Diameter in ultra‐small scale‐free random graphs

It is well known that many random graphs with infinite variance degrees are ultra‐small. More precisely, for configuration models and preferential attachment models where the proportion of vertices of degree at least k is approximately k −(τ − 1) with τ ∈ (2,3), typical distances between pairs of vertices in a graph of size n are asymptotic to 2loglogn|log(τ−2)| and 4loglogn|log(τ−2)|, respectively. In this paper, we investigate the behavior of the diameter in such models. We show that the diameter is of order loglogn precisely when the minimal forward degree d fwd of vertices is at least 2. We identify the exact constant, which equals that of the typical distances plus 2/logdfwd. Interestingly, the proof for both models follows identical steps, even though the models are quite different in nature.


Configuration model and main result
The configuration model CM n is a random graph with vertex set [n] ∶= {1, 2, … , n} and with prescribed degrees. Let = ( 1 , 2 , … , n ) be a given degree sequence, that is, a sequence of n positive integers with total degree n = ∑ i∈ [n] i , (1.1) assumed to be even. The configuration model (CM) on n vertices with degree sequence is constructed as follows: Start with n vertices and i half-edges adjacent to vertex i ∈ [n]. Randomly choose pairs of half-edges and match the chosen pairs together to form edges. Although self-loops may occur, these become rare as n → ∞ (see eg, [2,Theorem 2.16], [19]). We denote the resulting multi-graph on [n] by CM n , with corresponding edge set  n . We often omit the dependence on the degree sequence , and write CM n for CM n ( ).

Regularity of vertex degrees
Let us now describe our regularity assumptions. For each n ∈ N we have a degree sequence (n) = ( (n) 1 , … , (n) n ). To lighten notation, we omit the superscript (n) and write instead of (n) or ( (n) ) n∈N and i instead of (n) i . Let (p k ) k∈N be a probability mass function on N. We introduce the empirical degree distribution of the graph as We can define now the degree regularity conditions: Let F ,n be the distribution function of (p (n) k ) k∈N , that is, for k ∈ N, 1 { i ≤k} . (1.5) We suppose that satisfies the d.r.c. (a) and (b) with respect to some probability mass function (p k ) k∈N , corresponding to a distribution function F.

Condition 1.2 (Polynomial distribution condition)
We say that satisfies the polynomial distribution condition with exponent ∈ (2, 3) if for all > 0 there exist = ( ) > 1 2 , c 1 ( ) > 0 and c 2 ( ) > 0 such that, for every n ∈ N, the lower bound (1.6) holds for all x ≤ n , and the upper bound holds for all x ≥ 1.
There are two examples that explain Condition 1.2. Consider the case of i.i.d. degrees with P (D i > x) = cx −( −1) , then the degree sequence satisfies Condition 1.2 a.s. A second case is when the number of vertices of degree k is n k = ⌈nF(k)⌉ − ⌈nF(k − 1)⌉, and 1 − F(x) = cx −( −1) . Condition 1.2 allows for more flexible degree sequences than just these examples.
If we fix < min{ , 1 −1+ }, the lower bound (1.6) ensures that the number of vertices of degree higher than x = n is at least n 1− ( −1+ ) , which diverges as a positive power of n. If we take > 1 2 , these vertices with high probability form a complete graph. This will be essential for proving our main results. The precise value of is irrelevant in the sequel of this paper.
For an asymptotic degree distribution with asymptotic probability mass function (p k ) k∈N , we say that min = min {k ∈ N ∶ p k > 0} (1.8) is the minimal degree of the probability given by (p k ) k∈N . With these technical requests, we can state the main result for the configuration model: In fact, the result turns out to be false when p 1 + p 2 > 0, as shown by Fernholz and Ramachandran [12] (see also van der Hofstad and coworkers [17]), since then there are long strings of vertices with low degrees that are of logarithmic length.

Preferential attachment model and main result
The configuration model presented in the previous section is a static model, because the size n ∈ N of the graph was fixed.
The preferential attachment model instead is a dynamic model, because, in this model, vertices are added sequentially with a number of edges connected to them. These edges are attached to a receiving vertex with a probability proportional to the degree of the receiving vertex at that time plus a constant, thus favoring vertices with high degrees.
The idea of the preferential attachment model is simple, and we start by defining it informally. We start with a single vertex with a self-loop, which is the graph at time 1. At every time t ≥ 2, we add a vertex to the graph. This new vertex has an edge incident to it, and we attach this edge to a random vertex already present in the graph, with probability proportional to the degree of the receiving vertex plus a constant , which means that vertices with large degrees are favored. Clearly, at each time t we have a graph of size t with exactly t edges.
We can modify this model by changing the number of edges incident to each new vertex we add. If we start at time 1 with a single vertex with m ∈ N self loops, and at every time t ≥ 2 we add a single vertex with m edges, then at time t we have a graph of size t but with mt edges, that we call PA t (m, ). When no confusion can arise, we omit the arguments (m, ) and abbreviate PA t = PA t (m, ). We now give the explicit expression for the attachment probabilities.
(1. 10) where PA t,j−1 is the graph after the first j − 1 edges of vertex t have been attached, and correspondingly D t,j−1 (v) is the degree of vertex v. The normalizing constant c t,j in (1.10) is (2 + ∕m) + 1 + ∕m. (1.11) We refer to Section 4.1 for more details and explanations on the construction of the model (in particular, for the reason behind the factor j ∕m in the first line of (1.10)).
Consider, as in (1.2), the empirical degree distribution of the graph, which we denote by P k (t), where in this case the degrees are random variables. It is known from the literature [5,13] that, for every k ≥ m, as t → ∞, (1.12) where p k ∼ ck − , and = 3 + ∕m. We focus on the case ∈ (−m, 0), so that PA t has a power-law degree sequence with power-law exponent ∈ (2, 3).
In the proof of Theorem 1.5 we are also able to identify the typical distances in PA t : Theorem 1.6 (Typical distance in the preferential attachment model) Let V t 1 and V t 2 be two independent uniform random vertices in [t]. Denote the distance between V t 1 and V t 2 in PA t by H t . Then (1.14) Theorems 1.5 and 1.6 prove [17, Conjecture 1.8].

Structure of the paper and heuristics
The proofs of our main results on the diameter in Theorems 1.3 and 1.5 have a surprisingly similar structure. We present a detailed outline in Section 2 below, where we split the proof into a lower bound (Section 2.1) and an upper bound (Section 2.2) on the diameter. Each of these bounds is then divided into 3 statements, that hold for each model. In Sections 3 and 4 we prove the lower bound for the configuration model and for the preferential attachment model, respectively, while in Sections 5 and 6 we prove the corresponding upper bounds. In Caravenna and coworkers [6,Appendix], some proofs of technical results that are minor modifications of proofs in the literature are presented in detail. Even though the configuration and preferential attachment models are quite different in nature, they are locally similar, because for both models the attachment probabilities are roughly proportional to the degrees. The core of our proof is a combination of conditioning arguments (which are particularly subtle for the preferential attachment model), that allow to combine local estimates in order to derive bounds on global quantities, such as the diameter.
Let us give a heuristic explanation of the proof (see Figure 1 for a graphical representation). For a quantitative outline, we refer to Section 2. We write PA n instead of PA t to simplify the exposition, and denote by fwd the minimal forward degree, that is fwd = min − 1 for the configuration model and fwd = m for the preferential attachment model.
• For the lower bound on the diameter, we prove that there are so-called minimally connected vertices.
These vertices are quite special, in that their neighborhoods up to distance k − n ≈ log log n∕ log fwd are trees with the minimal possible degree, given by fwd + 1. This explains the first term in the right hand sides of (1.9) and (1.13).
Pairs of minimally connected vertices are good candidates for achieving the maximal possible distance, that is, the diameter. In fact, the boundaries of their tree-like neighborhoods turn out to be at distance equal to the typical distance 2k n between vertices in the graph, that is 2k n ≈ 2c dist log log n∕| log( − 2)|, where c dist = 1 for the configuration model and c dist = 2 for the preferential attachment model. This leads to the second term in the right hand sides of (1.9) and (1.13).
In the proof, we split the possible paths between the boundaries of two minimally connected vertices into bad paths, which are too short, and typical paths, which have the right number of edges in them, and then show that the contribution due to bad paths vanishes. The degrees along the path determine whether a path is bad or typical.
The strategy for the lower bound is depicted in the bottom part of Figure 1. • For the upper bound on the diameter, we perform a lazy-exploration from every vertex in the graph and realize that the neighborhood up to a distance k + n , which is roughly the same as k − n , contains at least as many vertices as the tree-like neighborhood of a minimally connected vertex. All possible other vertices in this neighborhood are ignored.
We then show that the vertices at the boundary of these lazy neighborhoods are with high probability quickly connected to the core, that is by a path of h n = o(log log n) steps. By core we mean the set of all vertices with large degrees, which is known to be highly connected, with a diameter close to 2k n , similar to the typical distances (see van der Hofstad and coworkers [17] for the configuration model and Dommers and coworkers [9] for the preferential attachment model).
The proof strategy for the upper bound is depicted in the top part of Figure 1.

