Random Graphs Associated to some Discrete and Continuous Time Preferential Attachment Models
aa r X i v : . [ m a t h . P R ] J a n Random Graphs Associated to some Discrete andContinuous Time Preferential Attachment Models
Angelica Pachon, Federico Polito & Laura Sacerdote
Mathematics Department “G. Peano”, University of Torino, Italy
August 29, 2018
Abstract
We give a common description of Simon, Barab´asi–Albert, II-PA and Price growth mod-els, by introducing suitable random graph processes with preferential attachment mechanisms.Through the II-PA model, we prove the conditions for which the asymptotic degree distribu-tion of the Barab´asi–Albert model coincides with the asymptotic in-degree distribution of theSimon model. Furthermore, we show that when the number of vertices in the Simon model(with parameter α ) goes to infinity, a portion of them behave as a Yule model with parame-ters ( λ, β ) = (1 − α, Keywords : Preferential attachment; Random graph growth; Discrete and continuous timemodels; Stochastic processes.
MSC2010 : 05C80, 90B15.
A large group of networks growth models can be classified as preferential attachment models .In the simplest preferential attachment mechanism an edge connects a newly created node toone of those already present in the network with a probability proportional to the number oftheir edges.Typically what is analyzed for these models are properties related both to the growth ofthe number of edges for each node and to the growth of the number of nodes.After the seminal paper by Barab´asi and Albert [1], models admitting a preferential attach-ment mechanism have been successfully applied to the growth of different real world networks,such as, amongst others, physical, biological or social networks. The typical feature revealinga preferential attachment growth mechanism is the presence of power-law distributions, e.g.,for the degree (or in-degree) of a node selected uniformly at random.Despite its present success, the preferential attachment paradigm is not new. In fact itdates back to a paper by Udny Yule [24], published in 1925 and regarding the development ofa theory of macroevolution. Specifically the study concerned the time-continuous process ofcreation of genera and the evolution of species belonging to them. Yule proved that when timegoes to infinity, the limit distribution of the number of species in a genus selected uniformlyat random has a specific form and exhibits a power-law behavior in its tail. Thirty years later,the Nobel laureate Herbert A. Simon proposed a time-discrete preferential attachment modelto describe the appearance of new words in a large piece of a text. Interestingly enough,the limit distribution of the number of occurrences of each word, when the number of wordsdiverges, coincides with that of the number of species belonging to the randomly chosen genusin the Yule model, for a specific choice of the parameters. This fact explains the designation
Yule–Simon distribution that is commonly assigned to that limit distribution. urthermore, it should be noticed that Barab´asi–Albert model exhibits an asymptoticdegree distribution that equals the Yule–Simon distribution in correspondence of a specificchoice of the parameters and still presents power-law characteristics for more general choicesof the parameters. The same happens also for other preferential attachment models.Yule, Simon and Barab´asi–Albert models share the preferential attachment paradigm thatseems to play an important role in the explanation of the scale-freeness of real networks.However, the mathematical tools classically used in their analysis are different. This makesdifficult to understand in which sense models producing very similar asymptotic distributionsare actually related one another. Although often remarked and heuristically justified, norigorous proofs exist clarifying conditions for such result. Different researchers from differentdisciplines, for example theoretical physicists and economists asked themselves about therelations between Simon, Barab´asi–Albert, Yule and also some other models closely relatedto these first three (sometimes confused in the literature under one of the previous names).Partial studies in this direction exist but there is still a lack of clarifying rigorous results thatwould avoid errors and would facilitate the extension of the models.The existing results refer to specific models and conditions but there is not a unitaryapproach to the problem. For instance, in [4], the authors compared the distribution of thenumber of occurrences of a different word in Simon model, when time goes to infinity, withthe degree distribution in the Barab´asi–Albert model, when the number of vertices goes toinfinity. In [21], an explanation relating the asymptotic distribution of the number of speciesin a random genus in Yule model and that of the number of different words in Simon modelappears. More recently, following a heuristic argument, Simkin and Roychowdhury [20] gavea justification of the relation between Yule and Simon models.The aim of this paper is to study rigorously the relations between these three models.A fourth model, here named II-PA model (second preferential attachment model), will bediscussed in order to better highlight the connections between Simon and Barab´asi–Albertmodels. Also we include the Price model that predates the Barab´asi–Albert model, and is infact the first model using a preferential attachment rule for networks.The idea at the basis of our study is to make use of random graph processes theory todeal with all the considered discrete-time models and to include in this analysis also thecontinuous-time Yule model through the introduction of two suitable discrete-time processesconverging to it. In this way we find a relationship between the discrete time models and thecontinuous time Yule model, which is easier to handle and extensively studied. Translatingresults from discrete models to their continuous counter-parts is usually a strong method toanalyze asymptotic properties. Thus, Theorems 4.3 and 4.4 provide an easy tool for this.The random graph process approach was used by Barab´asi and Albert to define theirpreferential attachment model of World Wide Web [1]. At each discrete-time step a newvertex is added together with m edges originating from it. The end points of these edges areselected with probability proportional to the current degree of the vertices in the network.Simulations from this model show that the proportion of vertices with degree k is c m k − γ ,with γ close to 3 and c m > k . A mathematically rigorous study of thismodel was then performed by Bollob´as, Riordan, Spencer and Tusnady [3] making use ofrandom graph theory. The rigorous presentation of the model allowed the authors to provethat the proportion of vertices with degree k converges in probability to m ( m + 1) B ( k,
