assume that d(q1,q2) 12δ. 1 ≥ ≥ (α, )-quasi-cycle of size Ω(δ). Now it is easy to see that we can find points q ,q [q1,q2] 3 1 2 ∈ δ with the following properties: (1) 6δ d(q1 ,q2 ) 12δ, and Case (ii). We have d([p1,q], [p2,p3]) > and ≤ ≤ 4 (2) there exist r ,r [p1,p3]orr ,r [p2,p3] such that δ 1 2 1 2 d([p2,q], [p1,p3]) > .Letq1 [p1,q] be the point clos- δ ∈ ∈ 4 d(q1 ,r1 ),d(q2 ,r2 ) δ. (Note that, for each x [q1,q2], ∈ δ 8 ≤ ≤ ∈ est to q such that d(q1, [p1,p3]) = 8 . Define q2 in the either d(x, [p1,p3]) δ or d(x, [p2,p3]) δ.) Hence, the cy- δ ≤ ≤ same way, where q2 [q, p2] and d(q2, [p2,p3]) = 8 .Let cle C formed by q1,r1,r2,q2 has length Θ(δ) and is a scaled ∈ δ version of Case (i). Thus, C is an (α, 1 )-quasi-cycle of size r1 [p1,p3] be the point such that d(q1,r1)= 8 and let 3 ∈ δ Ω(δ). r2 [p2,p3] be the point such that d(q2,r2)= .Asin ∈ 8 Case (i) we can derive that d(q ,q ) 7δ . We also get that 1 2 4 Remark. The result easily extends to graph metrics of 3δ ≥ d(r1,r2) 2 . Instead of using the geodesic [r1,r2], we find unweighted graphs (which are not geodesic). This follows ≥ δ points s1 [r1,p3] and s2 [r2,p3] such that d(s1,s2)= 16 because the proof works for the geodesic version of the graph ∈ δ ∈ and d([r1,s1], [r2,s2]) . Note that these points must ex- metric, and all distances involved in the proof change by at ≥ 16 5 most 1 when considering the original graph metric (since Proof. The proof follows from Theorem 4.1 in combi- each edge has length 1). nation with the following result of Bonk and Schramm [2] (with our notation3): Any δ-hyperbolic metric space ( ,d) In the preceding theorem, we considered (α,β)-quasi-cycles X 1 with bounded growth at some scale admits a good additive with β = , which is sufficient to bound the quasi-cyclicity k 3 embedding into H . of a strictly δ-hyperbolic metric space. But we can play with the constants to obtain the same result for other values of β, leading to a tradeoffbetween α,β, and the size of the quasi- 5. A LINEAR TIME ALGORITHM FOR cycle. Theorem 4.1 also implies that, if the quasi-cyclicity of a metric space ( ,d)isO(1), then ( ,d)isδ-hyperbolic. HYPERBOLIC EMBEDDING (We remind the readerX again that byX convention δ is con- We now address the algorithmic question of efficiently em- stant for δ-hyperbolicity.) bedding graph metrics into hyperbolic space. Throughout this section we assume that the input is a δ-hyperbolic graph 4.1 An Upper Bound for Distortion metric G =(V,E) with bounded degree. Our goal is to com- pute an embedding of G into k with small distortion, for Unfortunately, not all δ-hyperbolic metric spaces embed H some constant k. well into k, and we need some additional assumptions to H The result by Bonk and Schramm [2] is constructive but achieve low distortion embeddings into hyperbolic space. not algorithmic. Since we are interested in embedding large- In Euclidean space, good embeddings are associated with scale graphs, such as social networks, their size rules out the bounded doubling dimension: for example, the beacon-based feasibility of quadratic-time algorithms. Our main result in embedding of [9] requires bounded doubling dimension to this section is a simple randomized algorithm that runs in achieve constant (multiplicative) distortion. However, linear O( V + E ) time, and achieves constant additive dis- bounded doubling dimension alone does not guarantee low tortion for| all| but| | an ε fraction of the node-pairs of G. The distortion: the cycle metric has bounded doubling dimen- idea of ensuring good distortion for most of the node-pairs sion yet it embeds very poorly into hyperbolic space. On is widely used in practice [17, 18, 20], with a rigorous theo- the other hand, if we assume δ-hyperbolicity then a weaker retical treatment by Kleinberg et al. [9] for Euclidean space. condition than doubling dimension, called bounded growth We also follow this beacon-approach, however, we cannot at some scale and discussed below, suffices for a good upper apply their algorithm directly because our input graphs do bound