Links to the literature and comments
This paper studies the diameter in CM n and PA t when the degree power-law exponent satisfies ∈ (2, 3), which means the degrees have finite mean but infinite variance. Both in (1.9) and (1.13), the explicit constant is the sum of two terms, one depending on , and the other depending on the minimal forward degree (see (2.2)), which is min −1 for CM n and m for PA t . We remark that the term depending on is related to the typical distances, while the other is related to the periphery of the graph. There are several other works that have already studied typical distances and diameters of such models. van der Hofstad and coworkers [16] analyze typical distances in CM n for ∈ (2, 3), while van der Hofstad and coworkers [15] study > 3. They prove that for ∈ (2, 3) typical distances are of order log log n, while for > 3 is of order log n, and it presents the explicit constants of asymptotic growth. Van der Hofstad and coworkers [17] shows for > 2 and when vertices of degree 1 or 2 are present, that with high probability the diameter of CM n is bounded from below by a constant times log n, while when ∈ (2, 3) and the minimal degree is 3, the diameter is bounded from above by a constant times log log n. van der Hofstad and Komjáthy [18] investigate typical distances for configuration models and ∈ (2, 3) in great generality, extending the results in van der Hofstad and coworkers [17] beyond the setting of i.i.d. degrees. Interestingly, they also investigate the effect of truncating the degrees at n n for values of n → 0. It would be of interest to also extend our diameter results to this setting.
We significantly improve upon the result in van der Hofstad and coworkers [17] for ∈ (2, 3). We do make use of similar ideas in our proof of the upper bound on the diameter. Indeed, we again define a core consisting of vertices with high degrees, and use the fact that the diameter of this core can be computed exactly (for a definition of the core, see (2.8)). The novelty of our current approach is that we quantify precisely how far the further vertex is from this core in the configuration model. It is a pair of such remote vertices that realizes the graph diameter.
Fernholz and Ramachandran [12] prove that the diameter of CM n is equal to an explicit constant times log n plus o(log n) when ∈ (2, 3) but there are vertices of degree 1 or 2 present in the graph, by studying the longest paths in the configuration model that are not part of the 2-core (which is the part of the graph for which all vertices have degree at least 2). Since our minimal degree is at least 3, the 2-core is whp the entire graph, and thus this logarithmic phase vanishes. Dereich and coworkers [10] prove that typical distances in PA t are asymptotically equal to an explicit constant times log log t, using path counting techniques. We use such path counting techniques as well, now for the lower bound on the diameters. Van der Hofstad [14] studies the diameter of PA t when m = 1, and proves that the diameter still has logarithmic growth. Dommers and coworkers [9] prove an upper bound on the diameter of PA t , but the explicit constant is not sharp.
Again, we significantly improve upon that result. Our proof uses ideas from Dommers and coworkers [9], in the sense that we again rely on an appropriately chosen core for the preferential attachment model, but our upper bound now quantifies precisely how the further vertex is from this core, as for the configuration model, but now applied to the much harder preferential attachment model. CM n and PA t are two different models, in the sense that CM n is a static model while PA t is a dynamic model. It is interesting to notice that the main strategy to prove Theorems 1.3 and 1.5 is the same. In fact, all the statements formulated in Section 2 are general and hold for both models. Also the explicit constants appearing in (1.9) and (1.13) are highly similar, which reflects the same structure of the proofs. The differences consist in a factor 2 in the terms containing and in the presence of min −1 and m in the remaining term. The factor 2 can be understood by noting that in CM n pairs of vertices with high degree are likely to be at distance 1, while in PA t they are at distance 2. The difference in min − 1 and m is due to the fact that min − 1 and m play the same role in the two models, that is, they are the minimal forward degree (or "number of children") of a vertex that is part of a tree contained in the graph. We refer to Section 2 for more details.
While the structures of the proofs for both models are identical, the details of the various steps are significantly different. Pairings in the configuration model are uniform, making explicit computations easy, even when already many edges have been paired. In the preferential attachment model, on the other hand, the edge statuses are highly dependent, so that we have no rely on delicate conditioning arguments. These conditioning arguments are arguably the most significant innovation in this paper. This is formalized in the notion of factorizable events in Definition 4.4.
Typical distances and diameters have been studied for other random graphs models as well, showing log log behavior. Bloznelis [1] investigates the typical distance in power-law intersection random graphs, where such distance, conditioning on being finite, is of order log log n, while results on diameter are missing. Chung and Lu [7,8] present results respectively for random graphs with given expected degrees and Erdős and Rényi random graphs G(n, p), see also van den Esker, the last author and Hooghiemstra [11] for the rank-1 setting. The setting of the configuration model with finite-variance degrees is studied in Fernholz and Ramachandran [12]. In Chung and Lu [8], they prove that for the power-law regime with exponent ∈ (2, 3), the diameter is Θ(log n), while typical distances are of order log log n. This can be understood from the existence of a positive proportion of vertices with degree 2, creating long, but thin, paths. In [7], the authors investigate the different behavior of the diameter according to the parameter p.
An interesting open problem is the study of fluctuations of the diameters in CM n and PA t around the asymptotic mean, that is, the study of the difference between the diameter of the graph and the asymptotic behavior (for these two models, the difference between the diameter and the right multiple of log log n). In [16], the authors prove that in graphs with i.i.d. power-law degrees with ∈ (2, 3), the difference Δ n between the typical distance and the asymptotic behavior 2 log log n∕| log( − 2)| does not converge in distribution, even though it is tight (ie, for every > 0 there is M < ∞ such that P(|Δ n | ≤ M) > 1 − for all n ∈ N). These results have been significantly improved in van der Hofstad and Komjáthym [18].
In the literature results on fluctuations for the diameter of random graph models are rare. Bollobás in [3], and, later, Riordan and Wormald in [20] give precise estimates on the diameter of the Erdös-Renyi random graph. It would be of interest to investigate whether the diameter has tight fluctuations around c log log n for the appropriate c.

GENERAL STRUCTURE OF THE PROOFS
We split the proof of Theorems 1.3 and 1.5 into a lower and an upper bound. Remarkably, the strategy is the same for both models despite the inherent difference in the models. In this section we explain the strategy in detail, formulating general statements that will be proved for each model separately in the next sections. Throughout this section, we assume that the assumptions of Theorems 1.3 and 1.5 are satisfied and, to keep unified notation, we denote the size of the preferential attachment model by n ∈ N, instead of the more usual t ∈ N.
Throughout the paper, we treat real numbers as integers when we consider graph distances. By this, we mean that we round real numbers to the closest integer. To keep the notation light and make the paper easier to read, we omit the rounding operation.

Lower bound
We start with the structure of the proof of the lower bound in (1.9) and (1.13). The key notion is that of a minimally k-connected vertex, defined as a vertex whose k-neighborhood (ie, the neighborhood up to distance k) is essentially a regular tree with the smallest possible degree, equal to min for the configuration model and to m + 1 for the preferential attachment model. Due to technical reasons, the precise definition of minimally k-connected vertex is slightly different for the two models (see Definitions 3.2 and 4.2). Henceforth we fix > 0 and define, for n ∈ N, where fwd denotes the forward degree, or "number of children": Our first goal is to prove that the number of minimally k − n -connected vertices is large enough, as formulated in the following statement: Denote by M k − n the number of minimally k − n -connected vertices in the graph (either CM n or PA n ). Then, as n → ∞, The proof for the preferential attachment model makes use of conditioning arguments. Indeed, we describe as much information as necessary to be able to bound probabilities that vertices are minimally k connected. Particularly in the variance estimate, these arguments are quite delicate, and crucial for our purposes.
The bounds in (2.3) show that M k − n P − − → ∞ as n → ∞. This will imply that there is a pair of minimally k − n -connected vertices with disjoint k − n -neighborhoods, 1 hence the diameter of the graph is at least 2k − n , which explains the first term in (1.9) and (1.13). Our next aim is to prove that these minimally connected trees are typically at distance 2c dist log log n∕| log( − 2)|, where c dist = 1 for the configuration model and c dist = 2 for the preferential attachment model.

For this, let us now definek
The difference in the definition of c dist is due to fact that in CM n vertices with high degree are likely at distance 1, while in PA n are at distance 2. We explain the origin of this effect in more detail in the proofs.
It turns out that the distance between the k − n -neighborhoods of two minimally k − n -connected vertices is at least 2k n . More precisely, we have the following statement: Statement 2.2 (Distance between neighborhoods) Let W n 1 and W n 2 be two random vertices chosen independently and uniformly among the minimally k − n -connected ones. Denoting byH n the distance between the k − n -neighborhoods of W n 1 and W n 2 , we haveH n ≥ 2k n with high probability.
It follows immediately from Statement 2.2 that the distance between the vertices W n 1 and W n 2 is at least 2k − n + 2k n , with high probability. This proves the lower bound in (1.9) and (1.13). It is known from the literature that 2k n , see (2.4), represents the typical distance between two vertices chosen independently and uniformly in the graph. In order to prove Statement 2.2, we collapse 1 A justification for this fact is provided by the following Statement 2.2 (the randomly chosen vertices W n 1 and W n 2 have disjoint k − n -neighborhoods, becauseH n > 0 with high probability). For a more direct justification, see Remark 3.6 for the configuration model and Remark 4.7 for the preferential attachment model. the k − n -neighborhoods of W n 1 and W n 2 into single vertices and show that their distance is roughly equal to the typical distance 2k n . This is a delicate point, because the collapsed vertices have a relatively large degree and thus could be closer than the typical distance. The crucial point why they are not closer is that the degree of the boundary only grows polylogarithmically. The required justification is provided by the next statement: (2.6) Then, denoting the distance in the graph of size n by dist n , . (2.7) The proof of Statement 2.3 is based on path counting techniques. These are different for the two models, but the idea is the same: We split the possible paths between the vertices a and b into two sets, called good paths and bad paths. Here good means that the degrees of vertices along the path increase, but not too much. We then separately and directly estimate the contribution of each set. The details are described in the proof.