3) asthe number of vertices diverges, where B ( x, y ) is the Beta function.Here we reconsider all the models of interest in a random graph process framework. InSection 2 we introduce the necessary notations and basic definitions. Then, in Section 3, wepresent the four preferential attachment models of interest, i.e. Simon, II-PA, Price, Barab´asi–Albert, and Yule models, through a mathematical description that makes use of the randomgraphs approach. Such a description allows us to highlight an aspect not always well under-lined: the asymptotic distributions that in some cases coincide do not always refer to thesame quantity. For instance, the Barab´asi–Albert model describes the degree of the verticeswhile II-PA considers the in-degree . In Section 3 we also discuss the historical context and thelist of available mathematical results for each model. The proposed point of view by meansof random graphs processes then permits us to prove the novel results presented in Section 4.The theorems described and proved there clarify the relations between the considered asymp-totic distributions of the different models, specifying for which choice of the parameters thesedistributions coincide and when they are not related. n the concluding Section 5 we summarize the proved results and we illustrate with adiagram the cases in which the considered models are actually related. In this section we introduce some classical definitions, theorems and mathematical tools wewill use in the rest of the paper.Let us define a graph G = ( V, E ) as an ordered pair comprising a set of vertices V witha set of edges or lines E which are 2-elements subsets of V , so E ⊆ V × V . A graph G is directed if its edges are directed, i.e., if for every edge ( i, j ) ∈ E , ( i, j ) = ( j, i ), otherwise G iscalled an undirected graph.We say that G is a random graph , if it is a graph selected according with a probabilitydistribution over a set of graphs, or it is determined by a stochastic process that describesthe random evolution of the graph in time. A stochastic process generating a random graphis called a random graph process . In other words, a random graph process is a family ( G t ) t ∈T of random graphs (defined on a common probability space) where t is interpreted as time and T can be either countable or uncountable.A loop is an edge that connects a vertex to itself. The in-degree of a vertex v at time t ,denoted by ~d ( v, t ), is the number of incoming edges (incoming connections). Similarly, the degree of a vertex v at time t , denoted by d ( v, t ), is the total number of incoming and outgoingedges at time t (when an edge is a loop, it is counted twice). In this paper we also use theterm directed loop to indicate a loop that counts one to the in-degree.The random graphs studied in this paper are random graph processes starting at time t = 0, without any edge neither vertex, growing monotonically by adding at each discretetime step either a new vertex or some directed edges between the vertices already present,according to some law P ( v ti −→ v tj ) = P (( i, j ) ∈ G t ).We focus here on the analysis of the number of vertices with degree or in-degree k attime t , which we denote by N k,t and ~N k,t , respectively. In particular we are interested in theasymptotic degree or in-degree distribution of a random vertex, i.e., in the proportion N k,t /V t or ~N k,t /V t , as t goes to infinity, where V t denotes the total number of vertices at time t . Wewill add an upper index to N k,t or ~N k,t , for instance ~N Simon k,t , to indicate the process to whichwe refer, if necessary.Furthermore, we will make use of the following standard notation: for (deterministic)functions f = f ( t ) and g = g ( t ), we write f = O ( g ) if lim t →∞ f/g is bounded, f ∼ g iflim t →∞ f/g = 1, and f = o ( g ) if lim t →∞ f/g = 0.One of the methods used in the literature to study the asymptotic behavior of N k,t /V t or ~N k,t /V t is to prove that these random processes concentrate around their expectations. Inorder to do this, the Azuma and Hoeffding inequality is applied, when possible (see also [10],page 93). Lemma 2.1 (Azuma and Hoeffding inequality [12]) . Let ( X t ) nt =0 be a martingale with | X s − X s − | ≤ c for ≤ s ≤ t and c a positive constant. Then P ( | X t − X | > x ) ≤ exp( − x / c t ) . (2.1)One of the first authors to use this approach in preferential attachment random graphsstudies were Bollob´as, et. al in [3]. Here we apply this approach to study different randomgraph processes. In Section 3 we illustrate this technique by analyzing the Simon model,reporting the corresponding computations for the Barab´asi–Albert model. As stressed in the introduction, a number of models that make use of “preferential attach-ment” mechanisms are present in the literature. Here we consider some of them rigorouslyintroducing the corresponding random graph processes with the aim to allow a comparisonof their features. To this aim, it helps to present the most known models using a common no-tation. We first discuss the case of discrete time preferential attachment models, specificallySimon and Barab´asi–Albert models, and some others inspired by Simon model, the II-PA odel (second preferential attachment model) and Price model, which will help us to under-stand the relations between Simon and Barab´asi–Albert models. Moreover, we also discuss acontinuous time preferential attachment model, the Yule model, which is defined in terms ofindependent homogeneous linear birth processes. We rigorously prove that this model can berelated with Simon, and hence Barab´asi–Albert models.The Barab´asi–Albert model presented in [1] omits some necessary details to be formulatedin terms of a random graph process. Here we follow its description detailed as in Bollob´as et. al [3] where the rules for the growth of the random graph not mentioned in [1] are given.Furthermore, in order to make easier the understanding of each model, we follow the samescheme for its presentation, eventually specifying the absence of some results when not yetavailable.Our scheme considers:1. The mathematical description of the associated graph structure and its growth law.2. The historical context motivating the first proposal of the model and some successiveapplications.3. Available results on the degree or in-degree distribution with particular reference topower law behavior. We collect both, theorems and simulation results. Mathematical description:
