On the largest component of subcritical random hyperbolic graphs

We consider the random hyperbolic graph model introduced by [KPK + 10] and then formalized by [GPP12]. We show that, in the subcritical case $\alpha$>1, the size of the largest component is n^{1/(2$\alpha$)+o(1)} , thus strengthening a result of [BFM15] which gave only an upper bound of n^{1/$\alpha$+o(1)}.


Introduction and statement of result
In the last decade, the model of random hyperbolic graphs introduced by Krioukov et al. in [KPK + 10] was studied quite a bit due to its key properties also observed in large real-world networks. In [BnPK10] the authors showed empirically that the network of autonomous systems of the Internet can be very well embedded in the model of random hyperbolic graphs for a suitable choice of parameters. Moreover, Krioukov et al. [KPK + 10] gave empiric results that the model exhibits the algorithmic small-world phenomenon established by the groundbreaking letter forwarding experiment of Milgram from the '60s [TM67]. From a theoretical point of view, the model of random hyperbolic graphs has an elegant specification and is thus amenable to rigorous analysis by mathematicians. Informally, the vertices are identified with points in the hyperbolic plane, and two vertices are connected by an edge if they are close in hyperbolic distance.
A common way of visualizing the hyperbolic plane is via its native representation described in [BKL + 17] where the choice for ground space is R 2 . Here, a point of R 2 with polar coordinates (r, θ) has hyperbolic distance to the origin O equal to its Euclidean distance r and more generally, the hyperbolic distance d h (u, u ) between two points u = (r u , θ u ) and u = (r u , θ u ) is obtained by solving cosh d h (u, u ) := cosh r u cosh r u − sinh r u sinh r u cos(θ u −θ u ).
(1) In the native representation, an instance of the graph can be drawn by mapping a vertex v to the point in R 2 with polar coordinate (r v , θ v ) and drawing edges as straight lines (see Figure 1). The random hyperbolic model is defined as follows: for each n ∈ N, we consider a Poisson point process on the disk B h (O, R) of the hyperbolic plane. The radius is equal to R := 2 log(n/ν) for some positive constant ν ∈ R + (log denotes here and throughout the paper the natural logarithm). The intensity function at polar coordinates (r, θ) for 0 ≤ r < R and 0 ≤ θ < 2π is g(r, θ) := νe where f (r, θ) is the density function corresponding to the uniform probability on the disk B h (O, R) of the hyperbolic space of curvature −α 2 , that is θ is chosen uniformly at random in the interval [0, 2π) and independently of r which is chosen according to the density function Make then the following graph G = (V, E). The set of vertices V is the points set of the Poisson process and for u, u ∈ V , u = u , there is an edge with endpoints u and u provided the distance (in the hyperbolic plane) between u and u is at most R, i.e., the hyperbolic distance d h (u, u ) between u and u is such that d h (u, u ) ≤ R, where d h (u, u ) is obtained by solving Equation (1) For a given n ∈ N, we denote this model by Poi α,ν (n). Note in particular that g(r, θ)dθdr = νe R 2 = n, and thus E|V | = n. In the original model of Krioukov et al. [KPK + 10], n points, corresponding to vertices, are chosen uniformly and independently in the disk B h (O, R) of the hyperbolic space of curvature −α 2 , but since from a probabilistic point of view it is arguably more natural to consider the Poissonized version of this model, we consider the latter one; see also [GPP12] for the construction of the uniform model. The restriction α > 1 2 and the role of R guarantee that the resulting graph has bounded average degree (depending on α and ν only). If α < 1 2 , then the degree sequence is so heavy tailed that this is impossible (the graph is with high probability connected in this case, as shown in [BFM16]). Moreover, if α > 1, then as the number of vertices grows, the largest component of a random hyperbolic graph has sublinear order (see [BFM15,Theorem 1.4