on distortion. not have bounded doubling dimension, and our target space Given a metric space ( ,d) and a point x , define is hyperbolic space instead of Euclidean space. the ball B (r)= y X d(x, y) r . The∈Xdoubling x Our algorithm is inspired by the results of [2]. The main dimension of ( ,d) is{ the∈ smallestX | k such≤ that} every ball in idea is to consider the so-called Gromov boundary of a metric ( ,d) of radiusXr can be covered by at most 2k balls of radius rX k space. The boundary ∂G of G is doubling, and the bound- .Wealsosay( ,d)is2 -doubling, or simply doubling,if k k k 1 2 X ary ∂H of H resembles R − . We can therefore use the k is constant.2 A metric space ( ,d) has bounded growth X beacon-approach of Kleinberg et al. to compute an embed- at some scale if there exist constants R>r>0 such that k ding f of ∂G into ∂H . We then use results of [2] to extend every ball of radius R can be covered by K balls of radius r k f to an embedding F of G into H (see Fig. 5; more details for some constant K. It is easy to see that this is a weaker are explained in Section 5.1). Intuitively, the boundary of a condition than doubling. metric space ( ,d) can be seen as the set of geodesic rays In the context of graph metrics, it turns out that bounded of originatingX from a fixed base point in . Thus, f is node degree implies bounded growth at some scale, and so anX embedding of the set of geodesic rays of GXinto the set of that is the assumption we make for our graphs. (The k geodesic rays of H , and F extends this embedding to points bounded degree assumption is typically satisfied in most on the geodesic rays. Interestingly, if f is a good multiplica- practical graphs, including social networks, but can also tive embedding, then F is a good additive embedding. be theoretically justified: we argue in Lemma A.1 (Ap- Our algorithm consists of the following steps: pendix A) that without the bounded degree assumption, one cannot achieve constant (multiplicative) distortion, in 1. Compute the boundary ∂G. constant-dimensional hyperbolic space using only a constant factor scaling.) Finally, for our upper bound, we also allow 2. Compute an embedding f of ∂G into Euclidean space a constant-factor scaling of the input metric: this affects k 1 neither our lower bounds nor the intrinsic behavior of hy- (R − ). perbolic space. k For the sake of brevity, we define a good additive (multi- 3. Extend f to an embedding F of G into H . plicative) embedding as an embedding with constant additive (multiplicative) distortion subject to rescaling of the metric For step (3) we use the results of [2], which we describe in by only a constant factor. We can now state the main result Section 5.1. Since ∂G may contain a quadratic number of of this section. node-pairs, we cannot compute ∂G in linear time. Therefore, the key challenge is to compute f without fully constructing Theorem 4.2. Let G be a graph metric with bounded de- ∂G. Before we can describe this algorithm in detail, we first gree and constant quasi-cyclicity. Then, there exists a good need to review the key concepts used in the result of [2]. k additive embedding of G into H . 3The result of [2] actually requires the metric space to be 2The doubling dimension is closely related to the Assouad geodesic, but this restriction can be ignored using [2, Theo- dimension, see [1]. A metric space is doubling if and only if rem 4.1]. We point out that our graph metrics are 1-almost it has finite Assouad dimension. geodesic using their notation. 6 ∂Hk Hk G f k k 1 ∂G ∂H R − ≈ b b k 1 k G Con(∂G) Con(R − ) H g fˆ h F = h fˆ g ◦ ◦ k Figure 5: A schematic representation of the embedding of G into H . 