Abstract
A random graph evolution based on interactions of N vertices is studied. During the evolution both the preferential attachment rule and the uniform choice of vertices are allowed. The weight of an M-clique means the number of its interactions. The asymptotic behaviour of the weight of a fixed M-clique is studied. Asymptotic theorems for the weight and the degree of a fixed vertex are also presented. Moreover, the limits of the maximal weight and the maximal degree are described. The proofs are based on martingale methods.
1 Introduction
Network theory is currently one of the most popular research topics. Random graphs are used to describe real-life networks. For overviews of random graph models and their properties see [1–3]. It is known that many real-life networks (e.g. the WWW, biological and social networks) are scale-free, that is their asymptotic degree distributions follow power laws. To describe the evolution of such networks, in [4], the preferential attachment model was suggested. However, in [4], the description of the evolution of the graph was informal. A rigorous definition of the preferential attachment model was given in [5], where a mathematical proof of the power law degree distribution was presented for d ≤ n1/15 (d is the degree, n is the number of steps). For the recent development of the topic see [3, 6]].
Besides the degree distribution, other characteristics are also worth studying. The degree of a fixed vertex and the maximal degree in some preferential attachment models were investigated in [3, 7, 8]. In [9, 10]] the degree of a given vertex and the maximal degree were studied in a 2-parameter scale-free random graph model. A well-known technique to analyse the growth of the maximal degree is the martingale method (see [1, 3, 11]).
There are several versions of the preferential attachment model (see [3, 12]]). In [12] a general graph evolution procedure was introduced. In that procedure both the preferential attachment rule and the uniform choice of vertices are allowed, moreover, new links can be created between old vertices. The evolution method introduced in [13] in some sense resembles the one in [12]. However, the main feature of the model applied in [13] is the interaction of three vertices. Power law degree distribution in the three-interactions model was proved in [13, 14]]. The asymptotic behaviour of the weight and the degree of a fixed vertex, as well as the limits of the maximal weight and the maximal degree, were also described in [13, 14]]. Scale-free weight distributions, both for the edges and the triangles in the three-interactions model, were obtained in [15]. Instead of the three-interactions model, an evolution rule based on interactions of N vertices (N = 3 fixed) was studied in [16].
Throughout this paper we shall study the following N-interactions model (defined in [16]). Here N = 3 is a fixed integer. Usually, a complete graph with M vertices is called an M-clique. However, in this paper, we shall consider only that complete graph to be a clique, which is constructed by interactions of its vertices. We denote an M-clique by the symbol KM. At each step the evolution of the graph is based on the interaction of N vertices. More precisely, at each step n = 1; 2; : : : we consider N vertices which will interact during that step. It means that we draw all non-existing edges between those vertices, so we obtain a clique KN. Its complete subgraphs are also considered to be cliques. The weight of KN and the weights of all sub-cliques in KN are increased by 1. (That is we increase the weights of N vertices,
The details of the evolution are the following. At time n = 0 we start with N vertices, they interact, so they form an N-clique. Let the initial weight of this graph and the initial weights of its sub-cliques be one. After the initial step we start to increase the size of the graph. We select N vertices to interact. For the selection there are two possibilities at each step. On the one hand, with probability p, we add a new vertex that interacts with N – 1 old vertices (Step NEW). On the other hand, with probability (1 – p), we do not add any new vertex, but N old vertices interact (Step OLD). Here 0 < p ≤ 1 is fixed.
Step NEW: When we add a new vertex, then we choose N – 1 old vertices. The new vertex and the N – 1 old vertices interact, so they together form an N -clique. However, to choose the N – 1 old vertices we have two possibilities: Choice PA and Choice UNI.
Choice PA: With probability r we choose an (N – 1)-clique from the existing (N – 1)-cliques according to the weights of the (N – 1)-cliques. It means that an (N – 1)-clique of weight wt is chosen with probability wt /Σhwh (preferential attachment rule).
Choice UNI: On the other hand, with probability 1 – r, we choose N – 1 out of the existing vertices uniformly, that is, all groups of N – 1 vertices have the same chance to be chosen (uniform choice).
Step OLD: At a step when we do not add a new vertex, then N old vertices interact. As in the previous case, we have two options to choose the N old vertices: Choice PA and Choice UNI.
Choice PA: On the one hand, with probability q, we choose one N-clique KN out of the existing N-cliques according to their weights. It means that the probability that we choose KN is proportional to its weight (preferential attachment rule).
Choice UNI: On the other hand, with probability 1 – q, we choose from the existing vertices uniformly, that is all subsets consisting of N vertices have the same chance (uniform choice).
Power law degree and weight distributions for vertices in the general N-interactions model were obtained in [16, 17]]. The asymptotic behaviour of the weights of the N-cliques was examined and power law weight distribution for the N-cliques was obtained in [15].
In this paper we shall study the asymptotic behaviour of the weights and the degrees. Moreover, we shall also consider the limiting properties of the maximal weight and the maximal degree of vertices. In our proofs we follow some ideas of [13, 14]]. However, the combinatorial problems for general N are much more difficult than for N = 3.