The Simon model can be described as a random graphprocess in discrete time ( G tα ) t ≥ , so that G tα is a directed graph which starts at time t = 1 with a single vertex v and a directed loop. Then, given G tα , one forms G t +1 α by either adding with probability α a new vertex v i with a directed loop, i ≤ t + 1, oradding with probability (1- α ) a directed edge between the last added vertex v and v j ,1 ≤ j ≤ t , where the probability of v j to be chosen is proportional to its in-degree, i.e., P ( v −→ v j ) = (1 − α ) ~d ( v j , t ) /t, ≤ j ≤ t. (3.1)In Figure 1 we illustrate the growth law of this graph. v ( a ) v v ( b ) v v v ( c )(3 . α (3 . Figure 1:
Construction of ( G tα ) t ≥ . (a) Begin at time 1 with one single vertex and a directed loop. (b) Supposesome time has passed, in this case, the picture corresponds to a a realization of the process at time t = 4. (c) Given G α form G α by either adding with probability α a new vertex v with a directed loop, or adding a directed edgewith probability given by (3 . Historical context:
In [21], Simon considered a model to describe the growth of atext that is being written such that a word is added at each time t ≥
1. Different wordscorrespond to different vertices and repeated words to directed edges in the previousdescription. Simon introduced the two following conditions: For α ∈ (0 , P [( t + 1)th word has not yet appeared at time t ] = α (b) P [( t + 1)th word has appeared k times at time t ] = (1 − α ) k ~N k,t /t ,where ~N k,t is the number of different words that have appeared exactly k times at time t , or the number of vertices that have exactly k incoming edges (i.e. in-degree k ) at time t in G tα . Thus, at time t + 1 either with probability α a new word appears (i.e., a newvertex v i , i ≤ t + 1, with a directed loop appears), or with probability (1 − α ) the wordis not new, and if it has appeared k times at time t , a directed edge is added. The tarting point of this edge is the last vertex that has appeared in G tα , while its end pointis selected with probability (3.1) that corresponds in this case to k/t .3. Available results:
Simon was interested in getting results for the proportion of verticesthat have exactly in-degree k , with respect to the total number of vertices V t at time t .Thus, he proved asymptotic results for E ~N ,t / E V t as t −→ ∞ .Next, we will give a brief synopsis of the computations made by Simon in [21]. The ideais to condition on what has happened until time t and compute the expected value attime t + 1. For k = 1 it holds E ~N ,t +1 = α + (cid:16) − (1 − α ) t (cid:17) E ~N ,t , (3.2)and, for k > E ~N k,t +1 = (1 − α ) t (cid:2) ( k − E ~N k − ,t − k E ~N k,t (cid:3) + E ~N k,t . (3.3)Simon solved (3.2) and (3.3) (see also [10], pages 98–99) to get, as t −→ ∞ , E ~N ,t t −→ α − α , (3.4)and for k > E ~N k,t t −→ α − α Γ( k )Γ (cid:16) − α (cid:17) Γ (cid:16) k + 1 + − α (cid:17) , (3.5)where Γ is the gamma function.Observe now that the number of vertices appeared until time t , V t ∼ Bin( t, α ), so E V t = αt . Hence, using this and (3.4) and (3.5) for k = 1, E ~N ,t E V t −→ − α , (3.6)and for k > E ~N k,t E V t −→ − α Γ( k )Γ (cid:16) − α (cid:17) Γ (cid:16) k + 1 + − α (cid:17) = 11 − α B (cid:16) k, − α (cid:17) , (3.7)as t −→ ∞ , where B ( x, y ) is the Beta function.Now, let G r and G s denote the σ -fields generated by the appearance of directed edgesup to time r and s respectively, r ≤ s ≤ t . Since E (cid:2) E ( ~N k,t |G s ) | G r (cid:3) = E ( ~N k,t |G r ),then, Z Simon s = E ( ~N k,t |G s ) is a martingale, such that Z Simon t = ~N k,t and Z Simon0 = E ~N k,t .Furthermore, observe that at each unit of time, say s , either a new vertex appears orthe last one added, is attaching to another existing vertex v j , j ≤ s , but note this doesnot effect the in-degree of v = v j , or the probabilities these vertices will choose later,so it yields that | Z Simon s − Z Simon s − | ≤
1. Then, it is possible to use Azuma–Hoeffding’sinequality (2.1), and obtain that for every ǫ t ≫ t − / (for example take ǫ t = p ln t/t ), P (cid:16)(cid:12)(cid:12)(cid:12) ~N k,t t − E ~N k,t t (cid:12)(cid:12)(cid:12) ≥ ǫ t (cid:17) ≤ exp (cid:16) − ( tǫ t ) t (cid:17) −→ . (3.8)Now, using Chebyschev’s inequality, for every ε t >
0, such that tε t −→ ∞ as t −→ ∞ , P (cid:16)(cid:12)(cid:12)(cid:12) V t t − E V t t (cid:12)(cid:12)(cid:12) ≥ ε t (cid:17) ≤ tα (1 − α ) t ε t −→ . (3.9)Hence, by (3.8) and (3.9), ~N k,t /t −→ E ~N k,t /t and V t /t −→ E V t /t in probability. inally, since E ~N k,t /t and E V t /t converge as t goes to infinity to the constant values (3.4)and (3.5) respectively, and because V t is a random variable with binomial distribution,Bin( t, α ), then by properties of convergence in probability we obtain that ~N k,t V t −→ − α Γ( k )Γ (cid:16) − α (cid:17) Γ (cid:16) k + 1 + − α (cid:17) = 11 − α B (cid:16) k, − α (cid:17) , (3.10)in probability. Mathematical description:
In [16] a different model is analyzed. In that paper itis called Yule model and described in discrete time. The model is defined also as apreferential attachment model but in this case at each time step n a new vertex isadded with exactly m + 1 directed edges, m ∈ Z + . These edges start from the newvertex and are directed towards any of the previously existing vertices according to apreferential attachment rule. To define formally a random graph process, we can thinkfor a moment at an increasing time rescaled by 1 / ( m + 1) so that at each unit of time n , m + 1 scaling time steps happen. Let ( ˜ G tm ) t ≥ be a random graph process such thatfor all n ∈ Z + ∪ { } ,(a) at time t = n ( m + 1) + 1 add a new vertex v n +1 with a directed loop (it does countone for the in-degree), and(b) for i = 2 , . . . , m + 1 at each time t = n ( m + 1) + i add a directed edge from v n +1 to v j , 1 ≤ j ≤ n + 1, with probability P ( v n +1 −→ v j ) = ~d ( v j , t − t − . (3.11)Note then that ( ˜ G tm ) t ≥ starts at time t = 1 with a single vertex and one directedloop.In Figure 2 we illustrate the growth law of this graph. v ( a ) v ( b ) v v ( c ) (3 . . Figure 2:
Construction of ( ˜ G tm ) t ≥ for m = 2. (a) Begin at time 1 with one single vertex and a directed loop.(b) Suppose some time has passed, in this case, the picture corresponds to a a realization of the process at time t = 3. Keep in mind that here m = 2 and therefore m = 2 directed edge are added to the graph by preferentialattachment rule (but at this point the only possible choice is the vertex v ). (c) Here time is t = 5. A new vertex v already appeared at time t = 4 together with a directed loop. At time 5 instead the first of the m edges that mustbe added to the graph is chosen (red dashed directed edges) by means of the preferential attachment probabilities(3.11). Historical context:
In [16], Newman describes this model in terms of genus and speciesas follows.