Upper bound
We now describe the structure of the proof for the upper bound, which is based on two key concepts: the core of the graph and the k-exploration graph of a vertex. We start by introducing some notation. First of all, fix a constant ∈ (1∕(3 − ), ∞). We define Core n as the set of vertices in the graph of size n with degree larger than (log n) . More precisely, denoting by D t (v) = D t,m (v) the degree of vertex v in the preferential attachment model after time t, that is, in the graph PA t (see the discussion after (1.10)), we let The fact that we evaluate the degrees at time n∕2 for the PAM is quite crucial in the proof of Statement 2.4 below. In Section 6, we also give bounds on D n (v) for v ∈ Core n , as well as for v ∉ Core n , that show that the degrees cannot grow too much between time n∕2 and n. The first statement, that we formulate for completeness, upper bounds the diameter of Core n and is already known from the literature for both models: Next we bound the distance between a vertex and Core n . We define the k-exploration graph of a vertex v as a suitable subgraph of its k-neighborhood, built as follows: We consider the usual exploration process starting at v, but instead of exploring all the edges incident to a vertex, we only explore a fixed number of them, namely fwd defined in (2.2). (The choice of which edges to explore is a natural one, and it will be explained in more detail in the proofs. ) We stress that it is possible to explore vertices that have already been explored, leading to what we call a collision. If there are no collisions, then the k-exploration graph is a tree. In presence of collisions, the k-exploration graph is not a tree, and it is clear that every collision reduces the number of vertices in the k-exploration graph.
Henceforth we fix > 0 and, in analogy with (2.1), we define, for n ∈ N, (2.10) Our second statement for the upper bound shows that the k + n -exploration graph of every vertex in the graph either intersects Core n , or it has a bounded number of collisions:

Statement 2.5 (Bound on collisions)
There is a constant c < ∞ such that, with high probability, the k + n -exploration graph of every vertex in the graph has at most c collisions before hitting Core n . As a consequence, for some constant s > 0, the k + n -exploration graph of every vertex in the graph either intersects Core n , or its boundary has cardinality at least With a bounded number of collisions, the k + n -exploration graph is not far from being a tree, which explains the lower bound (2.11) on the cardinality of its boundary. Having enough vertices on its boundary, the k + n -exploration is likely to be connected to Core n fast, which for our purpose means in o(log log n) steps. This is the content of our last statement: Statement 2.6 There are constants B, C < ∞ such that, with high probability, the k + n -exploration graph of every vertex in the graph is at distance at most h n = ⌈B log log log n + C⌉ from Core n .
The proof for this is novel. For example, for the configuration model, we grow the k + n + h n neighborhood of a vertex, and then show that there are so many half-edges at its boundary that it is very likely to connect immediately to the core. The proof for the preferential attachment model is slightly different, but the conclusion is the same. This shows that the vertex is indeed at most at distance k + n +h n away from the core.
In conclusion, with high probability, the diameter of the graph is at most which gives us the expressions in (1.9) and (1.13) and completes the proof of the upper bound.

LOWER BOUND FOR CONFIGURATION MODEL
In this section we prove Statements 2.1 to2.3 for the configuration model. By the discussion in Section 2.1, this completes the proof of the lower bound in Theorem 1.3.
In our proof, it will be convenient to choose a particular order to pair the half-edges. This is made precise in the following remark: Remark 3.1 (Exchangeability in half-edge pairing) Given a sequence = ( 1 , … , n ) such that n = 1 + · · · + n is even, the configuration model CM n can be built iteratively as follows: ⊳ start with i half-edges attached to each vertex i ∈ [n] = {1, 2, … , n}; ⊳ choose an arbitrary half-edge and pair it to a uniformly chosen half-edge; ⊳ choose an arbitrary half-edge, among the n − 2 that are still unpaired, and pair it to a uniformly chosen half-edge; and so on.
The order in which the arbitrary half-edges are chosen does not matter in the above, by exchangeability (see also [13,Chapter 7]).

Proof of Statement 2.1
With a slight abuse of notation (see (1.8)), in this section we set Given a vertex v ∈ [n] and k ∈ N, we denote the set of vertices at distance at most k from v (in the graph CM n ) by U ≤k (v) and we call it the k-neighborhood of v. Definition 3.2 (Minimally k-connected vertex) For k ∈ N 0 , a vertex v ∈ [n] is called minimally k-connected when all the vertices in U ≤k (v) have minimal degree, that is, i = min for all i ∈ U ≤k (v), and furthermore there are no self-loops, multiple edges or cycles in U ≤k (v). Equivalently, v is minimally k-connected when the graph U ≤k (v) is a regular tree with degree min .
We denote the (random) set of minimally k-connected vertices by  k ⊆ [n], and its cardinality by M k = | k |, that is, M k denotes the number of minimally k-connected vertices.
Moreover, the number of vertices inside U ≤k (v) equals i k + 1. By (3.1), it is clear why min > 2, or min ≥ 3, is crucial. Indeed, this implies that the volume of neighborhoods of minimally k-connected vertices grows exponentially in k. Remark 3.4 (Collapsing minimally k connected trees) By Remarks 3.1 and 3.3, conditionally on the event {v ∈  k } that a given vertex v is minimally k-connected, the random graph obtained from CM n by collapsing U ≤k (v) to a single vertex, called a, is still a configuration model with n − i k vertices and with n replaced by n − 2i k , where the new vertex a has degree min ( min − 1) k .
Analogously, conditionally on the event {v ∈  k , w ∈  m , U ≤k (v)∩U ≤m (w) = ∅} that two given vertices v and w are minimally k and minimally m-connected with disjoint neighborhoods, collapsing U ≤k (v) and U ≤m (w) to single vertices a and b yields again a configuration model with n − i k − i m vertices, where n is replaced by n − 2i k − 2i m and where the new vertices a and b have degrees equal to min ( min − 1) k and min ( min − 1) m , respectively.
We denote the number of vertices of degree k in the graph by n k , that is, We now study the first two moments of M k , where we recall that the total degree n is defined by (1.1): Before proving Proposition 3.5, let us complete the proof of Statement 2.1 subject to it. We are working under the assumptions of Theorem 1.3, hence min ≥ 3 and the degree sequence satisfies the degree regularity condition, Condition 1.1, as well as the polynomial distribution condition Condition 1.2 with exponent ∈ (2, 3). Recalling (1.1)-(1.2), we can write n min = n p (n) min and n = n ∑ k∈N kp (n) k , so that, as n → ∞, On the other hand, applying (3.3), for some c ∈ (0, 1) one has Relations (3.7) and (3.8) show that (2.3) holds, completing the proof of Statement 2.1.

Remark 3.6 (Disjoint neighborhoods) Let us show that, with high probability, there are vertices
We proceed by contradiction: fix v ∈  k − n and assume that, for every vertex Then, for any w ∈  k − n there must exist a self-avoiding path from v to w of length ≤ 2k − n which only visits vertices with degree min (recall that U ≤k − n (v) and U ≤k − n (w) are regular trees). However, for fixed v, the number of such paths is

Proof of Proposition 3.5 To prove (3.3) we write
and since every vertex in the sum has the same probability of being minimally k-connected, are paired to half-edges incident to distinct vertices having minimal degree, without generating cycles. By Remark 3.1, we can start pairing a half-edge incident to v to a half-edge incident to another vertex of degree min . Since there are n min − 1 such vertices, this event has probability min (n min − 1) We iterate this procedure, and suppose that we have already successfully paired (i − 1) couples of half-edges; then the next half-edge can be paired to a distinct vertex of degree min with probability Indeed, every time that we use a half-edge of a vertex of degree min , we cannot use its remaining half-edges, and every step we make reduces the total number of possible half-edges by two. By (3.1), exactly i k couples of half-edges need to be paired for v to be minimally k-connected, so that which proves (3.3). If n min ≤ i k the right hand side vanishes, in agreement with the fact that there cannot be any minimally k-connected vertex in this case (recall (3.1)). To prove (3.4), we notice that We distinguish different cases: the k-neighborhoods of v and w might be disjoint or they may overlap, in which case w can be included in U ≤k (v) or not. Introducing the events we can write the right hand side of (3.13) as . (3.15) Let us look at the first term in (3.15). By Remarks 3.3 and 3.4, conditionally on {v ∈  k }, the probability of {w ∈  k , A c v,w } equals the probability that w is minimally k-connected in a new configuration model, with n replaced by n −2i k and with the number of vertices with minimal degree reduced from n min to n min − (i k + 1). Then, by the previous analysis (see (3.12)), By direct computation, the ratio in the right hand side of (3.16) is always maximized for i k = 0 (provided n ≥ 2n min − 3, which is satisfied since n ≥ min n min ≥ 3n min by assumption). Therefore, setting i k = 0 in the ratio and recalling (3.12), we get the upper bound Since there are at most n 2 min pairs of vertices of degree min , it follows from (3.17) that which explains the first term in (3.4). For the second term in (3.15), v and w are minimally k-connected with overlapping neighborhoods, , (3.19) and note that which explains the second term in (3.4).
For the third term in (3.15), v and w are minimally k-connected vertices with overlapping neighborhoods, but w ∉ U ≤k (v). This means that dist(v, w) = l + 1 for some l ∈ {k, … , 2k − 1}, so that U ≤k (v)∩U ≤l−k (w) = ∅ and, moreover, a half-edge on the boundary of U ≤(l−k) (w) is paired to a half-edge on the boundary of U ≤k (v), an event that we call F v,w;l,k . Therefore and we stress that in the right hand side w is only minimally (l − k)-connected (in case l = k this just means that w = min ). Then and U ≤l−k (w) to single vertices a and b with degrees respectively min ( min −1) k and min ( min − 1) l−k , getting a new configuration model with n replaced by n − 2i k − 2i l−k . Bounding the probability that a half-edge of a is paired to a half-edge of b, we get where we have used the definition (3.1) of i 2k−1 . Since ( min − 1)i 2k−1 ≤ i 2k , again by (3.1), we have obtained the third term in (3.4). ▪

Proof of Statement 2.2
We recall that W n 1 and W n 2 are two independent random vertices chosen uniformly in  k − n (the set of minimally k − n -connected vertices), assuming that  k − n ≠ ∅ (which, as we have shown, occurs with high probability). Our goal is to show that where we set We know from Statement 2.1 that as n → ∞ Therefore, (3.28) In analogy with (3.14), we introduce the event and show that it gives a negligible contribution. Recalling the proof of Proposition 3.5, in particular (3.20) and (3.24), the sum restricted to A v 1 ,v 2 leads precisely to the second term in the right hand side of (3.4): where we have used (3.6) and (3.8) (see also (3.5)). We can thus focus on the event whereP is the law of the new configuration model which results from collapsing the neighborhoods 2)). The degree sequencêof this new configuration model is a slight modification of the original degree sequence : two new vertices of degree O(log n) have been added, while 2(i k − n + 1) = O(log n) vertices with degree min have been removed (recall (3.6)). Consequentlŷstill satisfies the assumptions of Theorem 1.3, hence Statement 2.3 (to be proved in Section 3.3) holds forP and we We are ready to conclude the proof of Statement 2.2. By (3.28)-(3.29)-(3.30), , by the second relation in (2.3). Applying (3.31), it follows that P(E n ) = o(1), completing the proof of Statement 2.2.