]).
Notations: We say that an event holds asymptotically almost surely (a.a.s.), if it holds with probability tending to 1 as n → ∞. Given positive sequences (a n ) n≥1 and (b n ) n≥1 taking values in R, we write a n = o(b n ) to mean that a n /b n → 0 as n → ∞. Also we write a n = Θ(b n ) if |a n |/|b n | is bounded away from 0 and ∞, and a n = Ω(b n ) if |a n |/|b n | is bounded away from 0. Result: In this paper we study the size of the largest component of the graph in the case α > 1. In [BFM15, Theorem 1.4] it was shown that its size is a.a.s. at most n 1/α+o(1) . The main result of this paper is the following improvement, finding the exact exponent: Theorem 1. Let α > 1 and ν ∈ R + . Let G = (V, E) be chosen according to Poi α,ν (n), and let L 1 ⊆ G be the largest connected component of G. There is a constant C > 0, such that, a.a.s., the following holds: Remark 2. A careful inspection of the proofs shows that all results hold with probability at least 1 − o(n −1/2 ), and hence a Depoissonization argument (see [Pen03] for details) shows that Theorem 1 also holds for the original uniform model.

Related work:
The size of the largest component in random hyperbolic graphs was first studied in [BFM15]: it was shown that for α > 1 it is at most n 1/α+o(1) , whereas for α < 1 the largest component is linear. In the same paper the authors also showed that for α = 1 and ν sufficiently small there is a.a.s. no linear size component, whereas for α = 1 and ν sufficiently large a.a.s. there is a linear size component. In [FM18] the picture was made more precise: for α = 1 there is a critical intensity such that a.a.s. a linear size component exists iff ν is above a certain threshold. Also, for α < 1, for fixed α the size of the largest component is increasing in ν, and for fixed ν, it is decreasing in α. Furthermore, in [BFM16] it was shown that for α < 1/2 the graph is connected a.a.s., whereas for α = 1/2 the probability of being connected tends to 1 if ν ≥ π, and the probability of being connected is otherwise a monotone increasing function in ν that tends to 0 as ν tends to 0. For the case 1/2 < α < 1, it was shown in [KM19] that a.a.s. the second component is of size Θ((log n) 1/(1−α) ), whereas for α = 1/2 and ν sufficiently small it is Θ(log n) with constant probability, and for α = 1 it is a.a.s. Ω(n b ) for some b > 0. Starting with the seminal work of [KPK + 10], further aspects of random hyperbolic graphs have been discussed since then: the power law degree distribution, mean degree and clustering coefficient were analyzed in [GPP12]; the diameter was computed in [FK15, KM15, MS19], the spectral gap was analyzed in [KM18], typical distances were calculated in [ABF17], and bootstrap percolation in such graphs was considered in [CF16]. First passage percolation of random hyperbolic graphs (or more generally, geometric inhomogeneous random graphs) was analyzed in [KL]. Organization of the paper: In Section 2 we recall some well known properties of the random hyperbolic graph. Section 3 then describes the construction of the main tool of our proof: the separation zones. The existence of these zones shows that there is no long path of vertices with all vertices having roughly the same radial coordinates. Finally, in Section 4 we use the separation zones to control the size of the connected components of the graph which leads to the result of Theorem 1.