5.1 Embedding δ-hyperbolic Metric Spaces was shown in [5, p. 285]. k into H Lemma 5.1. There exists a constant 0 such that, if is X Gromov boundary. The boundary ∂ of a δ-hyperbolic δ-hyperbolic and δ 0,then X ≤ metric space ( ,d) is defined as follows. Given a point 1 X (x y)b (x y)b b , we say a sequence xi converges at infin- e− | d∂ (x, y) e− | x, y ∂ ∈X { }⊆X 2 ≤ X ≤ ∀ ∈ X ity if limi,j (xi xj )b = , where (y z)x is the Gromov →∞ | ∞ | Throughout the rest of the paper, we simply write d∂ for product of y and z at x. Two sequences xi and { }⊆X the boundary metric, implicitly assuming that our choiceX of yi that converge at infinity are called equivalent if { }⊆X satisfies Lemma 5.1, since all other relevant properties of limi (xi yi)b = . The elements of ∂ are the equiva- →∞ | ∞ X d∂ do not depend on the choice of b or . The following lence classes of sequences that converge at infinity. As men- X tioned above, the elements of ∂ correspond (roughly) to property of the boundary is crucial. X 2 geodesic rays originating from b. An example with = H Lemma 5.2 ([2, Proposition 6.2 & Theorem 9.2]). X (in the Poincar´edisc model) is shown on the right of Fig- If ( ,d) is δ-hyperbolic and has bounded growth at some X ure 5. Using limits we can easily extend the definition scale, then (∂ ,d∂ ) is doubling and has a bounded diame- X X of the Gromov product (y z)x to the boundary ∂ ,i.e., ter. y, z ∂ and x (see| [2] for details). X The∈X notion∪ X of a boundary∈X does not make sense for finite Hyperbolic extension. One can also define a metric space graphs, which are our primary interest, and so we extend a that acts as the “opposite” of the boundary. Let ( ,d)bea X graph metric G as follows. Let b V be an arbitrary node, bounded metric space with diameter D. The hyperbolic ex- ∈ called the base of G, and consider the shortest path tree of tension Con( )of( ,d) is the metric space ( (0,D],de) X X X× G rooted at b.LetL(G) V be the set of leaves of this tree. where ⊂ To each leaf u L(G) we attach a chain of nodes γu that d(x, x)+max(h, h) ∈ d ((x, h), (x,h)) = 2 log (7) extends to infinity, and let G be the resulting metric space e √hh (see Fig. 5 left). Each element of ∂G can be identified with for (x, h), (x,h) (0,D] γu for some u L(G). The definition of the Gromov product ∈ ∈X× implies that (γu γv)b =(u v)b, and so, considering only the It can easily be shown that Con( )isδ-hyperbolic. Fur- | | Gromov product, we can simply identify the boundary ∂G thermore, there exists a good additiveX embedding of into with L(G).4 Con(∂ ) and vice versa (see [2, Theorem 8.2]). Finally,X if X Returning to general boundaries, we can define a metric there is a good multiplicative embedding f of 1 into 2, X X d∂ = db, on a boundary ∂ . This metric depends on the then we can find a good additive embedding fˆ of Con( 1) X X choice of the base point b and a constant >0, and is into Con( ) (see [2, Theorem 7.4]). X ∈X 2 defined as follows: We areX now ready to describe step (3) of our algorithm. k m Given a good multiplicative embedding f of ∂G into ∂H k 1 ≈ (xi 1 xi)b − ˆ d∂ (x, y)=inf e− − | for x, y ∂ (6) R , we can compute a good additive embedding f of X ∈ X k i=1 Con(∂G) into Con(∂H ). We can further compute good ad- ditive embeddings g and h from G into Con(∂G) and from k k where the infimum is taken over all finite sequences x = Con(∂H )intoH , respectively. The extension F of f is x0,x1,...,xm = y in ∂ , and where e−∞ = 0. Note that ˆ X simply given by F = h f g, which is a good additive d∂ depends only on Gromov products in ∂ , and hence k ◦ ◦ X embedding of G into H (see Fig. 5). Furthermore, the em- extends to the set of leaves L(G) of a (finite)X graph metric bedding F can be computed using f in linear time, since fˆ, G, as described above. The following important property g, and h are explicitly stated in [2].5 4Using the terminology of [2], our construction implies that 5These embeddings require diameters of ∂G and f(∂G), the set of geodesic rays starting from b is cobounded in the which cannot be computed in linear time, but an approx- geodesic version of the graph metric (since each node is on imation of the diameters suffices for our purpose. In partic- a geodesic ray). This avoids the method employed in [2] ular, a 2-approximation of the diameter D of ( ,d)iseasily to deal with finite metrics, and results in a more natural computed in linear time, since D 2maxy Xd(x, y) 2D