Proof of Statement 2.3
In this section, we give a self-contained proof of Statement 2.3 for CM n , as used in the proof of Let us fix an arbitrary increasing sequence (g l ) l∈N 0 (that will be specified later). Define, for a, b ∈ R, a ∧ b ∶= min{a, b}. We say that a path ∈  k (a, b) is good when l ≤ g l ∧ g k−l for every l = 0, … , k, and bad otherwise. In other words, a path is good when the degrees along the path do not increase too much from 0 to k∕2 , and similarly they do not increase too much in the backward direction, from k to k∕2 .
For k ∈ N 0 , we introduce the event To deal with bad paths, we define , then there must be a path in  k (a, b) for some k ≤k, and this path might be good or bad. This leads to the simple bound We give explicit estimates for the two sums in the right hand side. We introduce the size-biased distribution function F * n associated to the degree sequence = ( 1 , … , n ) by If we choose uniformly one of the n half-edges in the graph, and call D * n the degree of the vertex incident to this half-edge, then F * n (t) = P(D * n ≤ t). We also define the truncated mean Now we are ready to bound (3.34).
Proposition 3.7 (Path counting for configuration model) Fix = ( 1 , … , n ) (such that n = 1 + … + n is even) and an increasing sequence (3.37) Proof Fix an arbitrary sequence of vertices = ( i ) 0≤i≤k ∈ [n] k+1 . The probability that vertex 0 is connected to 1 is at most because there are 0 1 ordered couples of half-edges, each of which can be paired with probability 1∕( n − 1) (recall Remark 3.1), and we use the union bound. By similar arguments, conditionally on a specific half-edge incident to 0 being paired to a specific half-edge incident to 1 , the probability that another half-edge incident to 1 is paired to a half-edge incident to 2 is by the union bound bounded from above by Iterating the argument, the probability that is a path in CM n is at most .
yields the first term in the right hand side of (3.37).
The bound for P( k (a)) is similar. Recalling (3.33)-(3.35), choosing 0 = a and summing (3.38) over and the same holds for P( k (b)). Plugging (3.39) and (3.40) into (3.34) proves (3.37). ▪ In order to exploit (3.37), we need estimates on F * n and n , provided by the next lemma: Lemma 3.8 (Tail and truncated mean bounds for D * n ) Assume that Condition 1.2 holds. Fix > 0, then there exist two constants C 1 = C 1 ( ) and C 2 = C 2 ( ) such that, for every x ≥ 0, Proof For every x ≥ 0 and t ≥ 0 we can see that where we recall that D n is the degree of a uniformly chosen vertex. This means that where we have used Condition 1.2 in the second last step (recall that 2 < < 3). For n , we can instead write where D n is again the degree of a uniformly chosen vertex. The claim now follows from (3.45)

▪
We are finally ready to complete the proof of Statement 2.3: Proof of Statement 2.3 As in (2.4), we takē and our goal is to show that, as n → ∞, We stress that ∈ (2, 3) and > 0 are fixed. Then we choose > 0 so small that We use the inequality (3.37) given by Proposition 3.7, with the following choice of (g k ) k∈N 0 : Let us focus on the first term in the right hand side of (3.37), that is Then observe that, by Lemma 3.8 and (3.49), for k ≤ 2k n hence the right hand side of (3.52) is n o(1) (since g 0 = (log n) log log n ). Then, for a , b ≤ log n, It remains to look at the second sum in (3.37): (3.54) Since g 0 = (log n) log log n while a , b ≤ log n, the right hand side of (3.58) is o (1). ▪

LOWER BOUND FOR PREFERENTIAL ATTACHMENT MODEL
In this section we prove Statements 2.1, 2.2 and 2.3 for the preferential attachment model. By the discussion in Section 2.1, this completes the proof of the lower bound in Theorem 1.5. We recall that, given m ∈ N and ∈ (−m, ∞), the preferential attachment model PA t is a random graph with vertex set [t] = {1, 2, … , t}, where each vertex w has m outgoing edges, which are attached to vertices v ∈ [w] with probabilities given in (1.10). In the next subsection we give a more detailed construction using random variables. This equivalent reformulation will be used in a few places, when we need to describe carefully some complicated events. However, for most of the exposition we will stick to the intuitive description given in Section 1.2.

Alternative construction of the preferential attachment model
We introduce random variables w,j to represent the vertex to which the jth edge of vertex w is attached, that is The graph PA t is a deterministic function of these random variables: two vertices v, w ∈ [t] with v ≤ w are connected in PA t if and only if w,j = v for some j ∈ [m]. In particular, the degree of a vertex v after the kth edge of vertex t has been attached, denoted by D t,k (v), is where we use the natural order relation Defining the preferential attachment model amounts to giving a joint law for the sequence = ( w,j ) (w,j)∈N× [m] . In agreement with (1.10), we set 1,j = 1 for all j ∈ [m], and for t ≥ 2 The factor j ∕m in the first line of (4.3) is commonly used in the literature (instead of the possibly more natural ). The reason is that, with such a definition, the graph PA t (m, ) can be obtained from the special case m = 1, where every vertex has only one outgoing edge: one first generates the random graph PA mt (1, ∕m), whose vertex set is [mt], and then collapses the block of vertices [m(i − 1) + 1, mi) into a single vertex i ∈ [t] (see also [13,Chapter 8]).

Remark 4.1
It is clear from the construction that PA t is a labeled directed graph, because any edge connecting sites v, w, say with v ≤ w, carries a label j ∈ [m] and a direction, from the newer vertex w to the older one v (see (4.1)). Even though our final result, the asymptotic behavior of the diameter, only depends on the underlying undirected graph, it will be convenient to exploit the labeled directed structure of the graph in the proofs.

Proof of Statement 2.1
We denote by U ≤k (v) the k-neighborhood in PA t of a vertex v ∈ [t], that is the set of vertices at distance at most k from v, viewed as a labeled directed subgraph (see Remark 4.1). We denote by D t (v) = D t,m (v) the degree of vertex v after time t, that is, in the graph PA t (recall (4.2)). We define the notion of minimally k-connected vertex in analogy with the configuration model (see Definition 3.2), up to minor technical restrictions made for later convenience.
and have degree D t (i) = m + 1, and there are no self-loops, multiple edges or cycles in U ≤k (v). The graph U ≤k (v) is thus a tree with degree m + 1, except for the root v which has degree m.
We denote the (random) set of minimally k-connected vertices by  k ⊆ [t] ⧵ [t∕2], and its cardinality by M k = | k |.
For the construction of a minimally k-connected neighborhood in the preferential attachment model we remind that the vertices are added to the graph at different times, so that the vertex degrees change while the graph grows. The relevant degree for Definition 4.2 is the one at the final time t. To build a minimally k-connected neighborhood, we need many vertices. The center v of the neighborhood is the youngest vertex in U ≤k (v), and it has degree m, while all the other vertices have degree m + 1.
Our first goal is to evaluate the probability P(v ∈  k ) that a given vertex v ∈ [t] ⧵ [t∕2] is minimally k-connected. The analogous question for the configuration model could be answered quite easily in Proposition 3.5, because the configuration model can be built exploring its vertices in an arbitrary order, in particular starting from v, see Remark 3.1. This is no longer true for the preferential attachment model, whose vertices have an order, the chronological one, along which the conditional probabilities take the explicit form (1.10) or (4.3). This is why the proofs for the preferential attachment model are harder than for the configuration model.
As it will be clear in a moment, to get explicit formulas it is convenient to evaluate the probability P(v ∈  k , U ≤k (v) = H), where H is a fixed labeled directed subgraph, that is, it comes with the specification of which edges are attached to which vertices. To avoid trivialities, we restrict to those H for which the probability does not vanish, that is, which satisfy the constraints in Definition 4.2, and we call them admissible.
Let us denote by H o ∶= H ⧵ H the set of vertices in H that are not on the boundary (ie, they are at distance at most k − 1 from v). With this notation, we have the following result:

7)
and D u−1 (H) = ∑ w∈H D u−1,m (w) is the total degree of H before vertex u is added to the graph, and the normalization constant c u,j is defined in (1.11).
Proof We recall that {a i → b} denotes the event that the ith edge of a is attached to b (see (4.1)). Since H is an admissible labeled directed subgraph, for all u ∈ H o and j ∈ [m], the jth edge of u is connected to a vertex in H, that we denote by H j (u). We can then write because the vertex H j (u) has degree precisely m (when u is not already present in the graph). For u ∉ H o , we have to evaluate the probability that its edges do no attach to H, which is (4.10) Using conditional expectation iteratively, we obtain (4.9) or (4.10) for every edge in the graph, depending on whether the edge is part of H or not. This proves (4.6) and (4.7). ▪ The event {v ∈  k , U ≤k (v) = H} is an example of a class of events, called factorizable, that will be used throughout this section and Section 6. For this reason we define it precisely.
It is convenient to use the random variable w,j , introduced in Section 4.1, to denote the vertex to which the jth edge of vertex w is attached (see (4.1)). Any event A for PA t can be characterized iteratively, specifying a set A s,i ⊆ [s] of values for s,i , for all (s, i) ≤ (t, m): Of course, the set A s,i is allowed to depend on the "past," that is,  on the event ≤(s,i−1) ∈ A ≤(s,i−1) . As a consequence, the chain rule for probabilities yields  4.3)).
Note that the event {v ∈  k } is not factorizable. This is the reason for specifying the realization of the k-neighborhood U ≤k (v) = H.
Henceforth we fix > 0. We recall that k − n was defined in (2.1). Using the more customary t instead of n, we have   .12), as t → ∞, Proof Similarly to the proof of (3.3), we write where the sum is implicitly restricted to admissible H (ie, to H that are possible realizations of U ≤k (v)).
Since we will use (4.5), we need a lower bound on (4.6) and (4.7). Recalling (1.11), it is easy to show, since the number of vertices in H o equals i k − m k = i k−1 , and u ≤ v for u ∈ H o , Note that for u ≤ t∕4 all the factors in the product in (4.  . Then, we obtain . (4.19) Choosing k = k − t as in (4.12) and bounding 1 − x ≥ e −2x for x small, as well as m + 1 ≤ 2m, we obtain , (4.20) where C is a constant and C ′ = 4C∕m. Recalling that i k is given by (4.4), and k − t by (4.12), hence and also (C ′ e 3 c m ) (1) . This implies that E[M k ] → ∞, as required. ▪ Remark 4.7 (Disjoint neighborhoods for minimally k-connected pairs) We observe that, on the event because if a vertex x is in U ≤k (v) ∩ U ≤k (w) and x ≠ v, w, this means that D x (t) = m + 2, because in addition to its original m outgoing edges, vertex x has one incident edge from a younger vertex in U ≤k (v) and one incident edge from a younger vertex in U ≤k (u), which gives a contradiction. Similar arguments apply when x = v or x = w.
We use the previous remark to prove the second relation in Statement 2.1 for the preferential attachment model.
be a preferential attachment model, with m ≥ 2 and ∈ (−m, 0). Then, for k ∈ N, Consequently, for k = k − t as in (4.12), as t → ∞, (4.24) By Remark 4.7, for v ≠ w we can write The crucial observation is that the event {v, w ∈  k , U ≤k (v) = H v , U ≤k (w) = H w } is factorizable (recall Definition 4.4 and Remark 4.5). More precisely, in analogy with (4.6) and (4.7): where now . (4.28) To prove (4.26), notice that in (4.27) and (4.28) every edge and every vertex of the graph appear. Further, (4.27) is the probability of the event {U ≤k (v) = H v , U ≤k (w) = H w }, while (4.28) is the probability that all vertices not in the two neighborhoods do not attach to the two trees.