Preliminaries
From now on, we suppose α > 1. In this section we collect some properties concerning random hyperbolic graphs. For notational convenience, for any point the radial distance to the boundary circle of radius R (instead of the distance to the origin O), and we identify a vertex v of the graph G with the coordinate pair v = (t v , θ v ). Moreover, we suppose throughout the paper that R is an integer.
By the hyperbolic law of cosines (1), the hyperbolic triangle formed by the geodesics between points p , p , and p, with opposing side segments of length d h , d h , and d h respectively, is such that the angle formed at p is: We state a very handy approximation for θ R (·, ·).
A direct consequence of this lemma is the following corollary: Corollary 4. For any R > 0, there is a function Throughout, we will need estimates for measures of regions of the hyperbolic plane, and more specifically, for regions obtained by performing some set algebra involving a few balls. For a point p of the hyperbolic plane H 2 , the ball of radius ρ centered at p will be denoted by B p (ρ), i.e., B p (ρ) := {q ∈ H 2 : d h (p, q) ≤ ρ}. Next, we collect a few standard results for such measures.
We also use classical Chernoff concentration bounds for Poisson random variables. See for instance ([BLM13] page 23).
Lemma 6 (Chernoff bounds). If X ∼ P(λ), then for any x > 0, In particular, for x ≥ λ, Lemma 6 together with Lemma 3.2 of [GPP12] yield the following lemma: Lemma 7. Let V be the vertex set of a graph chosen according to Poi α,ν (n), and let v be a vertex with t v > C log R for C sufficiently large. Then, a.a.s. |V ∩ B v (R)| = Θ(e 1 2 tv ).
The following observation is a simple consequence of Lemma 5: We then construct for each coordinate pair (t 0 , θ 0 ) ∈ (0, R/2) × [0, 2π), a zone that separates points to the left from points to the right in {(t, θ), t ≤ t 0 }. Precisely, define for t 0 < R/2 and θ 0 ∈ [0, 2π), the following separation zone: We thus have the following observation: t w ), i.e. v and w are not connected by an edge.
Proof. The function θ R (t v , t w ) is increasing in both of its arguments, hence we may assume that t v = t w = t. For this choice of t, v and w are connected by Corollary 4 iff |θ v − θ w | ≤ θ R (t, t). However, since A(t 0 , θ 0 ) = ∅ and θ v < θ 0 < θ w , we have |θ v − θ w | > 2θ R (t, t), i.e. v and w are not connected by an edge.

Figure 2: a separation zone
To use the previous observation, we need separation zones which do not contain any vertices. We prove below that this happens with large probability.
For different values of j, these sectors are disjoint, and thus the random variables |V ∩ A(t, 2jθ R (t, t))| are independent and Then, as t < R/2, Corollaries 4 and 8 give By choosing N = cR for some constant c > 0 sufficiently large (depending on K) the lemma follows.
We will now consider layers starting from the boundary of B O (R): set Let t max = 1 2α R (note that t max < R/2), the distance to circle of radius R roughly corresponding to the largest t for which we can find an element of V and set i max = min { i ≥ 0, t i ≥ t max }. We thus have i max ≤ R and t max ≤ t imax ≤ t max + 3 log R.
We also set t −1 = 0 and we define, for i, j ∈ {0, . . . , i max }, the angle and the consecutive layers Observation 11. For any i, j ∈ {0, . . . , i max }, We now define the following separation zones: for every i ∈ { 0, . . . , i max }, set k i max = 2π/(3cRθ i,i ) where c is the constant given in Lemma 10 for K = 1. For every 0 ≤ k < k i max , we find the (k + 1)-th separation zone to be the closest (to the right) empty region to the angle 3cRkθ i,i . More formally, define for 0 ≤ k < k i max , We assign A i,k then to be the closest region to 3cRkθ i,i : where min ∅ = ∞ and in this case A i,k = ∅. The set A i,k represents the (k + 1)-th separation zone of layer i. For notational convenience, we also set A i,k i max = A i,0 . We could have A i,k = A i,k+1 , and the two sets might not even be well defined. We will thus use Lemma 10 to show that asymptotically almost surely none of the two things happens.
In order to state the next lemma properly, we define the following (pseudo)distance between separation zones: Lemma 12. Let c be the constant given in Lemma 10 for K = 1 (depending only on α and ν). Then the event E R defined by Proof. Let c be the constant given in Lemma 10 for K = 1 and consider the event Clearly, F R ⊂ E R and it is sufficient to bound P F R . Then, for R large enough, using the definition of k i max , Since the last quantity goes to zero as R tends to infinity, the lemma is proven.
Hence, a.a.s. the distance between two consecutive separation zones A i,k and A i,k+1 is always of the order Rθ i,i . Define now, on E R , the area B i,k between two separation zones: for Figure 3: The separation zones We point out that every path of connected points from u ∈ V ∩ B i,k to v ∈ V ∩ B i, with k = has to go through a vertex w ∈ V such that t w > t i , i.e. points cannot be connected "below" as described in Figure 4. More formally, rewriting Observation 9 we obtain the following observation.
Then u and v can only be connected by a path that has at least one intermediate vertex w ∈ V such that t w > t i .
Proof. Note that t u , t v ≤ t i . Then by Observation 9, u and v are not connected by an edge. Since this holds for any k = and any u, v, there can be no path between u and v containing vertices w ∈ V such that t w ≤ t i (see also Figure 4). Figure 4: The green points are connected while the red ones are not.

