Limit theorems for the weights and the degrees in anN-interactions random graph model

The scale-free property of our model means the following (see [16]). Let N ≥ 3 be fixed. Let X(n, w) denote the number of vertices of weight w after n steps. Let 0 < p < 1, q > 0, r > 0 and (1 – r)(1 – q) > 0. Then for all w = 1, 2, : : : we have

almost surely, as n → ∞, where x_w, w = 1, 2, ..., are positive numbers satisfying

as w → ∞, withC0=pΓ(1+β+1α)/(αΓ(1+βα)). Let U(n, d) denote the number of vertices of degree d after n steps. Then, for any d ≥ N – 1 we have

a.s. as n → ∞, where u_d, d = N – 1, N, ..., are positive numbers with

as d → ∞, whereC1=pαα2−1+1αΓ1+β+1α/α2Γ1+βα.

In our model, besides the vertices, the cliques also have weights. It turns out that the weight distribution of the M -cliques is also power law. Let N ≥ 3 be fixed and let M be fixed with 1 < M ≤ N and denote by X_M(n, w) the number of M -cliques having weight w after n steps. If p > 0 and either r > 0 or (1 – p)q > 0, then

almost surely, as n → ∞, where x_M,_w, w = 1, 2, ..., are numbers satisfying

as w → ∞, with μ=p(N−1M−1)+p(1−r)(N−1M)+(1−p)(1−q)(NM). This result was presented in [15] for the case of M = 2, N = 3 and also for the case of M = N for arbitrary N > 2. The general case can be obtained by using the ideas of [18]. To this end one has to consider a slight modification of the general model of [18] and to prove appropriate versions of Theorems 2 and 3 in [18].

Now we turn to the asymptotic behaviour of a fixed clique and that of a fixed vertex. At time n = 0, the initial complete graph on N vertices contains (NM)M-cliques labelled by 0, –1, ..., −((NM)−1). When new M-cliques are born, they are labelled by 1, 2, .... Let j ≥ 0 be a fixed integer. Let us denote by W[n, M, j]  the weight of the jth M-clique after the nth step. Let us denote by D(n, j) the degree of the j th vertex after the nth step. (If n < j, then W[n, 1, j] = D[n, j] = 0.)

First, we study the weight of a fixed M-clique. At time n =0, the initial complete graph on N vertices is symmetric. Therefore and by the evolution mechanism of our graph, it is enough to describe W[n, M, j] and D[n, j] for j = 0, 1, 2, .... The following theorem describes the asymptotic behaviour of the weight of a fixed M -clique.

Let j ≥ 0 and 1 ≤ M ≤ N be fixed. Assume that α > 0. Then

almost surely as n → ∞, where γ_M,j is a positive random variable.

As a particular case with M = 1, the asymptotic behaviour of the weight of a fixed vertex is the following.

Let j ≥ 0 be fixed and let α > 0. Then

almost surely as n → ∞, where γ_1,jis a positive random variable.

We turn to the limit of the degree sequence of a fixed vertex.

Let j ≥ 0 be fixed and let α > 0. Then

almost surely as n → ∞, where the positive random variable γ_1,jis given in (8).

Now, we turn to the maximal weight and the maximal degree. Let us denote by 𝓦_n the maximum of the weights of the vertices after n steps, that is

Let α > 0. Then

where μ = sup{γ_1,j : j ≥ – (N –1)}is a finite positive random variable with γ_1,j defined in (8).

Let us denote by 𝓓_n the maximal degree after n steps, that is

Let α > 0. Then

where μ = sup{γ_1,j : j ≥ – (N –1)} is a finite positive random variable defined in Theorem 2.6.

Corollary 2.4 and Theorems 2.5, 2.6 and 2.7 are extensions of Theorem 4.1 in [13] and Theorems 5.2, 5.1, 5.3 in [14], respectively.

We see that the parameters α₁ and α₂ belong to the preferential attachment part of the model, while β₁ and β₂ are connected to the uniform choice part. Moreover, in the above Theorems 2.3-2.7 only α₁ and α₂ play role. Therefore the asymptotic behaviour in those theorems is not influenced by the uniform choice parameters. This phenomenon can be explained as follows. If j is fixed and n is large, then the j th vertex is ’old’ among the V_n np vertices. Therefore the degree and the weight of the j th vertex are relatively high compared to those of the ’young’ vertices. As there are lot of ’young’ vertices, therefore the uniform choice has minor influence on the degree and the weight of the j th vertex.