approach to deal with unweighted graph metrics. for all x . ≤ ∈X ≤ ∈X 7 δ Finally we can compute d¯(u, v)=minx w(u, x)+w(x, v) 5.2 Beacon-based Embedding for -hyperbolic ∈B Graphs for u L(G) and v . This is correct, since the ∈ \B ∈B ˜ In this section, we show how to efficiently compute the em- first node (after u) of the shortest path in G between u k 1 and v must be in (it can be v itself). This step runs in bedding f of ∂G into R − . In order to compute f, we adapt O( 2 L(G) ) time.B Since = O(1), the algorithm runs in the algorithm of Kleinberg et al. [9], where a fixed number of |B| | | B nodes (beacons) are used to embed a doubling metric space linear time. into a fixed-dimensional Euclidean space. Since we cannot ¯ efficiently compute a metric on ∂G, we need a modified ap- The metric d is the one for which we can compute the proach. Before we describe our method, however, let us beacon-based embedding of [9] in linear time. Because d¯(u, v) d∂G(u, v) 2d¯(u, v), the embedding also achieves briefly review the beacon-based scheme of [9]. ≤ ≤ Given a doubling metric space ( ,d), we pick a constant constant multiplicative distortion with respect to the metric number of beacons , and considerX only the distances d∂G(u, v), for all but an ε fraction of node-pairs in ∂G.Letf between pairs in EB⊆= X . Kleinberg et al. [9] show be this embedding, and let F be its extension, as described B B×X in Section 5.1. The embedding f essentially maps geodesic that, for a suitably chosen set ,iff is a good multiplicative k embedding of into EuclideanB space for pairs in E , then it rays (or shortest paths) in G =(V,E) to geodesic rays in H . X B For u ∂G,letA(u) be the set of nodes v V such that v is also a good multiplicative embedding on most pairs of . ∈ ∈ This can be formalized by the concept of an (ε,β)-base.AX is on the geodesic ray corresponding to u. Note that v can set of beacons is an (ε,β)-base if for all but an ε-fraction be on multiple geodesic rays; in that case we can assign v to of the pairs (u,B v) , there exists a b such that one of those rays arbitrarily. If f has constant multiplica- ∈X×X ∈B tive distortion on a pair (u1,u2) for u1,u2 dG, then F has min(d(u, b),d(v, b)) βd(u, v). If is s-doubling, then a ∈ random sample of O≤( 1 log 1 )( 2 )2logXs beacons is an (ε,β)- constant additive distortion on all pairs in A(u1) A(u2). ε ε β So, if A(u ) and A(u ) are both very large, and× (u ,u ) base with high probability. In addition, if is an (ε,β)- 1 2 1 2 is a pair with bad distortion, we may get many pairs with base, and f is an embedding of into EuclideanB space with bad distortion under F . To remedy this issue, we sample multiplicative distortion ∆ X1 on E , then f has O(∆) 4β B beacons based on the size of A(u), i.e., the probability of multiplicative distortion on≤ all but an ε fraction of node- choosing u ∂G as a beacon is A(u) / V . We now obtain pairs in . Furthermore, f can be computed efficiently using the following∈ result. | | | | only theX distance pairs in E . B We now describe how to utilize this beacon approach to Theorem 5.4. Given a δ-hyperbolic graph metric G with our setting. We pick an arbitrary node b V of the graph bounded degree, and any constant ε>0,wecancomputein G, and compute the shortest path tree rooted∈ at b, which linear time an embedding F of G into k such that, with takes linear time using breadth-first search. We identify H high probability, F is a good additive embedding for all but ∂G with the set of leaves L(G), using the metric in Equa- an ε fraction of all node-pairs in G. tion 6. We then randomly pick a constant number of bea- cons L(G), and compute distances from the beacons Proof. We use the algorithm described in Section 5.2. to allB the⊆ nodes of V , which also takes linear time. Let The running time directly follows from Lemma 5.3 and the E = L(G). Using the shortest paths trees from the discussion in Section 5.2. We need to show that F achieves B B× beacon nodes, we can also compute (u v)b for (u, v) E . constant additive distortion for all but an