Local degree distribution in scale free random graphs

In several scale free graph models the asymptotic degree distribution and the characteristic exponent change when only a smaller set of vertices is considered. Looking at the common properties of these models, we present sufficient conditions for the almost sure existence of asymptotic degree distribution constrained on the set of selected vertices, and identify the characteristic exponent belonging to it.


Introduction
Since the end of the nineties several complex real world networks and their random graph models have been investigated [4,5,6]. Many of them possess the scale free property: the tail of the degree distribution decreases polynomially fast, that is, if c d denotes the proportion of vertices of degree d, then c d ≈ C · d −γ holds for large values of d [1]. γ is called the characteristic exponent.
If the whole network is completely known, the empirical estimator of the characteristic exponent may have nice properties. However, real world networks usually are too large and complex, hence our knowledge of the graph is partial. For several models of evolving random graphs the degree distribution and the characteristic exponent change when attention is restricted to a set of selected vertices that are close to the initial configuration [10,11,13].
Starting from these phenomena, in this paper the degree distribution constrained on a set of selected vertices will be investigated, assuming that the graph model possesses the scale free property with characteristic exponent γ > 1, and the number of selected vertices grows regularly with exponent 0 < α ≤ 1. Sufficient conditions for the almost sure existence of the local asymptotic degree distribution will be given. It will be shown that under these conditions the characteristic exponent of the constrained degree distribution is α (γ − 1) + 1.
The proofs are based on the methods of martingale theory. Applications of the general results to different graph models (e.g. to the Albert-Barabási random tree [1]) will be shown.
In Section 2 we present the family of random graph models to be examined and formulate the sufficient conditions. In Sections 3 and 4 we mention some results about martingales and slowly varying sequences to be applied in the proofs. Section 5 contains the proof of the main results, and in Section 6 we give some examples and applications.

Main results
In this section we present sufficient conditions for the almost sure existence of asymptotic degree distribution constrained on the set of selected vertices, and we describe that distribution.
Let (G n = (V n , E n )) n∈N be a sequence of evolving simple random graphs. Some vertices are distinghuised; let S n ⊆ V n denote the set of selected vertices.
We start from a finite, simple graph G 0 = (V 0 , E 0 ), this is the initial configuration with V 0 = {u 1 , u 2 , . . . , u l }. S 0 ⊆ V 0 is arbitrarily chosen. For n ≥ 1, at the n th step • one new vertex, v n , is added to the graph: V n = V 0 ∪{v 1 , . . . , v n }; • the new vertex gets some random edges, thus E n−1 ⊆ E n , and every edge from E n \ E n−1 is connected to v n ; • the new vertex can be added to the set of selected vertices, v n ∈ S n is a random choice. The σ-field of events generated by the first n steps is denoted by F n .
For v ∈ V n let the degree of v in G n be denoted by deg n (v). Furthermore, for n ≥ 1, and d ≥ 0 define In some models it is possible that the new vertex does not get any edges at some steps. In other models the degree of the new vertex is fixed, for example, the degree of the new vertex is always 1 in random tree models. If the new vertex gets at least m edges at each step for some m > 0, then X [n, d] is at most |V 0 | for all n and d < m. Thus we denote the minimal initial degree of the new vertex by m, and we consider X [n, d] only for d ≥ m. Of course, m = 0 is also possible.

2.1.
Conditions on the graph model. We say that a discrete probability distribution (a n ) is exponentially decreasing if a n ≤ C · q n holds for all n ≥ 1 for some C > 0 and 0 < q < 1 . A sequence (a n ) is slowly varying if a [sn] /a n → 1 as n → ∞ for all s > 0.
Throughout this paper, for two sequences (a n ) , (b n ) of nonnegative numbers, a n ∼ b n means that b n > 0 except finitely many terms, and a n /b n → 1 as n → ∞. Now we can formulate the conditions on the graph model. This means that asymptotic degree distribution exists in this graph model. Note that X [n, d] → ∞ as n → ∞ almost surely.
This is the so called scale free property with characteristic exponent γ. That is, the asymptotic degree distribution decays polynomially with exponent γ. This implies that c d is positive for every d large enough, but we will need it for all d ≥ m, this is included in Condition 1.
Condition 3. For every n ≥ 0, if w 1 , w 2 ∈ V n and deg n (w 1 ) = deg n (w 2 ), then In other words, at each step, conditionally on the past, old vertices of the same degree get connected to the new vertex with the same probability.
is an exponentially decreasing probability distribution.
Loosely speaking, the degree of the new vertex has an exponentially decreasing asymptotic distribution. This trivially holds if the degree of the new vertex is fixed.
In many particular cases the following stronger condition is also met. There exists a random variable Z ≥ 0 with exponentially decreasing distribution such that This is a sort of upper bound for the initial degree of the new vertex.
We will see later that the nonnegativity of k d follows from the previous conditions; however, the positivity of k d cannot be omitted, as an example will show.