Proof of Statement 2.3
Fix > 0 and define, as in (2.4),k  Let us denote by u ↔ v the event that vertices u, v are neighbors in PA t , that is ) .
where C is an absolute constant. The history of (4.36) is that it was first proved by Bollobás and Riordan [4] for = 0 (so that = 1 − = 1∕2), and the argument was extended to all in Dommers and coworkers [9, Corollary 2.3].

Remark 4.10 Proposition 4.9 holds for every random graphs that satisfies (4.36).
We proceed in a similar way as in Section 3.3. Given two vertices x, y ∈ [t], we consider paths = ( 0 , 1 , … , k ) between x = 0 and y = k . We fix a decreasing sequence of numbers (g l ) l∈N 0 that serve as truncation values for the age of vertices along the path (rather than the degrees as for the configuration model). We say that a path is good when l ≥ g l ∧ g k−l for every l = 0, … , k, and bad otherwise. In other words, a path is good when the age of vertices does not decrease too much from 0 to k∕2 and, backwards, from k to k∕2 . Intuitively, this also means that their degrees do not grow too fast. This means that where  k (x, y) is the event of there being a good path of length k, as in (3.32), while  k (x) is the event of there being a path with i ≥ g i for i ≤ k − 1 but k < g k , in analogy with (3.33).

(4.39)
This is the starting point of the proof of Proposition 4.9.
We will show in [6, Appendix A] that the following recursive bound holds for suitable sequences ( k ) k∈N , ( k ) k∈N and (g k ) k∈N (see [6,Definition A.2]). We will prove recursive bounds on these sequences that guarantee that the sums in (4.39) satisfy the required bounds. We omit further details at this point, and refer the interested reader to [6, Appendix A].

Proof of Statement 2.2
Consider now two independent random vertices W t 1 and W t 2 that are uniformly distributed in the set of minimally k − t -connected vertices  k − t . We set and, in analogy with Section 3.2, our goal is to show that We know from Statement 2.1 that, as t → ∞, We also define the event and note that it is known (see [13,Theorem 8.13]) that lim t→∞ P(B t ) = 1. Therefore, (4.45) The contribution of the terms with v 1 = v 2 is negligible, since it gives → ∞ by Proposition 4.6. Henceforth we restrict the sum in (4.45) to v 1 ≠ v 2 . Summing over the realizations H 1 and H 2 of the random neighborhoods U ≤k − t (v 1 ) and U ≤k − t (v 2 ), and over paths from an arbitrary vertex x ∈ H 1 to an arbitrary vertex y ∈ H 2 , we obtain (4.46) The next proposition, proved below, decouples the probability appearing in the last expression:

Proposition 4.11
There is a constant q ∈ (1, ∞) such that, for all v 1 , v 2 , H 1 , H 2 and , The proof of Proposition 4.11 reveals that we can take q = 2 for t sufficiently large. Using (4.47) in (4.46), we obtain If we bound P ( ⊆ PA t ) ≤ p( ) in (4.48), as in (4.36), the sum over can be rewritten as the right hand side of (4.39) (recall (4.37)-(4.38)). We can thus apply Proposition 4.9 -because the proof of Proposition 4.9 really gives a bound on (4.39)-concluding that the sum over is at most p∕(log t) 2 , where the constant p is defined in Proposition 4.9. Since | H 1 | = | H 2 | = m k − t = (log t) 1− (recall (4.12)), we finally obtain is a given sequence of vertices with 0 ∈ H 1 and k ∈ H 2 . The event in (4.50) is not factorizable, because the degrees of the vertices in the path are not specified, hence it is not easy to evaluate its probability. To get a factorizable event, we need to give more information. For a vertex v ∈ [t], define its incoming neighborhood  (v) by The key observation is that the knowledge of  (v) determines the degree D s (v) at any time s ≤ t (for instance, at time t we simply have D t (v) = | (v)| + m).
We are going to fix the incoming neighborhoods  ( 1 ) = K 1 , …,  ( k−1 ) = K k−1 of all vertices in the path , except the extreme ones 0 and k (note that  ( 0 ) and  ( k ) reduce to single points in H o 1 and H o 2 , respectively, because 0 ∈ H 1 and k ∈ H 2 ). We emphasize that such incoming neighborhoods allow us to determine whether = ( 0 , … , k ) is a path in PA t . Recalling the definition of the event B t in (4.44), we restrict to (4.52) and simply drop B t from (4.50). We will then prove the following relation: for all v 1 , v 2 , H 1 , H 2 , = ( 0 , … , k ), and for all K 1 , … , K k−1 satisfying (4.52), we have Our goal (4.47) follows by summing this relation over all K 1 , … , K k−1 for which ⊆ PA t . The first line of (4.53) is the probability of a factorizable event. In fact, setting for short the event in the first line of (4.53) is the intersection of the following four events (see (4.8)): . Generalizing (4.9)-(4.10), we can rewrite the first line of (4.53) as follows, recalling (1.10): (4.54) We stress that D u,j−1 ( i ) is non-random, because it is determined by K i . Analogous considerations apply to D u,j−1 (H 1 ∪ H 2 ∪ o ). We have thus obtained a factorizable event.
Next we evaluate the second line of (4.53). Looking back at (4.26)-(4.28), we have (4.55) On the other hand, (4.56) Using the bound (1 − (a + b)) ≤ (1 − a)(1 − b) in the second line of (4.54), and comparing with (4.55)-(4.56), we only need to take into account the missing terms in the product in the last lines. This shows that relation (4.53) holds if one sets q = C 1 C 2 therein, where To complete the proof, it is enough to give uniform upper bounds on C 1 and C 2 , that does not depend on H 1 , H 2 , . We start with C 1 . In the product we may assume u > t∕4, because the terms with u ≤ t∕4 are identically one, since for x small and recalling that < 0, it follows that Since i k is given by (4.4), for k = k − t as in (4.12) we have Recalling also (4.52) and bounding m + 1 ≤ 2m, we obtain again by (4.52), we get It follows that C 1 C 2 is bounded from above by some constant q. This completes the proof. ▪

Proof of Theorem 1.6
Dereich and coworkers [10] have already proved the upper bound. For the lower bound we use Proposition 4.9. In fact, fork t as in (4.33), (4.59) If v 1 and v 2 are both larger or equal than g 0 = ⌈ t (log t) 2 ⌉, then we can apply Proposition 4.9. The probability that V 1 < g 0 or V 2 < g 0 is and this completes the proof of Theorem 1.6.

UPPER BOUND FOR CONFIGURATION MODEL
In this section we prove Statements 2.5 and 2.6 for the configuration model. By the discussion in Section 2.2, this completes the proof of the upper bound in Theorem 1.3, because the proof of Statement 2.4 is already known in the literature, as explained below Statement 2.4. Throughout this section, the assumptions of Theorem 1.3 apply. In particular, we work on a configuration model CM n , with ∈ (2, 3) and min ≥ 3.

Proof of Statement 2.5
We first recall what Core n is, and define the k-exploration graph. Recall from (2.8) that, for CM n , Core n is defined as where > 1∕(3− ). Since the degrees i are fixed in the configuration model, Core n is a deterministic subset.
For any v ∈ [n], we recall that U ≤k (v) ⊆ [n] denotes the subgraph of CM n consisting of the vertices at distance at most k from v. We next consider the k-exploration graphÛ ≤k (v) as a modification of U ≤k (v), where we only explore min half-edges of the starting vertex v, and only min − 1 for the following vertices: Starting from the vertex v, we pair successively one half-edge after the other, as described in Definition 5.1 (recall also Remark 3.1). In order to buildÛ ≤k + n (v), we need to make a number of pairings, denoted by  , which is random, because collisions may occur. In fact, when there are no collisions,  is deterministic and takes its maximal value given by i k + n in (3.1), therefore Introducing the event C i ∶= "there is a collision when pairing the ith half-edge," we can write ) .

(5.3)
Let E be the event that the first half-edges are paired to vertices with degree ≤ (log n) (ie, the graph obtained after pairing the first half-edges is disjoint from Core n ). Then On the event E i−1 , before pairing the ith half-edge, the graph is composed by at most i − 1 vertices, each with degree at most (log n) , hence, for i ≤ 3(log n) 1+ , for some c ∈ (0, ∞), thanks to n = n (1 + o(1)) (recall (3.5)). The same arguments show that which completes the proof. ▪

Corollary 5.4 (Large boundaries)
Under the assumptions of Theorem 1.3 and on the eventÛ ≤k + n (v)∩ Core n = ∅, with high probability, the boundaryÛ =k + n (v) of the k + n -exploration graph of any vertex v ∈ [n] contains at least ( min − 2)( min − 1) k + n −1 ≥ 1 2 (log n) 1+ vertices, each one with at least two unpaired half-edges.
Proof By Proposition 5.3, with high probability, every k + n -exploration graph has at most one collision before hitting Core n . The worst case is when the collision happens immediately, that is, a half-edge incident to v is paired to another half-edge incident to v: in this case, removing both half-edges, the k + n -exploration graph becomes a tree with ( min − 2)( min − 1) k + n −1 vertices on its boundary, each of which has at least ( min − 1) ≥ 2 yet unpaired half-edges. Since ( min − 2)∕( min − 1) ≥ 1 2 for min ≥ 3, and moreover ( min − 1) k + n = (log n) 1+ by (5.1), we obtain the claimed bound. If the collision happens at a later stage, that is, for a half-edge incident to a vertex different from the starting vertex v, then we just remove the branch from v to that vertex, getting a tree with ( min − 1)( min − 1) k + n −1 vertices on its boundary. The conclusion follows. ▪ Together, Proposition 5.3 and Corollary 5.4 prove Statement 2.5.