ε fraction of dis- | ∈ B k 1 ˜ 1 (u v)b tance pairs in G.Letf be the embedding of ∂G into − Let d(u, v)= 2 e− | , for small enough. By Lemma 5.1 R ˜ ˜ ˜ used to construct F . Now imagine replacing each u dG we have d(u, v) d∂G(u, v) 2d(u, v). Unfortunately d ∈ may not be a metric,≤ but we can≤ easily fix that problem by by the set A(u) placed arbitrarily close to the original u in considering the weighted graph G˜ =(L(G),E ) with the the metric space, without changing the doubling constant B weights given by d˜, using the shortest path metric d¯ on G˜. (we can use a metric representing points on a line for the distances among A(u)). Let ( ,d) be the resulting metric X ¯ space. We can run the beacon-based algorithm on ;letf Lemma 5.3. We can compute d for (u, v) E in linear X ¯ ¯ ∈ B be the corresponding embedding. Let be a sample set of time and d(u, v) d∂G(u, v) 2d(u, v). B ≤ ≤ beacons of (chosen uniformly to construct f ), and let X B Proof. First, it is clear that d¯(u, v) d˜(u, v) d∂G(u, v). be the corresponding sample set of beacons of ∂G (to con- ≤ ≤ Now let u = u0,u1,...,um = v be the shortest path from u struct f) such that u if A(u) = . Note that the ∈B ∩B ∅ to v in G˜. Then we get the following: probability of choosing is the same as the probability of choosing (assuming weB can choose the same beacon mul- m m B d∂G(u, v) d∂G(ui 1,ui) 2 d˜(ui 1,ui)=2d¯(u, v) tiple times), since the probability of choosing u ∂G as a ≤ − ≤ − beacon is A(u) / . ∈ i=1 i=1 | | |X | Let u1,u2 ∂G where u1 = u2, and let v1 A(u1) and ¯ ˜ ∈ ∈ Next we compute d by updating the weights w(u, v)ofG. v2 A(u2). We know that all but an ε fraction of distance ∈ First we compute new weights defined by pairs (v1,v2)of have constant multiplicative distortion X w(u, v)=minx L(G) w(u, x)+w(v, x) for u, v . This under embedding f . Let∆be the multiplicative distortion ∈ 2 \B ∈B step runs in O( L(G) ) time. If w(u, v) 8 O(∆) multiplicative distortion under f. As a result, the of δ-hyperbolic geodesic spaces and graphs. In Proc. corresponding pair (v1,v2)ofG must have constant additive 24th Annual Symposium on Computational Geometry distortion under F . Thus, F is a good additive embedding (SoCG ’08), pp. 59–68, 2008. for all but an ε fraction of all node-pairs in G. [4] J. Fakcharoenphol, S. Rao, and K. Talwar. A tight bound on approximating arbitrary metrics by tree Remark. An astute reader may have noticed that our em- metrics. Journal of Computer and System Sciences, bedding algorithm requires knowledge of δ of the graph G, 69(3):485–497, 2004. just as the beacon-based Euclidean embedding of [9] requires [5] E. Ghys and P. de la Harpe (eds.). Sur les groupes knowledge of the doubling constant of the input metric. In Hyperboliques d’apr`es Mikhael Gromov, volume 83 of practice, the value of δ is neither known nor computable in Progress in Mathematics. Birkh¨auser, Boston, 1990. linear time, but empirical evidence suggests that it is typi- [6] M. Gromov. Hyperbolic groups. Mathematical cally small, and so randomly choosing a sufficiently large but Sciences Research Institute Publications, Springer, constant number of beacons ∂G is enough for comput- 8:75–263, 1987. B⊆k ing an embedding F of G into H . Indeed, our mathematical [7] W. B. Johnson and J. Lindenstrauss. Extensions of framework can be viewed as lending theoretical justification Lipschitz mappings into a Hilbert space. for beacon-based hyperbolic-space embeddings, like the one Contemporary Mathematics,26:189–206,1984. implemented in [20]. [8] R. Kleinberg. Geographic routing using hyperbolic space. In Proc. 26th INFOCOM, pp. 1902–1909, 2007. 6. CONCLUSION [9] J. Kleinberg, A. Slivkins, and T. Wexler. In this paper, we explored the problem of embedding undi- Triangulation and embedding using small sets of rected graphs into hyperbolic space with low distortion. The beacons. Journal of the ACM,56(6):32,2009. wide-spread use of hyperbolic space for network analysis and [10] B. Knaster, C. 