Species are added to genera by “speciation”, the splitting of one species intotwo, [ . . . ] . If we assume that this happens at some stochastically constant rate,then it follows that a genus with k species in it will gain new species at a rateproportional to k , since each of the k species has the same chance per unit timeof dividing in two. Let us further suppose that occasionally, say once every m peciation events, the new species produced is, by chance, sufficiently differentfrom the others in its genus as to be considered the founder member of anentire new genus. (To be clear, we define m such that m species are addedto preexisting genera and then one species forms a new genus. So m + 1 newspecies appear for each new genus and there are m + 1 species per genus onaverage.) This description is linked to the model proposed by Simon; the difference is that theoriginal Simon model does not fix m speciation events, instead it assumes that thenumber of speciation events is random and follows a probability distribution Geo( α ),with 0 < α < Available results:
Note that the number of vertices with in-degree equal to k is equiv-alent to the number of genera that have k species, thus, the number of vertices within-degree equal to k at time t = n ( m + 1), corresponds to the number of genera thathave k species, when the number of genera is n .Let ~N k,t be the number of vertices with in-degree equal to k in ( ˜ G tm ) t ≥ . In [16] anheuristic analysis of the II-PA model shows that the proportion of vertices that haveexactly in-degree k , with respect to the total number of vertices at time t = n ( m + 1),is in the limitlim n −→∞ ~N k,t n = (1 + 1 /m )Γ( k )Γ(2 + 1 /m )Γ( k + 2 + 1 /m ) = (1 + 1 /m ) B ( k, /m ) , (3.12)We prove this in Theorem 4.2 where the result is obtained with probability one. Mathematical description:
In [15], the Price model is described as a random graphprocess in discrete time ( ˜ G nm ) n ≥ , so that ˜ G nm is a direct graph and the process startsat time n = 1 with a single vertex, v , and M + k directed loops, where k > M a random variable with expectation m . New vertices are continuallyadded to the network, though not necessarily at a constant rate. Each added vertexhas a certain out-degree, and this out-degree is fixed permanently at the creation of thevertex. The out-degree may vary from one vertex to another, but the mean out-degree,which is denoted m , is a constant over time. Thus, given ˜ G nm form ˜ G n +1 m by adding anew vertex v n +1 with k directed loops, and from it a random number of directed edges, M n +1 to different old vertices with probability proportional to their in-degrees at time n , i.e., P ( v n +1 −→ v j | M = m , . . . , M n = m n ) = ~d ( v n , n ) nk + P ni =1 m i , ≤ j ≤ n, (3.13)where M , . . . , M n +1 are taken independent and identically distributed, with E ( M i ) = m , and m a positive rational number. Note that in this model, the update of theprobabilities (3.13) every single time an edge is added, is not taken into account.2. Historical context:
In [7], Price describes empirically the nature of the total worldnetwork of scientific papers, and it is probably the first example of what is now calleda scale-free network. In [8], he formalizes a model giving rise to what he calls thecumulative advantage distribution . He finds a system of differential equations describingthe process, and solves them under specific assumptions. All the derivations are madefor k = 1.3. Available results:
Let ~N k,n be the number of vertices with in-degree equal to k in( ˜ G nm ) n ≥ . Newman [15] analyzes this model by using the method of master-equationsfor the case k = 1, and finds the same system as in [16] for the analysis of the II-PAmodel. Thus, he obtains the same limit solution for the proportion of vertices within-degree k , as in the II-PA model, i.e.,lim n −→∞ ~N k,n n = (1 + 1 /m )Γ( k )Γ(2 + 1 /m )Γ( k + 2 + 1 /m ) = (1 + 1 /m ) B ( k, /m ) . (3.14)A rigorous analysis of (3.14) can be made using Chebyschev’s inequality and followingthe same lines as in the proof of Theorem 4.2 for the II-PA model (see Section 4.1.1). .4 Barab´asi–Albert model Mathematical description:
In [3], Bollob´as, Riordan, Spencer, and Tusnady makethe Barab´asi and Albert model precise in terms of a random graph process. We followtheir description in this paragraph. Add at each time step a new vertex with m , m ∈ Z + ,different directed edges. For the case m = 1, let ( G t ) t ≥ be a random graph processso that G t is a directed graph which starts at time t = 1 with one vertex v and oneloop. Then, given G t form G t +11 by adding the vertex v t +1 together with a single edgedirected from v t +1 to v j , 1 ≤ j ≤ t + 1, with probability P ( v t +1 −→ v j ) = ( d ( v j ,t )2 t +1 , ≤ j ≤ t, t +1 , j = t + 1 . (3.15)For m > G tm ) t ≥ by running the process ( G t ) on the sequenceof imaginary vertices v ′ , v ′ , . . . , then form the graph G tm from G mt by identifying thevertices v ′ , v ′ , . . . , v ′ m to form v , v ′ m +1 , v ′ m +2 . . . , v ′ m to form v and so on.We can also define this model in a similar manner as we did for the II-PA model. Think-ing once more that the time increases with a scaling of 1 / ( m + 1), then let us define theprocess ( G tm ) t ≥ , such that for every n ∈ Z + ∪ { } ,(a) at time t = n ( m + 1) + 1 add a new vertex v n +1 ,(b) for i = 2 , . . . , m + 1 at each time t = n ( m + 1) + i add an edge from v n +1 to v ,where v is chosen with P ( v n +1 −→ v ) = ( d ( v,t − mn + i − − , v = v n +1 , d ( v,t − mn + i − − , v = v n +1 . (3.16)Observe that ( G tm ) t ≥ starts at time t = 1 just with a single vertex, without loops.2. Historical context:
Barab´asi and Albert, in [1] proposed a random graph model of thegrowth of the world wide web, where the vertices represent sites or web pages, and theedges links between sites. In this process the vertices are added to the graph one at a timeand joined to a fixed number of earlier vertices, selected with probability proportionalto their degree. This preferential attachment assumption is originated from the ideathat a new site is more likely to join popular sites than disregarded sites. The model isdescribed as follows.