2.2.
Conditions on the set of selected vertices. Recall that S n ⊆ V n is the set of selected vertices in G n . We emphasize that deg n (v) always denotes the degree of vertex v in G n , not in S n .
We will need the following notations. The σ-field generated by the first n steps and adding the edges of v n+1 at the (n + 1)st step is denoted by F + n . Furthermore, for n ≥ 1 and d ≥ m let 1 if v n ∈ S n and deg n (v n ) = d, 0 otherwise; The conditions on the set of selected vertices are the following.
Vertices cannot be deleted from the set of selected vertices.
Condition 8. I * (n + 1) is F + n -measurable for all n ≥ 0. At each step we have to decide whether the new vertex is to be selected immediately after choosing its neighbours. Selecting the neighbours of a fixed vertex is an example.

Condition 9.
There exists a sequence of positive random variables (ζ n ) that are slowly varying as n → ∞, and |S n | = n i=1 I * (i) ∼ ζ n · n α for some α > 0, with probability 1.
This means that the size of the set of selected vertices is regularly growing with exponent α > 0.
holds a.s. as n → ∞, with some exponentially decreasing probability distribution (q d ) d≥m .
The last condition holds if the degree of the new vertex v n is fixed, or its degree and I * (n) are independent conditionally on F n−1 . In that case sequence q d = p d satisfies the condition. It is also possible that the asymptotic degree distribution of the new selected vertices is different from (p d ) if only it decays exponentially fast.

2.3.
Description of the local degree distribution. Now we formulate the main results.
Theorem 1. Suppose that Conditions 1-10 hold for a random graph model (G n , S n ), then the limits The constants x d satisfy the following recursive equations.

Remark 1.
From the proof it is clear that with Condition 2 dropped the limits x d still exist and the recursive equations remain valid. The role of the scale free property of the graph is just to guarantee that the asymptotic degree distribution constrained on the set of selected vertices is also polynomially decaying.

Martingales
We will extensively use the following propositions that are based on well-known facts of martingale theory.
that is, the predictable increasing process in the Doob decomposition of M 2 n . Then M n = o A 1/2 n log A n holds almost surely on the event {A ∞ = ∞}, and M n converges to a finite limit, as n → ∞, almost surely on the event This is a corollary of Propositions VII-2-3 and VII-2-4 of [14].
Proposition 2. Let (M n , G n ) be a square integrable nonnegative submartingale, and This is easy to prove applying Proposition 1 to the martingale part of the Doob decomposition of M n .
holds almost everywhere on the event { ∞ n=1 Y n = ∞ }. This proposition follows from the Lévy generalization of the Borel-Cantelli lemma that can be found in [14] (Corollary VII-2-6).

Slowly varying sequences
In the proofs we will use the basic results of the theory of regularly varying sequences, see e.g. [2,3,7].
We say that a sequence of positive numbers (β n ) is regularly varying with exponent µ if the following holds: where (γ n ) is slowly varying.
is slowly varying as n → ∞, and n −λ β n → 1 as n → ∞ for some λ > −1. Then the following holds.
This is a consequence of the results of Bojanić and Seneta [2,3].
All sequences are nonnegative, hence we have as n → ∞. Combining this with (2) we get that the second term on the right-hand side of (3) is o (α n γ n n µ ) as n → ∞.
The first term is asymptotically equal to α n γ n n µ as n → ∞. Thus we get that Next, let δ differ from 0. Let α n = κ n n δ , and B n = γ n n µ with slowly varying sequences (κ n ) and (γ n ). We have (γ n ) is slowly varying, and λ = δ + µ − 1 > −1, thus Proposition 4 applies, and we obtain that Let us apply the already proved particular case to (κ n ) and n δ β n . Then we get that We can suppose that B n is increasing, since for every regularly varying sequence with positive exponent one can find another, increasing one, which is equivalent to it.
hence it is regularly varying with exponent µ. By part a) we have After subtraction we obtain that n i=1 α i β i = o(α n B n ). Proposition 6. Let a 1 , a 2 . . . and b 1 , b 2 , . . . be nonnegative numbers satisfying is slowly varying as n → ∞ for every bounded sequence of real numbers (s n ).
The first sum on the right-hand side tends to 0, since The second sum is K log t + o (1), the third one is r m − r n−1 = o (1) , and the last one also converges to 0.
By supposition We can complete the proof similarly to (4).