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Lower bounds on the Our work suggests a number of open problems and re- distortion of embedding finite metric spaces in graphs. search directions. First, is the converse of Theorem 4.1 also Discrete & Computational Geometry,19(1):79–94, true, i.e., is an (α,β)-quasi-cycle of size n always strictly 1998. Ω(n)-hyperbolic? This can easily be shown for α> 1 , but 2 [16] R. Sarkar. Low distortion Delaunay embedding of the problem is open for other values of α (and β). On a re- trees in hyperbolic plane. In Proc. 19th International lated note, Theorem 3.5 and its counterpart Theorem 4.2 are Symposium on Graph Drawing, pp. 355–366, 2012. not exactly converses. Is it possible to obtain a tight upper bound on the distortion as a function of the quasi-cyclicity [17] Y. Shavitt and T. Tankel. Big-bang simulation for of a metric space? Finally, it would be interesting to eval- embedding network distances in Euclidean space. uate how well our embedding algorithm works in practice. ACM Trans. on Networking,12(6):993–1006,2004. Can it improve upon existing, more direct algorithms to em- [18] Y. Shavitt and T. Tankel. Hyperbolic embedding of bed graphs into hyperbolic space, like the one implemented internet graph for distance estimation and overlay in [20]? construction. ACM Trans. on Networking, 16(1):25–36, 2008. [19] X. Zhao, A. Sala, C. Wilson, H. Zheng, and 7. REFERENCES B. Y. Zhao. Orion: shortest path estimation for large n [1] P. Assouad. Plongements lipschitziens dans R . social graphs. In Proc. 3rd Workshop on Online Social Bulletin de la Soci´et´eMath´ematique de France, Networks, pp. 9–9, 2010. 111:429–448, 1983. [20] X. Zhao, A. Sala, H. Zheng, and B. Y. Zhao. Efficient [2] M. Bonk and O. Schramm. Embeddings of Gromov shortest paths on massive social graphs. In Proc. 7th hyperbolic spaces. Geometric & Functional Analysis, International Conference on Collaborative Computing, 10(2):266–306, 2000. pp. 77–86, 2011. [3] V. Chepoi, F. Dragan, B. Estellon, M. Habib, and Y. Vax`es. Diameters, centers, and approximating trees 9 APPENDIX A. ADDITIONAL PROOFS k k Lemma 3.2. Let p H and let P :[0, 1] H be a ∈ → curve such that ∠P (0)pP (1) = α and dH(p, P (t)) R for 0 t 1.ThenthelengthofP is at least α sinh(R≥). ≤ ≤ Proof. As for Euclidean space, we can define polar coor- k dinates for H , using the distance and angles w.r.t. a fixed k point in H . We need the formula to compute the arc length of a curve P (t)=(r(t),φ(t)) (0 t 1) in polar coordinates 2 ≤ ≤ in H , which is as follows: 1 dr 2 dφ 2 P = +sinh2(r(t)) dt (8) | | dt dt 0 Now consider the hyperbolic plane through p, P (0), and P (1). Choose polar coordinates (r, φ) for this hyperbolic plane such that p is at the center (r = 0). We can extend k these polar coordinates to H , but the remaining coordinates can be ignored for our bound. If we write P (t)=(r(t),φ(t)), then we obtain the claimed bound using Equation 8: 1 dr 2 dφ 2 P +sinh2(r(t)) dt | |≥ dt dt 0 1 dφ 2 sinh2(R) dt ≥ dt 0 φ(1) = sinh(R) dφ φ(0) α sinh(R) ≥ Lemma A.1. Consider the metric space (Sn,λdS ),where Sn is the star graph with center v0 and leaves v1,...,vn, and λ is a scaling factor of the standard metric. If there ex- k ists an embedding f of (Sn,λdS ) into H with only constant multiplicative distortion, then λk =Ω(logn). k Proof. Let pi = f(vi) be the embedded points in H . Since f is non-expansive, we get that dH(p0,pi) λ for 1 i n. If the embedding f has multiplicative distortion≤ ≤ ≤ k λ ρ, then the balls in H with radius ρ centered at the points pi (1 i n) must be interior disjoint. Furthermore, all these≤ balls≤ must lie inside the ball with radius 2λ centered k at p0. The volume of a ball with radius r in H is Vr = r k 1 G(k) 0 sinh − (s)ds, where G is a function depending only Θ(rk) on k. This can also be written as Vr = G(k)2 . Since ρ λ is constant and n balls with radius ρ must fit in a ball with radius 2λ, there must be constants c2 >c1 > 0 such that the following inequality holds: G(k)n2c1λk G(k)2c2λk ≤ log n + c1λk c2λk ≤ From the above inequality we can directly conclude that λk =Ω(logn). 10