Proof of Statement 2.6
Consider the k + n -exploration graphÛ =Û ≤k + n (v) of a fixed vertex v ∈ [n], as in Definition 5.1, and let x 1 , … , x N be the (random) vertices on its boundary. We stress that, by Corollary 5.4, with high probability N ≥ 1 2 (log n) 1+ . Set where B, C are fixed constants, to be determined later on. Henceforth we fix a realization H ofÛ =Û ≤k + n (v) and we work conditionally on the event {Û = H}. By Remark 3.1, we can complete the construction of the configuration model CM n by pairing uniformly all the yet unpaired half-edges. We do this as follows: for each vertex x 1 , … , x N on the boundary ofÛ, we explore its neighborhood, looking for fresh vertices with higher and higher degree, up to distance h n (we call a vertex fresh if it is connected to the graph for the first time, hence it only has one paired half-edge). We now describe this procedure in detail: Definition 5.5 (Exploration procedure) Let x 1 , … , x N denote the vertices on the boundary of a k + n -exploration graphÛ =Û ≤k + n (v). We start the exploration procedure from x 1 . ⊳ Step 1. We set v (1) 0 ∶= x 1 and we pair all its unpaired half-edges. Among the fresh vertices to which v (1) 0 has been connected, we call v 1 the one with maximal degree. ⊳ When there are no fresh vertices at some step, the procedure for x 1 stops. ⊳ Step 2. Assuming we have built v (1) 1 , we pair all its unpaired half-edges∶ among the fresh connected vertices, we denote by v (1) 2 the vertex with maximal degree.
⊳ We continue in this way for (at most) h n steps, defining v (1) j for 0 ≤ j ≤ h n (recall (5.5)). After finishing the procedure for x 1 , we perform the same procedure for Definition 5.6 (Success) Let x 1 , … , x N be the vertices on the boundary of a k + n -exploration grapĥ U =Û ≤k + n (v). We define the event S x i ∶= "x i is a success" by Here is the key result, proved below: Proposition 5.7 (Hitting the core quickly) There exists a constant > 0 such that, for every n ∈ N and for every realization H ofÛ,

6)
and, for each i = 2, … , N, This directly leads to the proof of Statement 2.6, as the following corollary shows: Proof By Corollary 5.4, with high probability, every vertex v ∈ [n] either is at distance at most k + n from Core n , or has a k + n -exploration graphÛ =Û ≤k + n (v) with at least N ≥ 1 2 (log n) 1+ vertices on its boundary. It suffices to consider the latter case. Conditionally onÛ = H, the probability that none of these vertices is a success can be bounded by Proposition 5.7: = o(1∕n).
This is uniform over H, hence the probability that no vertex is a success, without conditioning, is still o(1∕n). It follows that, with high probability, every v ∈ [n] has at least one successful vertex on the boundary of its k + n -exploration graph. This means that the distance of every vertex v ∈ [n] from Core n is at most k + n + h n = k + n + o(log log n), by (5.5). Recalling (5.1), we have completed the proof of Corollary 5.8 and thus of Statement 2.6. ▪ To prove Proposition 5.7, we need the following technical (but simple) result: Lemma 5.9 (High-degree fresh vertices) Consider the process of building a configuration model CM n as described in Remark 3.1. Let  l be the random graph obtained after l pairings of half-edges and let V l be the random vertex incident to the half-edge to which the lth half-edge is paired. For all l, n ∈ N and z ∈ [0, ∞) such that l ≤ n 4 (1 − F ,n (z)), (5.10) the following holds∶ In particular, when Conditions 1.1 and 1.2 hold, for every > 0 there are c > 0, n 0 < ∞ such that Proof By definition of CM n , the (l + 1)st half-edge is paired to a uniformly chosen half-edge among the n − 2l − 1 that are not yet paired. Consequently Since | l | ≤ 2l ≤ n 2 (1 − F ,n (z)) by (5.10), we obtain which proves (5.11). Assuming Conditions 1.1 and 1.2, we have n = n(1 + o(1)), with ∈ (0, ∞), see (3.5), and there are c 1 > 0 and > 1∕2 such that 1 − F ,n (z) ≥ c 1 z −( −1) for 0 ≤ z ≤ n . Consequently, for 0 ≤ z ≤ n 1∕3 , the right hand side of (5.10) is at least n 4 c 1 n ( −1)∕3 . Note that ( − 1)∕3 < 2∕3 (because < 3), hence we can choose n 0 so that n 4 c 1 n ( −1)∕3 ≥ n 1∕3 for all n ≥ n 0 . This directly leads to (5.12). ▪ With Lemma 5.9 in hand, we are able to prove Proposition 5.7: Proof of Proposition 5. 7 We fix v ∈ [n] and a realization H ofÛ =Û ≤k + n (v). We abbreviate The vertices on the boundary ofÛ are denoted by x 1 , … , x N . We start proving (5.6), hence we focus on x 1 and we define v (1) (1) h n as in Definition 5.5, with v (1) 0 = x 1 . We first fix some parameters. Since 2 < < 3, we can choose , > 0 small enough so that Next we define a sequence (g ) ∈N 0 that grows doubly exponentially fast: Then we fix B = 1∕ and C = log( ∕ log 2) in (5.5), where is the same constant as in Core n , see (2.8). With these choices, we have g h n = e e ⌈log log log n⌉ > e log log n = (log n) , while g h n −1 < (log n) . (5.18) Roughly speaking, the idea is to show that, with positive probability, one has v (1) j > g j . As a consequence, v (1) h n > g h n ≥ (log n) , that is v (1) h n belongs to Core n and x 1 is a success. The situation is actually more involved, since we can only show that v (1) j > g j before reaching Core n .
Let us make the above intuition precise. Recalling (5.15), let us set Then we introduce the events In words, the event T means that one of the vertices v (1) 0 , … , v (1) has already reached Core n , while the event W means that the degrees of vertices v (1) 0 , … , v (1) grow at least like g 0 , … , g and, furthermore, each v k is a fresh vertex (this is actually already implied by Definition 5.5, otherwise v k would not even be defined). We finally set Note that T h n coincides with S x 1 = "x 1 is a success." Also note that Consequently, if we prove that P * (E h n ) ≥ , then our goal P * (S x 1 ) ≥ follows (recall (5.6)).
The reason for working with the events E j is that their probabilities can be controlled by an induction argument. Recalling (5.15), we can write The key point is the following estimate on the conditional probability, proved below: with > 0 is defined in (5.16) and c > 0 is the constant appearing in relation (5.12). This yields which leads us to Since ∑ j≥0 j < ∞ and j < 1 for every j ≥ 0, by (5.21) and (5.17), the infinite product is strictly positive. Also note that P * (E 0 ) = P * ( v (1) 0 ≥ 2) = 1, because g 0 = 2 and v (1) 0 ≥ min ≥ 3. Then > 0, as required.
It remains to prove (5.21). To lighten notation, we rewrite the left hand side of (5.21) as Note that, on the event D j ⊆ W j , vertex v (1) j is fresh (ie, it is connected to the graph for the first time), hence it has m = v (1) j − 1 unpaired half-edges. These are paired uniformly, connecting v (1) j to (not necessarily distinct) vertices w (1) , … , w (m) . Let us introduce for 1 ≤ ≤ m the event the left hand side of (5.21) can be estimated by ) .

(5.24)
We claim that we can apply relation (5.12) from Lemma 5.9 to each of the probabilities in the last line of (5.24). To justify this claim, we need to look at the conditioning event D j ∩ C k−1 , recalling (5.23), (5.22) and (5.19). In order to produce it, we have to do the following: ⊳ First we build the k + n -exploration graphÛ ≤k + n (v) = H, which requires to pair at most O(( min − 1) k + n ) = O((log n) 1+ ) half-edges (recall Definition 5.1); ⊳ Next, starting from the boundary vertex x 1 , we generate the fresh vertices v (1) 0 , … , v (1) j all outside Core n , because we are on the event T c j , and this requires to pair a number of half-edges which is at most (log n) j ≤ (log n) h n = O((log n) +1 ); ⊳ Finally, in order to generate w (1) , … , w (k−1) , we pair exactly k − 1 half-edges, and note that It follows that the conditioning event D j ∩ C k−1 is in the -algebra generated by  l for l ≤ O((log n) 1+ + ) (we use the notation of Lemma 5.9). In particular, l ≤ n 1∕3 . Also note that z = g j+1 ≤ g h n = O((log n) ), see (5.18), hence also z ≤ n 1∕3 . Applying (5.12), we get ) because 1 − x ≤ e −x and n − 1 ≥ n∕2 for all n ≥ 2 (note that g j ≥ g 0 = 2). Since g j+1 = (g j ) e , by (5.17), we finally arrive at which is precisely (5.21). This completes the proof of (5.6).
In order to prove (5.7), we proceed in the same way: for any fixed 2 ≤ i ≤ N, we start from the modification of (5.15) given by P * ( ⋅ ) ∶= P( ⋅ |Û = H, S c x 1 , … , S c x i−1 ) and we follow the same proof, working with the vertices v (i) (1) h n (recall Definition 5.5). We leave the details to the reader. ▪

UPPER BOUND FOR PREFERENTIAL ATTACHMENT MODEL
In this section we prove Statements 2.5 and 2.6 for the preferential attachment model. By the discussion in Section 2.2, this completes the proof of the upper bound in Theorem 1.5, because the proof of Statement 2.4 is already known in the literature, as explained below Statement 2.4.