Starting with a small number ( m ) of vertices, at every time step add a newvertex with m ( ≤ m ) edges that link the new vertex to m different verticesalready present in the system. To incorporate preferential attachment, assumethat the probability that a new vertex will be connected to a vertex i dependson the connectivity k i of that vertex, so it would be equal to k i / P j k j . Thus,after t steps the model leads to a random network with t + m vertices and mt edges. To write a mathematical description of the process given above it is necessary to clarifysome details. First, since the model starts with m vertices and none edges, then thevertices degree are initially zero, so the probability that the new vertex is connected toa vertex i , 1 ≤ i ≤ m , is not well defined. Second, to link the new vertex to m differentvertices already present, it should be necessary to repeat m times the experiment ofchoosing an old vertex, but the model does not say anything on changes of attachmentprobabilities at each time, i.e. it is not explained if the m old vertices are simultaneouslyor sequentially chosen. These observations were made by Bollob´as, Riordan, Spencer,and Tusnady in [3], where after noted the problems in the Barab´asi–Albert model, theygive an exact definition of a random graph process that fits to that description.3. Available results:
In [1], Barab´asi and Albert obtain through simulation that aftermany time steps the proportion of vertices with degree k obeys a power law Ck − γ , where C is a constant and γ = 2 . ± .
1, and by a heuristic argument they suggest that γ = 3.Let N k,t be the number of vertices with degree equal to k in ( G tm ) t ≥ . In [3] Bollob´as,Riordan, Spencer, and Tusnady analyzed mathematically this model. Their first result is hat, for t = n ( m +1), i.e., when the total number of vertices is n , and m ≤ k ≤ m + n / (the bound k ≤ m + n / is chosen to make the proof as easy as possible), E N k,t n ∼ m ( m + 1) k ( k + 1)( k + 2) = α k , uniformly in k .The authors consider F s , the σ -field generated by the appearance of directed edgesup to time s , s ≤ t , and define Z s = E ( N k,t | F s ) and see it is a martingale satisfying | Z s − Z s − | ≤ Z t = N k,t and Z = E N k,t , k = 1 , , . . . (at time t = 0 the randomgraph is the empty graph). Using Azuma–Hoeffding inequality (2.1) they obtain that P (cid:16)(cid:12)(cid:12)(cid:12) N k,t n − E N k,t n (cid:12)(cid:12)(cid:12) ≥ p ln t/t (cid:17) ≤ exp (cid:16) − ln t (cid:17) −→ , as t goes to infinity. Hence, it follows that, for every k in the range m ≤ k ≤ m + n / , N k,t n −→ α k , in probability. Thus, the proportion of vertices with degree k , N k,t n −→ m ( m + 1) B ( k,
3) (3.17)in probability as t −→ ∞ . Note that2 m ( m + 1) k ( k + 1)( k + 2) = m ( m + 1) B ( k, . Furthermore, since the Beta function satisfies the asymptotics B ( x, y ) −→ x − y for x large enough, then N k,t /n ∼ m ( m + 1) k − as k −→ ∞ and obeys a power law forlarge values of k , with γ = 3 as Barab´asi and Albert suggested. Hence, it is provedmathematically that when vertices are added to the graph one at a time and joined to afixed number of existing vertices selected with probability proportional to their degree,the degree distribution follows a power law behavior only in the tail (for k big enough),with an exponent γ = 3. A second proof of this result is given in [23] (see Theorem 8.2). Differently from the previous models, this model evolves in continuous time. We do notdescribe this model in terms of random graph processes, however in subsection 4.2 we discussits relation with Simon model and conclude that the Yule model can be interpreted as acontinuous time limit of Simon model, a model with a random graph interpretation.1.