The main results of the paper are the following. In Theorem 2.3, we describe the asymptotic behaviour of the weight of a fixed M-clique (1 ≤ M ≤ N fixed). The limit of the weight of a fixed vertex follows as a particular case with M = 1 (Corollary 2.4). The asymptotic behaviour of the degree of a fixed vertex is also described (Theorem 2.5). Moreover, we find the limits of the maximal weight and the maximal degree (Theorems 2.6 and 2.7). Corollary 2.4, Theorems 2.5, 2.6 and 2.7 are extensions of the results in [13, 14]], where the 3-interactions model was studied.
The general structure of our results is limn→∞n–γXn = ν almost surely, where is a positive random variable. Therefore our results are in line with the corresponding results of [3, 9–11]. The theorems are listed in Section 2. All the proofs and some auxiliary results are presented in Section 3.
2 Main results
Let 1 ≤ M ≤ N be a fixed integer. We introduce the following notation.
We see that when M = 1, then α′ = α and β′ = β. First we list some results concerning the scale-free property.
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 xw, w = 1, 2, ..., are positive numbers satisfying
as w → ∞, with
a.s. as n → ∞, where ud, d = N – 1, N, ..., are positive numbers with
as d → ∞, where
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 XM(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 xM,w, w = 1, 2, ..., are numbers satisfying
as w → ∞, with
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
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 γM,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 γ1,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 γ1,jis given in (8).
Now, we turn to the maximal weight and the maximal degree. Let us denote by 𝓦n the maximum of the weights of the vertices after n steps, that is
Let α > 0. Then
where μ = sup{γ1,j : j ≥ – (N –1)}is a finite positive random variable with γ1,j defined in (8).
Let us denote by 𝓓n the maximal degree after n steps, that is
Let α > 0. Then
where μ = sup{γ1,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 Vn 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 𝓦n/𝓓n ~ α/α2.
Let τn denote the maximum of the labels of those vertices where the maximal weight is attained, that is let
Then the sequence τn(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.
3 Proofs and auxiliary lemmas
Introduce the following notation. Let 𝓔n –1 denote the -algebra of observable events after the (n – 1)th step. Let Vn denote the number of vertices after the nth step. Let j ≥ 0 and 1 ≤ M ≤ N be fixed integers. Let I[n, M, j] be the indicator of the event that the j th M-clique exists after n steps, that is
Let J[n, M, j] be the indicator of the event that the j th M-clique is born at the nth step. Then J[n, M, j] = I[n, M, j] – I[n – 1, M, j]. For all fixed positive integers j, k, l, M, 0 ≤ j ≤ l, 1 ≤ k, 1 ≤ M ≤ N, we consider the following sequences:
We can see that b[n, M, k] and en,M are deterministic, while d[n, M, k, j] is an 𝓔n – 1-measurable random variable for any n, M, k and j. Using the definition of b[n, M, k] and the Stirling-formula for the Gamma function, we can show that
where bM,k = Γ(1 + α′k) > 0, k and M are fixed. Moreover, we can easily see that
In the following lemma we introduce a martingale which will play an important role in the paper.
Let j, k, l, M, 0 ≤ j ≤ l, 1 ≤ M ≤ N be fixed nonnegative integers and let
Then (Z[n, M, k, j]; 𝓔n) 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 𝓔n-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 en+1,M, we obtain the result. □
The proof contains two parts. First, we will show that the result is valid with non-negative γM, j. Then we will show that γM, j is positive with probability 1.
Let Bn+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 (Bn, n ∈ ℕ) is adapted to the sequence of σ-algebras (𝓔n; n ∈ ℕ). Using Corollary VII-2-6 of [19] and (25), we have
Consider the martingale (Z[n, M, k, j], 𝓔n) 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 γM,j.
Now we will show that γM,j is positive with probability 1. Consider the supermartingale
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
where
Proof. Consider the conditional expectation E{D[n + 1, j] – D[n, j]|𝓔n} 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
Proof of Theorem 2.5. Consider the following bounded random variable:
provided that the jth vertex exists after k steps. As limk→∞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 Ck 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
In the last step we applied that
Now, we give upper bounds on the events
(Here 𝕀A 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:
Above we applied that, by(18),
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] = maxm ≤ j ≤ nW[n, 1, j]. Then 0 ≤ 𝓦n – 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 Lk for all kα > 1. Hence, this submartingale converges almost surely and in Lk for every
for
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. □
Acknowledgement
The authors wish to thank the referees for their helpful comments. According to the hints of the referees the authors were able to improve the paper considerably.
István Fazekas was supported by the TÁMOP-4.2.2.C-11/1/KONV-2012-0001 project. The project has been supported by the European Union, co-financed by the European Social Fund.
Bettina Porvázsnyik was supported by the European Union and the State of Hungary, co-financed by the European Social Fund in the framework of TÁMOP 4.2.4.A/2-11-1-2012-0001 ‘National Excellence Program’.