Proof of Statement 2.5
Recall the definition of Core t in (2.8). It is crucial that in Core t , we let D t∕2 (v) be large. We again continue to define what a k-exploration graph and its collisions are, but this time for the preferential attachment model: Definition 6.1 (k-exploration graph) Let (PA t ) t≥1 be a preferential attachment model. For v ∈ [t], we call the k-exploration graph of v to be the subgraph of PA t , where we consider the m edges originally incident to v, and the m edges originally incident to any other vertex that is connected to v in this procedure, up to distance k from v. Definition 6.2 (Collision) Let (PA t ) t≥1 be a preferential attachment model with m ≥ 2, and let v be a vertex. We say that we have a collision in the k-exploration graph of v when one of the m edges of a vertex in the k-exploration graph of v is connected to a vertex that is already in the k-exploration graph of v.
Now we want to show that every k-exploration graph has at most a finite number of collisions before hitting the Core t , as we did for the configuration model. The first step is to use Dommers and coworkers [9, Lemma 3.9]: Lemma 6.3 (Early vertices have large degree) Fix m ≥ 1. There exists a > 0 such that for some > 1∕(3 − ). As consequence, [t a ] ⊆ Core t with high probability.
In agreement with (2.10) (see also (4.12)), we set We want to prove that the exploration graphÛ ≤k + t (v) has at most a finite number of collisions before hitting Core t , similarly to the case of CM n , now for PA t . As it is possible to see from (2.8), Core t ⊆ [t∕2], that is, is a subset defined in PA t when the graph has size t∕2. As a consequence, we do not know the degree of vertices in [t∕2] when the graph has size t. However, in Dommers and coworkers [9,Appendix A.4] the authors prove that at time t all the vertices t∕2 + 1, … , t have degree smaller than (log t) .
We continue by giving a bound on the degree of vertices that are not in Core t . For vertices i ∈ [t∕2]⧵Core t we know that D t∕2 (i) < (log t) , see (2.8), but in principle their degree D t (i) at time t could be quite high. We need to prove that this happens with very small probability. Precisely, we prove that, for some B > 0, This inequality implies that when a degree is at most (log t) at time t∕2, then it is unlikely to grow by B(log t) between time t∕2 and t. This provides a bound on the cardinality of incoming neighborhoods that we can use in the definition of the exploration processes that we will rely on, in order to avoid Core t . We prove (6.3) in the following lemma that is an adaptation of the proof of Dommers and coworkers [9,Lemma A.4]. Its proof is deferred to [6, Appendix B]: Lemma 6.4 (Old vertex not in Core t ) There exists B ∈ (0, ∞) such that, for every i ∈ [t∕2], We can now get to the core of the proof of Statement 2.5, that is we show that there are few collisions before reaching Core t : Lemma 6.5 (Few collisions before hitting the core) Let (PA t ) t≥1 be a preferential attachment model, with m ≥ 2 and ∈ (−m, 0). Fix a ∈ (0, 1) and l ∈ N such that l > 1∕a. With k + t as in (6.2), the probability that there exists a vertex v ∈ [t] such that its k + t -exploration graph has at least l collisions before hitting Core t ∪ [t a ] is o(1).
Next we give a lower bound on the number of vertices on the boundary of a k + n -exploration graph. First of all, for any fixed a ∈ (0, 1), we notice that the probability of existence of a vertex in  Together, Lemmas 6.3, 6.5 and 6.6 complete the proof of Statement 2.5.
The rest of this section is devoted to the proof of Lemma 6.5. We first need to introduce some notation, in order to be able to express the probability of collisions. We do this in the next subsection.

Ulam-Harris notation for trees
Define where W 0 ∶= ∅. We use W ≤k as a universal set to label any regular tree of depth k, where each vertex has m children. This is sometimes called the Ulam-Harris notation for trees. Given y ∈ W and z ∈ W m , we denote by (y, z) ∈ W +m the concatenation of y and z. Given Given a finite number of points z 1 , … , z m ∈ W ≤k , abbreviate ⃗ z m = (z 1 , … , z m ), and define W (⃗ z m ) ≤k to be the tree obtained from W ≤k by cutting the branches starting from any of the z i 's (including the z i 's themselves): (6.6) Remark 6.7 (Total order) The set W ≤k comes with a natural total order relation, called shortlex order, in which shorter words precede longer ones, and words with equal length are ordered lexicographically. More precisely, given x ∈ W and y ∈ W m , we say that x precedes y if either < m, or if = m and x i ≤ y i for all 1 ≤ i ≤ . We stress that this is a total order relation, unlike the descendant relation ⪰ which is only a partial order. (Of course, if y ⪰ x, then x precedes y, but not vice versa).

Collisions
We recall that, given z ∈ [t] and j ∈ [m], the jth half-edge starting from vertex z in PA t is attached to a random vertex, denoted by z,j . We can use the set W ≤k to label the exploration graphÛ ≤k (v), as follows:Û where V ∅ = v and, iteratively, V z = V x ,j for z = (x, j) with x ∈ W ≤k−1 and j ∈ [m].
The first vertex generating a collision is V Z 1 , where the random index Z 1 ∈ W ≤k is given by where "min" refers to the total order relation on W ≤k as defined in Remark 6.7. Now comes a tedious observation. Since V Z 1 = V y for some y which precedes Z 1 , by definition of Z 1 , then all descendants of Z 1 will coincide with the corresponding descendants of y, that is V (Z 1 ,r) = V (y,r) for all r. In order not to over count collisions, in defining the second collision index Z 2 , we avoid exploring the descendants of index Z 1 , that is we only look at indices in W (Z 1 ) ≤k , see (6.6). The second vertex representing a (true) collision is then V Z 2 , where we define Iteratively, we define so that V Z i is the ith vertex that represents a collision. The procedure stops when there are no more collisions. Denoting by  the (random) number of collisions, we have a family of random elements of W ≤k , such that (V Z i ) 1≤i≤ are the vertices generating the collisions.

Proof of Lemma 6.5
Recalling (6.7) and (6.6), given arbitrarily z 1 , … , z l ∈ W ≤k , we definê that is, we consider a subset of the full exploration graphÛ ≤k (v), consisting of vertices V z whose indexes z ∈ W ≤k are not descendants of z 1 , … , z l . The basic observation is that In words, this means that to recover the full exploration graphÛ ≤k (v), it is irrelevant to look at vertices V z for z that is a descendant of a collision index z 1 , … , z l .
We will bound the probability that there are l collisions before reaching Core t ∪ [t a ], occurring at specified indices z 1 , … , z l ∈ W ≤k , for k = k + t as in (6.2), as follows: where for the constant B given by Lemma 6.4, we define (6.11) Summing (6.10) over z 1 , … , z l ∈ W ≤k we get Since, for k = k + t as in (6.2), we can bound the probability of having at least l collisions, before reaching Core t ∪ [t a ], is O( (t) l (log t) 2l ) = o(1∕t), because l > 1∕a by assumption. This completes the proof of Lemma 6.5. It only remains to show that (6.10) holds true.

6.1.4
Proof of (6.10): case l = 1 We start proving (6.10) for one collision. By (6.9), we can replaceÛ ≤k (v) byÛ (z 1 ) ≤k (v) in the left hand side of (6.10), that is, we have to prove that (6.13) Since v, k and z 1 are fixed, let us abbreviate, and recalling (6.8), (6.14) Note that V z 1 is the only collision precisely whenÛ is a tree and V z 1 ∈Û. Then (6.13) becomes We will actually prove a stronger statement: for any fixed deterministic labeled directed tree H ⊆ [t] and for any y ∈ H, This yields (6.15) by summing over y ∈ H-note that |H| ≤ |W ≤k | ≤ 2(log t) 1+ by (6.12)-and then summing over all possible realizations of H.
It remains to prove (6.16). We again use the notion of a factorizable event, as in the proof of the lower bound. Since the events in (6.16) are not factorizable, we will specify the incoming neighborhood  (y) (recall (4.51)) of all y ∈ H. More precisely, by labeling the vertices of H, see (6.14), as (v x , j) ∈ N v̄s . (6.21) Let us summarize where we now stand: When we fix a family of (N v s ) s∈ that is compatible and satisfies the constraints (6.19) and (6.21), in order to prove (6.16) it is enough to show that The probability on the left-hand side of (6.22) can be factorized, using conditional expectations and the tower property, as a product of two kinds of terms: ⊳ For every edge (u, r) ∈ N-say (u, r) ∈ N v s , with s ∈ -we have the term corresponding to the fact that the edge needs to be connected to v s ; ⊳ On the other hand, for every edge (u, r) ∉ N, we have the term corresponding to the fact that the edge may not connect to any vertex in H.
(We emphasize that all the degrees D ⋅,⋅ ( ⋅ ) appearing in (6.24) and (6.25) are deterministic, since they are fully determined by the realizations of the incoming neighborhoods (N v s ) s∈ .) We can obtain the right-hand side in (6.22) by replacing some terms in the product.
⊳ Among the edges (u, r) ∈ N, whose contribution is (6.24), we have the one that creates the collision, namely (v x , j). If we want this edge to be connected outside H, as in the right-hand side in (6.22), we need to divide the left hand side of (6.22) by We also have to replace some other terms corresponding to edges (u, r) ∈ N v̄s , because the degree of vertex vs is decreased by one after connecting (v x , j) outside H. More precisely, for every edge (u, r) ∈ N v̄s that is younger than (v x , j), that is (u, r) > (v x , j), we can reduce the degree of vs by one by dividing the left-hand side of (6.22) by Finally, the contribution of the edges (u, r) ∈ N v s for s ≠s is unchanged. ⊳ For every edge (u, r) ∉ N, the probability that such edge is not attached to H, after we reconnect the edge (v x , j), becomes larger, since the degree of H is reduced by one.
Since c v,j ≥ m(v − 1), the first relation in (6.19) yields Hence, since D t (vs) ≤ (1 + B)(log t) by the second relation in (6.19), we can bound Likewise, since D t (H) ≤ |H|(1 + B)(log t) , for k = k + t we get, by (6.12), where the last inequality holds for t large enough. Recalling (6.11), This completes the proof of (6.22), and hence of (6.10), in the case where l = 1.