If we compare Corollary 2.4 and Theorem 2.5, we see that the asymptotic ratio of the weight and the degree of vertex j is

And the last expression in (14) is nothing else but the ratio of the expected growth of the weights of the ’old’ vertices to the expected growth of the degrees of the ’old’ vertices during one step when the choice is PA. Similar observation is true for the asymptotic ratio of the maximal weight to the maximal degree. That is, by Theorems 2.6 and 2.7, we have 𝓦_n/𝓓_n ~ α/α₂.

Let τ_n denote the maximum of the labels of those vertices where the maximal weight is attained, that is let

Then the sequence τ_n(ω), n = 1, 2, ... is bounded for almost all fixed elementary events ω. It is a simple consequence of Theorem 2.6 and its proof. A similar statement is valid for the degrees.

Let j, k, l, M, 0 ≤ j ≤ l, 1 ≤ M ≤ N be fixed nonnegative integers and let

Then (Z[n, M, k, j]; 𝓔_n) is a martingale for n ≥ l.

Proof. At each step, the weight of a fixed M-clique is increased by 1 if and only if it takes part in an interaction. The total weight of N-cliques after n steps is n + 1. The total weight of (N – 1)-cliques after n steps is N(n + 1). When a new vertex is born and we choose N – 1 vertices uniformly, the probability that the vertices of a given M-clique are selected is

When we choose N vertices uniformly at random, the probability that the vertices of a given M-clique are chosen is

Therefore, as in [15], it is easy to show that the probability that the j th M-clique takes part in interaction at step (n + 1) is

provided that the j th M-clique exists at the lth step. Using this fact, we can see for n ≥ l

Multiplying both sides by b(n + 1, M, k, we see that

Using that d[n + 1, M, k, j] is 𝓔_n-measurable, we obtain the desired result. □

The following sequence is a nonnegative supermartingale

Proof. In a similar way as in the proof of Lemma 3.1, we have for n ≥ k

Multiplying both sides of (24) by e_n+1,M, we obtain the result. □

The proof contains two parts. First, we will show that the result is valid with non-negative γ_{M, j}. Then we will show that γ_{M, j} is positive with probability 1.

Let B_n+1 = {W[n + 1, M, j] = W[n, M, j] + 1}. Consider the event that the jth M-clique exists after n steps. On this event, by (21),

The sequence (B_n, n ∈ ℕ) is adapted to the sequence of σ-algebras (𝓔_n; n ∈ ℕ). Using Corollary VII-2-6 of [19] and (25), we have

Consider the martingale (Z[n, M, k, j], 𝓔_n) introduced in Lemma 3.1 and let k = 1. Then

By the Marcinkiewicz strong law of large numbers, we have

almost surely, for any ɛ > 0. By this fact and (18), we obtain that

Using that α′ > 0 and M > 0, we see that d[n, M, 1, j]  converges as n → ∞. Therefore the martingale Z[n, M, 1, j] is bounded from below. Now, we shall see that the martingale Z[n, M, 1, j] has bounded differences. The sequence b[n, M, 1] is decreasing, hence

It is also easy to compute that

So the martingale Z[n, M, 1, j] is bounded from below and it has bounded differences. Therefore, by Proposition VII-3-9 of [19], it is convergent almost surely as n → ∞. By the definition of Z[n, M, 1; j], we see that b[n, M, 1]W[n, M, j] also converges almost surely on the event {W[l, M, j] > 0}. This fact, (26) and (18) imply that (8) is true with non-negative γ_M,j.

Now we will show that γ_M,j is positive with probability 1. Consider the supermartingale (en,MI[k,M,j]W[n,M,j]−1,Fn), n ≥ j, in Lemma 3.2. This supermartingale is nonnegative therefore, according to the submartingale convergence theorem, it converges almost surely. lim_t→∞I[l, M, j] = 1 almost surely, hence en,MW[n,M,j]−1 also converges almost surely as n → ∞. This and (19) imply that γ_M,j is positive almost surely. □

Contrary to the weight, the degree of a fixed vertex can grow by 0, 1, ..., N – 1 at each step. Hence 0 ≤ D[n, j] ≤ D[n – 1, j] N – 1 for all fixed j ≥ 0. Moreover, the degree of a fixed vertex does not change at steps when we do not add a new vertex and the choice is done according to the preferential attachment rule.