Proofs
For sake of convenience, instead of X * [n, d], we consider the number Z * [n, d] of selected vertices with degree greater than or equal to d. That is, for n ≥ 1 and d ≥ m let We also need the following notations. First we show that Theorem 1 is implied by the following proposition. For all d ≥ m we have Z * [n, d] ∼ z d |S n | a.s. as n → ∞ with some positive constants z d . In addition, It is clear that It is easy to derive the recursive equations for x d = z d − z d+1 from z m = 1 and equation (6). The denominators are positive, because Conditions 1, 6, and 9 guarantee that c d is nonnegative, α is positive, and k d is positive.
It is also easy to check that sequence (x d ) is a probability distribution. We have Summing up the equations above we get that since, by Conditions 9 and 10, α > 0 and the sequence (q d ) is a probability distribution. The next step is solving the recursion for (x d ). Set It is easy to check that the recursive equations of Theorem 1 are satisfied by the sequence By Condition 2, c d ∼ K · d −γ holds as d → ∞, and by Condition 4 the sequence (p j ) is exponentially decreasing. Hence it follows, as d → ∞, that for some K ′ > 0. By Condition 10 the sequence (q d ) is exponentially decreasing, thus the series in the expression converges. Using the asymptotics of (a d ) and (t d ) we get that Consequently, the degree distribution constrained on the set of selected vertices decays polynomially, and the new characteristic exponent is determined by α and γ, namely, γ * = α (γ − 1) + 1, as stated.
Therefore Theorem 1 is indeed a consequence of (6). Let us continue with the proof of (6). We proceed by induction on d.
The case d = m is obvious, because the initial degree is never less than m, and the degree of a vertex cannot decrease, thus every vertex in S n \ S 0 has at least m edges.
Suppose that holds for some z d−1 > and d ≥ m + 1 almost surely. First we determine the expected number of vertices of degree ≥ d in S n+1 , given F n , for n ≥ 1. Every vertex in S n counts if its degree is at least d in G n , or if its degree is equal to d − 1 in G n and it gets a new edge from v n+1 . The new vertex v n+1 counts if it falls into S n+1 and its degree is ≥ d in G n+1 . Thus the following equality holds for every n ≥ 1. Taking conditional expectations with respect to F n we obtain that By Condition 3, vertices of the same degree are connected to v n+1 with the same conditional probability. This implies that X [n, d] may be equal to zero, then Y [n, d] = 0 as well. We will consider all quotients of the form 0/0 as 1.
The middle term on the right-hand side of (8) can be transformed by the help of (9).
, hence from equation (10) we obtain that Set c [1, d] = 1 and for n ≥ 2 define Then for n large enough we have For several particular models it is quite easy to compute the conditional expectations E (Y [i, d − 1]| F i ), and hence, to determine the asymptotics of c [n, d]. In the present general case the conditional expectation is not specified. However, as the following sequence of lemmas shows, the asymptotics of the partial sums can be described, and one can calculate the asymptotics of c [n, d]. The proof of the lemmas will be postponed to the second part of this section. We emphasize that in the lemmas the induction hypothesis is assumed all along.
Consider the partial sums with probability 1. By equation (11), the process is a submartingale. Let A [n, d] denote the increasing process in the Doob decomposition of V [n, d]; it is given by First we describe the asymptotics of A [n, d]. Therefore Proposition 2 implies that V [n, d] ∼ A [n, d] almost surely as n → ∞. Finally, by Lemma 2 and Lemma 3 we obtain the asymptotics

Consequently, we have
The size of S n is asymptotically equal to ζ n n α by Condition 9. Thus the proof of (6) can be completed by using Lemmas 1-4. Now we continue with the proofs of Lemmas 1-4.
Proof of Lemma 1. Similarly to equation (7), but considering all vertices, we see that for every i ≥ 0 and j ≥ m. Adding up for i = 1, . . . , n we obtain that for every j ≥ m and n ≥ 1. By Conditions 1 and 4, from (17) it follows that holds almost surely, as n → ∞, for every j ≥ m. Adding this up for j = m, . . . , d we get a.s., as n → ∞. Therefore it is sufficient to prove that It is clear that (M n , G n ) is a martingale. Using Condition 5 we will derive an upper bound for the corresponding increasing process A n introduced in Proposition 1.
The infinite sum of these terms converges, thus the second sum on the right-hand side of (19) is bounded for fixed d.
On the other hand, Y [i, d] ≤ |V i | ≤ i + l follows from the definition, therefore Thus A n = O (n 2 log n). This bound can be further improved as follows.
Applying Proposition 1 to the martingale (M n ) we get that M n = O (n 1+η ) a.s. for all η > 0. Equation We obtain that A n = O (n 1+η log n). Hence by Proposition 1 we have M n = o n 1 2 +η log n a.e. on the event {A ∞ = ∞}, for all η > 0. Therefore M n = o(n) holds almost surely, and this completes the proof of Lemma 1.
Proof of Lemma 2. Fix an arbitrary d ≥ m. Lemma 1 and the induction hypothesis imply that Thus in (12) we can apply the approximation 1 − x = e −x+O(x 2 ) (x → 0). Set if X [i, d] > 0, and a i = b i = 0 otherwise. Then a i and b i are nonnegative. In addition, as n → ∞, by Lemma 1. According to the induction hypothesis, X [n, d] ∼ c d · n, which implies that nb n → 1 as n → ∞. Therefore Proposition 6 applies to the sequences (a n ) and (b n ) with K = k d /c d . Thus, due to part a), is regularly varying with exponent K.