6.1.5
Proof of (6.10): general case l ≥ 2 The proof for the general case is very similar to that for l = 1, so we only highlight the (minor) changes.
In analogy with (6.13), we can replaceÛ ≤k (v) byÛ in the left-hand side of (6.10), thanks to (6.9). Then, as in (6.14), we write The extension of (6.16) becomes that for any fixed deterministic labeled directed tree H ⊆ [t] and for all y 1 , … , y l ∈ H, As in (6.17), we can write H = {v s } s∈ and y 1 = vs 1 , … , y l = vs l for somes 1 , … ,s l ∈ .
To obtain a factorizable event, we must specify the incoming neighborhoods  v s = N v s for all s ∈ , which must be compatible with H and satisfy the constraint (6.19). If we write for some then we also impose the constraint that obviously generalizes (6.21), namely The analogue of (6.22) then becomes When we define N as in (6.23), the probability in the left-hand side of (6.31) can be factorized in a product of terms of two different types, which are given precisely by (6.24) and (6.25). In order to obtain the probability in the right-hand side of (6.31), we have to divide the left-hand side by a product of factors analogous to (6.26) and (6.27). More precisely, (6.26) becomes We define accordingly, namely we take the product for i = 1, … , l of (6.28) with x, j,s replaced respectively by x i , j i ,s i . Then it is easy to show that arguing as in the case l = 1. This completes the proof of (6.31).

Proof of Statement 2.6
The next step is to prove that the boundaries of the k + t -exploration graphs are at most at distance h t = ⌈B log log log t + C⌉ (6.33) from Core t , where B, C are constants to be chosen later on. Similarly to the proof in Section 5.2, we consider a k + t -exploration graph, and we enumerate the vertices on the boundary as x 1 , … , x N , where N ≥ s(m, l)m k + t from Lemma 6.6 and l is chosen as in Lemma 6.5. We next define what it means to have a success: Definition 6.8 (Success) Consider the vertices x 1 , … , x N on the boundary of a k + t -exploration graph. We say that x i is a success when the distance between x i and Core t is at most 2h t .
The next lemma is similar to Lemma 5.7 (but only deals with vertices in [t∕2]): Lemma 6.9 (Probability of success) Let (PA t ) t≥1 be a preferential attachment model, with m ≥ 2 and ∈ (−m, 0). Consider v ∈ [t∕2] ⧵ Core t and its k + t -exploration graph. Then there exists a constant > 0 such that

34)
and for all j = 2, … , N, The aim is to define a sequence of vertices w 0 , … , w h that connects a vertex x i on the boundary with Core t . In order to do this, we need some preliminar results. We start with the crucial definition of a t-connector: Definition 6.10 (t-connector) Let (PA t ) t≥1 be a preferential attachment model, with m ≥ 2. Consider two subsets A, B ⊆ [t∕2], with A ∩ B = ∅. We say that a vertex j ∈ [t] ⧵ [t∕2] is a t-connector for A and B if at least one of the edges incident to j is attached to a vertex in A and at least one is attached to a vertex in B.
The notion of t-connector is useful, because, unlike in the configuration model, in the preferential attachment model typically two high-degree vertices are not directly connected. From the definition of the preferential attachment model, it is clear that the older vertices have with high probability large degree, and the younger vertices have lower degree. When we add a new vertex, this is typically attached to vertices with large degrees. This means that, with high probability, two vertices with high degree can be connected by a young vertex, which is the t-connector.
A further important reason for the usefulness of t-connectors is that we have effectively decoupled the preferential attachment model at time t∕2 and what happens in between times t∕2 and t. When the sets A and B are appropriately chosen, then each vertex will be a t-connector with reasonable probability, and the events that distinct vertices are t-connectors are close to being independent. Thus, we can use comparisons to binomial random variables to investigate the existence of t-connectors. In order to make this work, we need to identify the structure of PA t∕2 and show that it has sufficiently many vertices of large degree, and we need to show that t-connectors are likely to exist. We start with the latter.
In more detail, we will use t-connectors to generate the sequence of vertices w 1 , … , w h between the boundary of a k + n -exploration graph and the Core t , in the sense that we use a t-connector to link the vertex w i to the vertex w i+1 . (This is why we define a vertex x i to be a success if its distance from Core t is at most 2h t , instead of h t .) We rely on a result implying the existence of t-connectors between sets of high total degree: Lemma 6.11 (Existence of t-connectors) Let (PA t ) t≥1 be a preferential attachment model, with m ≥ 2 and ∈ (−m, 0 , (6.36) where D t∕2 (A) = ∑ v∈A D t∕2 (v) is the total degree of A at time t∕2.
Proof The proof of this lemma is present in the proof of Dommers and coworkers [9, Proposition 3.2]. ▪ Remark 6.12 Notice that this bound depends on the fact that the number of possible t-connectors is of order t.
A last preliminary result that we need is a technical one, which plays the role of Lemma 5.9 for the configuration model and shows that at time t∕2 there are sufficiently many vertices of high degree, uniformly over a wide range of what 'large' could mean: We now use a concentration result on the empirical degree distribution (for details, see [13,Theorem 8.2]), which assures us that there exists a second constant C > 0 such that, with high probability, for every x ∈ N, (6.39) Fix now > 0, then from this last bound we can immediately write, for a suitable constantc as in (6.38), This is clearly true for x ≤ (log t) q , for any positive q. Taking c =c∕2 completes the proof. ▪ With the above tools, we are now ready to complete the proof of Lemma 6.9: Proof of Lemma 6.9 As in the proof of Proposition 5.7, we define the super-exponentially growing sequence g as in (5.17), where > 0 is chosen small enough, as well as > 0, so that (5.16) holds. The constants B and C in the definition (6.33) of h t are fixed as prescribed below (5.17). We will define a sequence of vertices w 0 , … , w h such that, for i = 1, … , h, D t (w i )(t) ≥ g i and w i−1 is connected to w i . For this, we define, for i = 1, … , h − 1, so that we aim for w i ∈ H i .
We define the vertices recursively, and start with w 0 = x 1 . Then, we consider t-connectors between w 0 and H 1 , and denote by w 1 the vertex in H 1 with minimal degree among the ones that are connected to w 0 by a t-connector. Recursively, consider t-connectors between w i and H i+1 , and denote by w i+1 the vertex in H i+1 with minimal degree among the ones that are connected to w i by a t-connector. Recall (5.18) to see that g h t ≥ (log t) , where h t is defined in (6.33). The distance between w 0 and Core t is at most 2h t = 2⌈B log log log t + C⌉. If we denote the event that there exists a t connector between w i−1 and H i by {w i−1 ∼ H i }, then we will bound from below . (6.43) In Lemma 6.11, the bound on the probability that a vertex j ∈ [t] ⧵ [t∕2] is a t-connector between two subsets of [t] is independent of the fact that the other vertices are t-connectors or not. This means that, with  i the -field generated by the path formed by w 0 , … , w i and their respective t-connectors, where D t (H i ) = ∑ u∈H i D t∕2 (u). This means that . (6.45) We have to bound every term in the product. Using Lemma 6.13, for i = 1, 1 − e − D t∕2 (w 0 )D t∕2 (H 1 )∕t ≥ 1 − e − D t∕2 (w 0 )g 1 P ≥g 1 (t∕2) , (6.46) while, for i = 2, … , h − 1 Applying (6.37) and recalling (5.25)-(5.26), the result is ) , (6.48) for some constantc. Since h t = ⌈B log log log t + C⌉, and with high probability as t → ∞, we can find a constant such that > > 0, (6.50) which proves (6.34).
To prove (6.35), we observe that all the lower bounds that we have used on the probability of existence of t-connectors only depend on the existence of sufficiently many potential t-connectors. Thus, it suffices to prove that, on the event S c x 1 ∩ · · · ∩ S c x j−1 , we have not used too many vertices as t-connectors. On this event, we have used at most h t ⋅(j−1) vertices as t-connectors, which is o(t). Thus, this means that, when we bound the probability of S x j , we still have t − h t ⋅ (j − 1) possible t-connectors, where j is at most (log t) 1+ . Thus, with the same notation as before, so that we can proceed as we did for S x 1 . We omit further details. ▪ We are now ready to identify the distance between the vertices outside the core and the core: Proposition 6.14 (Distance between periphery and Core t ) Let (PA t ) t≥1 be a preferential attachment model with m ≥ 2 and ∈ (−m, 0). Then, with high probability and for all v ∈ [t] ⧵ Core t , Proof We start by analyzing v ∈ [t∕2]. By Lemma 6.3, with high probability there exists a ∈ (0, 1] such that [t a ] ⊆ Core t . Consider l > 1∕a, and fix a vertex v ∈ [t∕2]. Then, by Lemma 6.5 and with high probability, the k + t -exploration graph starting from v has at most l collisions before hitting Core t . By Lemma 6.6 and with high probability, the number of vertices on the boundary of the k + t -exploration graph is at least N = s(m, l)(log t) 1+ . It remains to bound the probability that none of the N vertices on the boundary is a success, meaning that it does not reach Core t in at most 2h t = 2⌈B log log t + C⌉ steps.
By Lemma 6.9, 53) thanks to the bound N ≥ s(m, l)(log t) 1+ . This means that the probability that there exists a vertex v ∈ [t∕2] such that its k + n -exploration graph is at distance more than A log log log t from Core t is o(1). This proves the statement for all v ∈ [t∕2].
Next, consider a vertex v ∈ [t] ⧵ [t∕2]. Lemma 6.5 implies that the probability that there exists a vertex v ∈ [t] ⧵ [t∕2] such that its k + t -exploration graph contains more than one collision before hitting Core t ∪ [t∕2] is o (1). As before, the number of vertices on the boundary of a k + t -exploration graph starting at v ∈ [t] ⧵ [t∕2] is at least N ≥ s(m, 1)m k + n = s(m, 1)(log t) 1+ . We denote these vertices by x 1 , … , x N . We aim to show that, with high probability, (6.54) For every i = 1, … , N, there exists a unique vertex y i such that y i is in the k + t -exploration graph and it is attached to x i . Obviously, if y i ∈ [t∕2] then also x i ∈ [t∕2], since x i has to be older than y i . If y i ∉ [t∕2], then and this bound does not depend on the attaching of the edges of the other vertices {y j ∶ j ≠ i}. This means that we obtain the stochastic domination , (6.56) where we write that X ⪰ Y when the random variable X is stochastically larger than Y. By concentration properties of the binomial, Bin Thus, the probability that none of the vertices on the boundary intersected with [t∕2] is a success is bounded by We conclude that the probability that there exists a vertex in [t] ⧵ [t∕2] such that it is at distance more than k + t + 2h t from Core t is o(1). This completes the proof of Statement 2.6, and thus of Theorem 1.5. ▪