References
[1] Durrett R., Random graph dynamics, Cambridge University Press, Cambridge, 200710.1017/CBO9780511546594Search in Google Scholar
[2] Janson S., Łuczak T., Rucinski A., Random graphs, Wiley-Interscience, New York, 200010.1002/9781118032718Search in Google Scholar
[3] van der Hofstad R., Random Graphs and Complex Networks, Eindhoven University of Technology, The Netherlands, rhofs-tad@win.tue.nl, 2013, available at http://www.win.tue.nl/~rhofstad/NotesRGCN2013.pdf10.1017/9781316779422Search in Google Scholar
[4] Barabási A. L., Albert R., Emergence of scaling in random networks, Science, 1999, 286, 509–51210.1515/9781400841356.349Search in Google Scholar
[5] Bollobás B., Riordan O., Spencer J., Tusnády G., The degree sequence of a scale-free random graph process, Random Structures Algorithms, 2001, 18, 279–29010.1515/9781400841356.384Search in Google Scholar
[6] Grechnikov E., An estimate for the number of edges between vertices of given degrees in random graphs in the Bollobás-Riordan model, Mosc. J. Comb. Number Theory, 2011, 1(2), 40–73Search in Google Scholar
[7] Katona Zs., Móri T. F., A new class of scale free random graphs, Statist. Probab. Lett., 2006, 76(15), 1587–159310.1016/j.spl.2006.04.017Search in Google Scholar
[8] Lindholm M., Vallier T., On the degree evolution of a fixed vertex in some growing networks, Statist. Probab. Lett., 2011, 81(6), 673–67710.1016/j.spl.2011.02.015Search in Google Scholar
[9] Móri T. F., On a 2-parameter class of scale-free random graphs, Acta Math. Hungar., 2007, 114(1-2), 37–4810.1007/s10474-006-0511-0Search in Google Scholar
[10] Móri T. F., Degree distribution nearby the origin of a preferential attachment graph, Electron. Commun. Probab., 2007, 12, 276-28210.1214/ECP.v12-1299Search in Google Scholar
[11] Móri T. F., The maximum degree of the Barabási-Albert random tree, Combin. Probab. Comput., 2005, 14(3), 339–34810.1017/S0963548304006133Search in Google Scholar
[12] Cooper C., Frieze A., A general model of web graphs, Random Structures Algorithms, 2003, 22, 311–33510.1002/rsa.10084Search in Google Scholar
[13] Backhausz Á., Móri T. F., A random graph model based on 3-interactions, Ann. Univ. Sci. Budapest. Sect. Comput., 2012, 36, 41–52Search in Google Scholar
[14] Backhausz Á., Móri T. F., Weights and degrees in a random graph model based on 3-interactions, Acta Math. Hungar., 2014, 143(1), 23–4310.1007/s10474-014-0390-8Search in Google Scholar
[15] Fazekas I., Noszály Cs., Perecsényi A., Weights of cliques in a random graph model based on three-interactions, Lith. Mat. J., 2015, 55(2), 207–22110.1007/s10986-015-9274-zSearch in Google Scholar
[16] Fazekas I., Porvázsnyik B., Scale-free property for degrees and weights in an N-interactions random graph model, J. Math. Sci. (N.Y.), 2016, 214(1), 69-8210.1007/s10958-016-2758-5Search in Google Scholar
[17] Fazekas I., Porvázsnyik B., Scale-free property for degrees and weights in a preferential attachment random graph model, J. Probab. Stat., 2013, Article ID 707960, DOI: 10.1155/2013/707960Search in Google Scholar
[18] Ostroumova L., Ryabchenko A. and Samosvat E., Generalized preferential attachment: tunable power-law degree distribution and clustering coefficient, In: A. Bonato, M. Mitzenmacher, P. Prałat, (Eds.), Algorithms and models for the web graph: 10th international workshop, WAW 2013, Cambridge, MA, USA, December 14-15, 2013, Proceedings, (14-15 December 2013, Cambridge, MA, USA), Springer, Lecture Notes in Computer Science 2013, 8305, 185–20210.1007/978-3-319-03536-9_15Search in Google Scholar
[19] Neveu J., Discrete-parameter martingales, North-Holland, Amsterdam, 1975Search in Google Scholar
[20] Backhausz Á., Analysis of random graphs with methods of martingale theory, PhD thesis, Eötvös Loránd University, Budapest, Hungary, 2012Search in Google Scholar
[21] Hoeffding W., Probability inequalities for sums of bounded random variables, J. Amer. Statist. Assoc., 1963, 58, 13–3010.1080/01621459.1963.10500830Search in Google Scholar
[22] Fazekas I., Porvázsnyik B., The asymptotic behaviour of the weights and the degrees in an N-interactions random graph model, preprint available at arXiv: http://arxiv.org/pdf/1405.1267.pdf, 2014Search in Google Scholar
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