Mathematical description:
In the description of the Yule model we use T to denotecontinuous time, instead of t that denotes discrete time, i.e. T ∈ R + ∪ { } and t ∈ Z + .Consider a population starting at time T = 0 with one individual. As time increases,individuals may give birth to new individuals independent of each other at a constantrate λ >
0, i.e., during any short time interval of length h each member has probability λh + o ( h ) to create an offspring. Since there is no interaction among the individuals,then if at epoch T the population size is k , the probability that an increase takes placeat some time between T and T + h equals kλh + o ( h ). Formally, let N ( T ) be the numberof individuals at time T with N (0) = 1, then if N ( T ) = k , k ≥
1, the probability of anew birth in (
T, T + h ) is kλh + o ( h ), and the probability of more than one birth is o ( h ),i.e., P ( N ( T + h ) = k + ℓ | N ( T ) = k ) = kλh + o ( h ) , ℓ = 1 ,o ( h ) , ℓ > , − kλh + o ( h ) , ℓ = 0 . Thus, { N ( T ) } T ≥ is a pure birth process and with the initial condition P ( N (0) = k ) = δ k, ; this linear birth process is called the Yule process . onsider now two independent Yule processes, { N β ( T ) } T ≥ and { N λ ( T ) } T ≥ , with pa-rameters β > λ > β , a new Yule process with parameter λ starts. In a randomgraph context, a Yule model can be characterized through Yule processes of differentparameters as described in the following. The first Yule process denoted by { N β ( T ) } T ≥ , β >
0, accounts for the growth of the number of vertices. As soon as the first vertex iscreated, a second Yule process, { N λ ( T ) } T ≥ , λ >
0, starts describing the creation of in-links to the vertex. The evolution of the number of in-links for the successively createdvertices, proceeds similarly. Specifically, for each of the subsequent created vertices, anindependent copy of { N λ ( T ) } T ≥ , modeling the appearance of the in-links is initiated.Let us define Y = 0 and for k ≥ Y k = inf { T : N λ ( T ) = k + 1 } , so that Y k is the time of the k th birth, and W ∗ k = Y k − Y k − is the waiting time betweenthe ( k − k th birth. In a Yule process it is well-known that the waitingtimes W ∗ k , k ≥
1, are independent, each exponentially distributed with parameter λk .Conversely, it is possible to reconstruct { N λ ( T ) } T ≥ from the knowledge of the W ∗ j , j ≥
1, by defining Y k = k X j =1 W ∗ j , N λ ( T ) = min { k : Y k > T } . (3.18)Thus if the W ∗ j are independently distributed exponential random variables, of parameter λj , then { N λ ( T ) } T ≥ is a Yule process of parameter λ .2. Historical context:
Yule in [24] observed that the distribution of species per genus inthe evolution of a biological population typically presents a power law behavior, thus,he proposed a stochastic model to fit these data. In the original paper [24] the processis described as follows:
Let the chance of a species “throwing” a specific mutation, i.e., a new speciesof the same genus, in some small assigned interval of time be p , and supposethe interval so small that p may be ignored compared with p . Then, puttingaside generic mutations altogether for the present, if we start with N primespecies of different genera, at the end of the interval we will have N (1 − p ) which remain monotype and Np genera of two species. The new species as wellas the old can now throw specific mutation. Yule proceeded to the limit, taking the time interval ∆ T as indefinitely but the number ofsuch intervals n as large, so that n ∆ T = T is finite, and he wrote p = λ ∆ T = λT /n . Yulenot only studied this process. In [24], he furthermore constructed a model of evolutionby considering two independent Yule processes, one for species with a constant rate λ > β >
0. In other words, at time T = 0 the process starts with asingle genus composed by a single species. As time goes on, new genera (each composedby a single species) develop as a Yule process of parameter β , and simultaneously andindependently new species evolve as a Yule process with rate λ . Furthermore, since anew genus appears with a single species, then each time a genus births, a Yule processwith rate λ starts.3. Available results:
Let N g ( T ) and N s ( T ), T ≥
0, be the counting processes measuringthe number of genera and species created until time T , respectively. It is well-knownthat the probability distribution of the number of individuals in a Yule process withparameter λ is geometric, Geo( e − λT ). Thus, the distribution of the number of species N s ( T ) in a genus during the interval of time [0 , T ] is P ( N s ( T ) = k ) = e − λT (1 − e − λT ) k − , k ≥ , T ≥ . (3.19)On the other hand, it is also known that by conditioning on the number of genera presentat time T , the random instants at which creation of novel genera occurs are distributedas the order statistics of iid random variables with distribution function P ( T ≤ τ ) = e βτ − e βt − , ≤ τ ≤ t (3.20) see [13] and the references therein). The authors in [13] take into account that thehomogeneous linear pure birth process lies in the class of the so-called processes withthe order statistic property, see [14], and use [6, 11, 18] and [22] to get (3.20).Thus, let N T be the size of a genus chosen uniformly at random at time T . Then, P ( N T = k ) = Z T P ( N s ( T ) = k | N s ( τ ) = 1) P ( T ∈ dτ )= Z T e − λ ( T − τ ) (1 − e − λ ( T − τ ) ) k − β e βτ e βT − dτ = β − e − βT Z T e − βy e − λy (1 − e − λy ) k − dy. (3.21)The interest now is in the limit behavior when T −→ ∞ :lim T −→∞ P ( N T = k ) = β Z ∞ e − βy e − λy (1 − e − λy ) k − dy. (3.22)Letting ρ = β/λ it is possible to recognize the integral as a beta integral to obtain (see[24], page 39) lim T −→∞ P ( N T = k ) = ρ Γ( k )Γ(1 + ρ )Γ( k + 1 + ρ ) = ρB ( k, ρ ) , k ≥ . (3.23) In [4], Bornholdt and Ebel pointed out that the asymptotic power law of the Barab´asi–Albertmodel with m = 1 coincides with that of the Simon model characterized by α = 1 / m = 1 Barab´asi–Albertmodel could be mapped to the subclass of Simon models with α = 1 / Theorem 4.1.