16ÁGNES BACKHAUSZ AND TAMÁS F. MÓRI
The remainder terms produce a slowly varying function, because by part b) of Proposition 6 we get that Here we already know the asymptotics of S [n, d − 1] and c [n, d − 1] from Lemmas 1 and 2. In addition, Z * [n, d − 1] ∼ z d−1 |S n | ∼ z d−1 ζ n n α a.s., due to the induction hypothesis and Condition 9. It is clear from the definition that Y [n, d − 1] is nonnegative, thus we have Let us apply part a) of Proposition 5 in the following setting.
On the other hand, for a fixed j ≤ d − 1 we have by Lemma 2. In this case α n remains the same as before, and we set β n = E ( I * [n + 1, j]| F n ). Using Condition 10 and equation (21) we obtain that almost surely as n → ∞. Thus we can apply part a) or part b) of Proposition 5 with µ = α, according that q j vanishes or it is positive.

18ÁGNES BACKHAUSZ AND TAMÁS F. MÓRI
Then we get that almost surely as n → ∞. Hence we conclude that almost surely as n → ∞. Since (q d ) is a probability distribution by Condition 10, it follows that 1 − d−1 j=m q j = ∞ j=d q j . This completes the proof.
Proof of Lemma 4. From equation (14) it follows that By equation (7) we have We will estimate B 1 and B 2 separately. Similarly to the proof of Lemma 1, fix a positive ε > 0 such that κ = E e εZ d−1 < ∞, and set C i = 2 ε log i. Using Condition 5 and holds. For estimating the first term on the right-hand side we make use of equation (9).
To the second term we can apply (20 . From all these we obtain that Returning to (23) we conclude that Now the proof can be completed by comparing this with Lemma 3. Loosely speaking, the degree of a typical vertex is asymptotically larger than or equal to the degree of the new vertex. This is in accordance with the fact that the degree of a fixed vertex cannot decrease.
Similarly to Lemma 1, one can prove that d j=0 which means the same for the selected vertices.

Graph models
In this section we briefly review some scale free random graph models and sets of selected vertices to which the results of the previous section can be applied.
6.1. Generalized plane oriented recursive tree. We start from one edge, and at each step one new vertex and one new edge are added to the graph. At the nth step the probability that a given vertex of degree d is connected to v n is (d + β) /T n−1 , where β > −1 is the parameter of the model, and T n−1 = (2 + β) (n + 1) + β. These kind of random trees are widely examined, see for example [5,15]. β = 0 gives the Albert-Barabási tree [1].
We fix an integer j ≥ 1. At the nth step v n is added to the set of selected vertices if it is at distance j from u 1 in G n . Thus S n is the jth level of the tree G n .
6.2. Independent edges. We start from one edge. At the nth step, independently of each other, every old vertex is connected to the new one with probability λd/T n−1 , where d is the degree of the old vertex in G n−1 , 0 < λ < 2 is a fixed parameter, and T n−1 denotes the sum of degrees in G n−1 . The restriction on λ guarantees that the probability given above belongs to [0, 1]. It is clear that m = 0. We fix one vertex, v, and S n consists of its neighbours in G n .
In [8,Theorem 3.1.] it is proven that the asymptotic degree distribution is given by Clearly, c d ∼ 2λ (2 + λ) d −3 (d → ∞). Thus the first two conditions are satisfied, and γ = 3. Condition 3 holds, because the probability that a given vertex gets a new edge depends only on its actual degree. It is also clear that Conditions 7 and 8 hold. Condition 9 is a corollary of [11, Theorem 2.1], and we have α = 1/2. In this case the initial degree of the new vertex is not fixed. It is proven in [8] that Note that the conditional distribution of Y [n, d] is binomial of order X [n, d] and parameter λd/T n < 1. One can check Condition 5 with Z d having a suitable Poisson distribution.
Condition 10 can be verified basing on the fact that the degree distribution of a new selected vertex is similar to the distribution of a new vertex because of the independent random choices, and the following results. Theorem 2.1 in [12] states that T n = 2λn + o (n 1−ε ) almost surely if ε > 0 is sufficiently small. Moreover, Theorem 2.2 there implies that the maximum degree after n steps is O ( √ n) almost surely.