Covering component
On a high level, the advantage of separation zones is that it is impossible to stay in the same connected component going from right to left (or the other direction) remaining always at the same radius or going towards the boundary. We will thus construct, starting from a certain vertex, a covering component, that is, a component which covers a.a.s. the whole connected component of the vertex if this vertex is the vertex closest to the center of its connected component.
We describe now in detail the iterative construction process of the covering component. Suppose that the event E R holds. This happens a.a.s. according to Lemma 12. Consider a vertex v ∈ V . If v is in the layer L 0 , we define and (see also Figure 5) Denote now by k the unique integer such that v ∈ L i ∩ B i,k . The covering component of v is then defined as Proof. Suppose that the event E R holds. This happens a.a.s. according to Lemma 12.
By contradiction, consider a vertex u in the connected component of v that is not contained in C v , and a shortest path v 0 = v, . . . , v m = u. Hence there exists a smallest k ≥ 1 such that the vertex v k is not in C v . Let L i k be the layer of v k .
Suppose now there exists k < k so that for some i k > i k , v k ∈ L i k . We may then choose the largest k , so that ∀ ∈ {k + 1, . . . , k } , ∃i ≤ i k , v ∈ L i (see Figure 6). Hence, the v are in the same zone B i k ,s as v k and as v k or in a layer closer to the boundary. Since v is by hypothesis the vertex such that t v = max { t u | u ∈ Conn(v)}, we must have v ∈ L i k . Therefore, by Observation 13, v and v k must be in the same zone B i k ,s for some s and thus v k ∈ C v .
Lemma 15. A.a.s., . For each 0 ≤ j < i ≤ i max , divide layer L j into π/(2θ i,j ) sectors of angle (at most) 4θ i,j . For any such sector S (i,j) k , for any 0 ≤ j < i ≤ i max , from Lemma 6 we have P |S By a union bound over all π/(2θ i,j ) sectors S (i,j) k and then over all i, j, we have Hence, since for each vertex v ∈ V ∩ L i , the set Θ i,j (v) can intersect at most two adjacent sectors S (i,j) k , we have P (∃0 ≤ j < i ≤ i max , ∃v ∈ V ∩ L i , |Θ i,j (v)| > 2d i,j ) = o(1).
Therefore, P E R ∩ ∃i ∈ {0, . . . , i max } , ∃k ∈ 1, . . . , k i max , |V ∩ L i ∩ B i,k | ≥ e By Lemma 14 we obtain the upper bound for |L 1 | in the theorem. For the lower bound, by Lemma 5, for any function ω tending to infinity with n arbitrarily slowly, µ(B O (r max + ω) 1/n, and hence a.a.s. we find a vertex v with t v ≥ t max − ω. In such case, the degree of v is, by Lemma 7, a.a.s. Θ(e 1 2 (tmax−ω)) ) = n 1 2α +o(ω/n) . The degree of a vertex is a lower bound on the size of its component, and hence Theorem 1 follows.