Let m = 1 . Then, the in-degree distribution of the II-PA model and the degreedistribution of the Barab´asi–Albert model at time t , t ≥ , are the same, i.e., if at time t thereare n vertices in the processes, then for any k ∈ Z + , ~N II-PA k,t n = N BA k,t n , where ~N II-PA k,t and N BA k,t denote the number of vertices with in-degree and degree equal to k in ( ˜ G t ) and ( G t ) at time t , i.e., in the II-PA and Barab´asi–Albert models, respectively.Proof. We follow the mathematical description of the II-PA and Barab´asi–Albert models interms of the random graph processes ( ˜ G tm ) t ≥ and ( G tm ) t ≥ , presented in Sections 3.2 and3.4, respectively. Let us divide each unit of time in two sub-units. At each instant of time t = 2 n + 1 a new vertex v n +1 is created in both models; in the II-PA model this vertex iscreated together with a directed loop. Furthermore, at each time t = 2 n +2 = 2( n +1) an edge(a directed edge in the II-PA model) is added from v n +1 to v j , j ≤ n + 1, with probabilitiesgiven by (3.11) and (3.16) for the II-PA and Barab´asi–Albert models, respectively. Hence ourthesis corresponds to show that (3.11) and (3.16) coincide under our hypotheses. e see that the denominator for both probabilities (3.11) and (3.16) is 2 n +1, and althoughthe two numerators count different quantities, the in-degree for the II-PA and the degree forBarab´asi–Albert models, their values also coincide. This is easy to check when v j = v n +1 andthe directed edge created at time t = 2( n + 1) is to v n +1 . In fact the numerators of (3.11)and (3.16) become both one. Let us now show that the two numerators coincide also when v j = v n +1 .Let us suppose v j = v n +1 , and let t = 2( n + 1) − n + 1. Observe that in the Barab´asi–Albert model the degree of v j at time t = 2 n + 1, d ( v j , n + 1), j ∈ { , , . . . , n } , is the sum ofthe number of incoming edges from time t = 2 j + 1 (when v j +1 is added) to time 2 n + 1, plusthe degree corresponding to the edge added at time t = 2 j from v j , that is two if the edge wasa loop and one otherwise. On the other hand, in the II-PA model the in-degree of v j at time t = 2 n + 1, ~d ( v j , n + 1), j ∈ { , , . . . , n } is the sum of the number of incoming edges addedin the interval of time t ∈ [2 j + 1 , n + 1] (so this part coincides with Barab´asi–Albert model),plus the in-degree corresponding to the directed edge added at time t = 2 j from v j . Thus, ifit is a directed loop to v j , the in-degree of v j at time t = 2 j is two (since when v j appeared,it did together with a directed loop), otherwise the in-degree is one. This concludes the proofand (3.11) and (3.16) coincide. Remark 4.1.
The proof of Theorem 4.1 enlightens the advantage given by the re-definitionof existing models in terms of random graph processes. In particular this reading shows im-mediately that the two models can be related only when m = 1 . Remark 4.2.
The in-degree distribution of the II-PA model and the degree distribution of theBarab´asi–Albert model are different when m > . In fact take for example m = 2 and supposethe first directed edge from v n +1 is not a loop, i.e., a vertex v j , ≤ j ≤ n is chosen. Then,at time t = 3 n + 1 , ~d ( v n +1 , n + 1) = 1 in the II-PA model, while d ( v n +1 , n + 1) = 2 in theBarab´asi–Albert model. Thus at time t = 3 n + 2 , (3.11) and (3.16) are different because thecorresponding numerators differ. Next we discuss the relationship between Simon and the II-PA models, which allows us torelate Barab´asi–Albert and Simon models. Before writing such a result, observe the followingfact. Let Y i be a random variable that counts the number of direct edges originated in theSimon model by the i th vertex v i , until the appearing of the ( i + 1)th vertex. Note that Y i follows a Geometric distribution with parameter α . So, if α = 1 / ( m + 1), then E Y i = m , andthat is the number of out-going links from a vertex in the the II-PA and Barab´asi–Albertmodels.What we will establish in the following theorem is that the asymptotic in-degree distri-bution of the II-PA and Simon models coincide when α = 1 / ( m + 1). To do that, first weintroduce the following definition. Definition 4.1.
We say that a vertex v i appears “complete” when it has appeared in theprocess together with all the directed edges originated from it. Thus, at time t = n ( m + 1) , theII-PA model has exactly n “complete” vertices. Now we are ready to enunciate the theorem.
Theorem 4.2.
Let m ∈ Z + be fixed. If ( ˜ G tm ) t ≥ is the random graph process defining theII-PA model, and ~N II-PA k,t the number of vertices with in-degree equal k , k ≥ , at time t in ( ˜ G tm ) t ≥ , then, at time t = n ( m + 1) ~N II-PA k,t n −→ (1 + 1 /m )Γ( k )Γ(2 + 1 /m )Γ( k + 2 + 1 /m ) (4.1) almost surely as n −→ ∞ . Remark 4.3.
Observe that by (4.1) and (3.10) if α = 1 / ( m + 1) in the Simon model, thenthe asymptotic in-degree distribution of Simon and II-PA models coincide. Moreover, fromTheorem 4.1, (4.1) and (3.10), it follows that the asymptotic degree distribution of Barab´asi–Albert model coincides with the asymptotic in-degree distribution of Simon model only when m = 1 and α = 1 / ( m + 1) , so that α = 1 / . We conjecture that some other properties ofBarab´asi–Albert model when m = 1 , for example the diameter, should be also related with theanalogous features of Simon model when α = 1 / . emark 4.4. Observe that (4.1) coincides with (3.12). Thus, the previous theorem gives arigorous formalization to the heuristic result in [16].
Remark 4.5.
Theorem 4.2 can be compared with the recent model-free approach of Os-troumova, Ryabchenko and Samostav (see Section 3, Theorem 2 in [17], with A = m/ ( m + 1) and B = 0 ). However, in that work a preferential attachment rule proportional to the degreeisconsidered, and Theorem 2 in [17] makes use of the initial condition that the degree of anexisting vertex should be at least equal to m . Instead, in the II-PA model, it is considered apreferential attachment rule proportional to the in-degree with the initial condition that thein-degree of an existing vertex should be at least equal to one. Therefore, the II-PA modeldoes not fit into the general setup of Theorem 2 in [17] and we cannot directly apply it to getthe result of Theorem 4.2 given in this paper. We believe however that, following these newideas, but considering the in-degree and the corresponding initial condition we can obtain theasymptotic in-degree distribution for the II-PA model.However, in this paper we use the master equations approach for consistency with thetheory used to study Simon model. Before proving Theorem 4.2 we need to prove the following lemmas.
Lemma 4.1.
Let r, s, t ∈ Z + and b ∈ R such that | b/r | < , then t Y r = s +1 (cid:16) − br (cid:17) = (cid:16) st (cid:17) b (cid:16) O (cid:16) t − sst (cid:17)(cid:17) . Proof.