Let j ≥ 0 be fixed. For n ≥ k, we have

where0≤Rn≤(N−1)pβVn.

Proof. Consider the conditional expectation E{D[n + 1, j] – D[n, j]|𝓔_n} provided that the jth vertex exists after k steps.

When the model evolves according to the preferential attachment rule the degree of a fixed vertex can be increased by 0 or 1 at each step. Therefore, the expected growth of the degree of the jth vertex in the (n +1)th step when the choice is PA is W(n,1,j]n+1. At steps when the choice is UNI the degree of a fixed vertex can be increased at most by N – 1. Moreover, using(21), the probability that the growth of the degree of the jth vertex is positive when the choice is UNI is not greater than pβVn. Therefore, the expected growth of the degree of the jth vertex in the (n + 1)th step when the choice is UNI is less than or equal to (N – 1) pβVn. □

Proof of Theorem 2.5. Consider the following bounded random variable: ξn=I[k,1,j]N−1(D[n,j]−D[n−1,j]). By the Remark 3.3, we have 0 ≤ ξ_n ≤ 1. Applying an appropriate version of Corollary VII-2-6 of [19] (see Proposition 2.4 of [20]), then using Lemma 3.4 and (8), we have

provided that the jth vertex exists after k steps. As lim_k→∞W[k, 1, j] = ∞ a.s., we obtain the statement. □

The following lemma will be used to study the maximal weight. It is an extension of Lemma 5.2 in [14].

For all fixed nonnegative integers k ≥ 0, 1 ≤ m ≤ n, let

Then there exists a positive constant C_k such that

Proof. We use induction on k. Let k = 0. Then

Suppose that the statement is true for k – 1. By Lemma 3.1, Z[n, 1, k, j] is a martingale. The difference of two martingales is also a martingale. So, in the definition of Z[n, 1, k, j] changing I[l, 1, j] for J[l, 1, j], we obtain again a martingale. Using the definitions of J[n, 1, j] and Z[n, 1, k, j], we have

In the last step we used that W(l, 1, j) = 1 if J[l, 1, j] = 1. Now, we give upper bounds for the two terms in (32) separately. We have already seen that, for a fixed k, the sequence b[n, 1, k] is decreasing. Therefore, applying also (18),

For the second term in (32), changing the order of summations and using that I[i,1,j]=∑l=jiJ[l,1,j], we have

In the last step we applied that W[i,1,j]+k−1W[i,1,j]≤k if I[i, 1, j] > 0.

Now, we give upper bounds on the events {Vi2<pi2} and {Vi<pi2} separately. Using the induction hypothesis, we have

(Here 𝕀_A is the indicator of the set A.) On the other hand, by(18),

as i → ∞. In the above computation we used Hoeffding’s exponential inequality (Theorem 2 in [21]) to obtain the following upper bound: P{Vi<pi2}≤e−εi, where ε=p22. Therefore, by(34),(36) and(35), we have

Above we applied that, by(18), b[i+1,1,k]b[i,1,k−1]=O(i−α) as i → ∞. Finally,(32),(33) and(37) give the result. □

Proof of Theorem 2.6. Let M[m, n] = max{W[n, 1, j] : –(N – 1)j < m}, where 1 ≤ m ≤ n fixed. From Corollary 2.4, we have

almost surely. By (22), the following process is a submartingale:

Let Q[m, n] = max_{m ≤ j ≤ n}W[n, 1, j]. Then 0 ≤ 𝓦_n – M[m, n] ≤ Q[m, n]. The maximum of increasing numbers of submartingales is also a submartingale, so

is a submartingale. For non-negative numbers the maximum is majorized by the sum. Therefore, and by Lemma 3.5, we obtain

Since

we see that the submartingale b[n, 1, 1]Q[m, n] is bounded in L^k for all kα > 1. Hence, this submartingale converges almost surely and in L^k for every k>1α. Moreover, by (18), (39) and (40), we have

for k>1α. As Q[m, n] is decreasing as m increases, so

Therefore, as 0 ≤ 𝓦n – M[m, n] ≤ Q[m, n],

This relation and (38) imply (11). By relation μ is a.s. finite. □

The evolution mechanism of the graph implies that D[n, j] ≤ (N –1)W[n, 1, j]. Therefore we have

Multiplying both sides by n^-α and then considering the limit as n → ∞, Theorem 2.5 implies

as n → ∞. As m → ∞, by (42), we obtain the desired result. □