Since | b/r | <
1, then using Taylor expansion for ln(1 − b/r ) we get t Y r = s +1 (cid:16) − br (cid:17) = exp h t X r = s +1 (cid:18) − br + O (cid:16) b r (cid:17)(cid:19) i . Now, by Euler–Maclaurin it is possible to obtain that (see [19])1. P tr =1 1 r = ln t + 1 − R t y −⌊ y ⌋ y dy ,2. P tr =1 1 r = t − R r ⌊ y ⌋ y dy .Using these expressions and the fact that y − ≤ ⌊ y ⌋ ≤ y , where ⌊ y ⌋ indicates the integerpart of y , we obtain ln t − ln s − t − sst < t X r =1 r < ln t − ln s, (4.2) t − sst − t − s ( st ) < t X r =1 r < t − sst , (4.3)or, P tr =1 /r = ln t − ln s − | δ | , where | δ | < ( t − s ) / ( st ), and P tr =1 /r = ( t − s ) / ( st ) − | δ | ,where | δ | < ( t − s ) / ( st ) . Thus, t Y r = s +1 (cid:16) − br (cid:17) = exp h b ln (cid:16) st (cid:17) + O (cid:16) t − sst (cid:17)i = (cid:16) st (cid:17) b (cid:16) O (cid:16) t − sst (cid:17)(cid:17) . Lemma 4.2.
Let ~N II-PA k,t and ~N ( k, n ) denote the number of vertices with in-degree equal k , k ≥ , at time t , and the number of vertices with in-degree k when there are exactly n completevertices in the II-PA model, respectively. Then, • for m = 1 and k = 1 , E ~N (1 , n + 1) = (cid:16) − m + 1)( n + 1) − (cid:17) + (cid:16) − m + 1)( n + 1) − (cid:17) E ~N (1 , n );(4.4) for m > and k = 1 , E ~N (1 , n + 1) = 1 + (cid:16) − m ( n + 1)( m + 1) − (cid:17) E ~N (1 , n ) + O (cid:16) n (cid:17) ; (4.5) • for m = 1 and k ≥ , E ~N ( k, n + 1) = ( k − E ~N ( k − , n )( n + 1)( m + 1) − (cid:16) − k ( n + 1)( m + 1) − (cid:17) E ~N ( k, n )= ( k − E ~N ( k − , n )2( n + 1) − (cid:16) − k n + 1) − (cid:17) E ~N ( k, n ); (4.6) • for m > and k ≥ , E ~N ( k, n + 1) = ( k − m E ~N ( k − , n )( n + 1)( m + 1) − (cid:16) − km ( n + 1)( m + 1) − (cid:17) E ~N ( k, n ) + O (cid:16) kn (cid:17) . (4.7) Proof.
Let m = 1 and k = 1 we start at time t = ( m + 1) n = 2 n , i.e., when there are exactly n complete vertices. To see what happens when exactly ( n + 1) complete vertices appear, weneed to check what happens in two steps of the process, at time 2 n +1, when deterministicallyappears a new vertex with a directed loop, and at time 2 n +2 = 2( n +1), when a new directededge is added by preferential attachment, and the last vertex added becomes complete. Thus,conditioning on what happens until time t + 1, we have E ( ~N II-PA1 ,t +2 ) = E h ( ~N II-PA1 ,t + 1) (cid:16) − ~N II-PA1 ,t + 1 t + 1 (cid:17) + ~N II-PA1 ,t (cid:16) ~N II-PA1 ,t + 1 t + 1 (cid:17)i = (cid:16) − t + 1 (cid:17) + (cid:16) − t + 1 (cid:17) E ( ~N II-PA1 ,t ) . (4.8)Thus, if ~N ( k, n ) denotes the number of vertices with in-degree k when there are exactly n complete vertices in the process, then we can write the previous equation as (4.4).Let m > k = 1. Now we need a bit more attention, since we have to consider twodifferent situations, when t is multiple of ( m + 1) and when t is not. In the first situation t has the form t = n ( m + 1), so we are in the instant of time when there are exactly n complete vertices, and as we did above, to see what happens later we check what happensin the two subsequent steps of the process, at time n ( m + 1) + 1 when a deterministic eventhappens, a new vertex with a directed loop appears, and at time n ( m +1)+2 when somethingprobabilistic happens, a new directed edge is added by preferential attachment. In the firstcase equation (4.8) still holds. In the second situation observe that if m >
1, in order to seecomplete the vertex added at time t = n ( m + 1) + 1, we have to check what happens from n ( m + 1) + 1 until n ( m + 1) + ( m + 1) = ( n + 1)( m + 1), when this vertex becomes complete.Thus, when t is not multiple of ( m + 1) we have the following equation. E ( ~N II-PA1 ,t +1 ) = E h ~N II-PA1 ,t (cid:16) − ~N II-PA1 ,t t (cid:17) + ( ~N II-PA1 ,t − (cid:16) ~N II-PA1 ,t t (cid:17)i = (cid:16) − t (cid:17) E ( ~N II-PA1 ,t ) . (4.9)Now, we may use simultaneously (4.8) and (4.9) to get the corresponding equation of whathappens in ( m + 1) steps of the process. We start at time t = ( n + 1)( m + 1) −
1, so at time t + 1 the process will have exactly ( n + 1) complete vertices, and since t is not multiple of( m + 1), we need to begin using (4.9) ( m −
1) times, and then use (4.8). Iterating m times,we obtain E ( ~N II-PA1 ,t +1 ) = h E ( ~N II-PA1 ,t − m ) i m − Y j =0 (cid:16) − t − j (cid:17) = h E ( ~N II-PA1 ,t − m ) i t Y r = t − ( m − (cid:16) − r (cid:17) = (cid:16) − mt + O (cid:16) t (cid:17)(cid:17)h E ( ~N II-PA1 ,t − m ) i , (4.10) here we have used in the the last two steps that r = t − j and Lemma 4.1. Finally, usingthe notation ~N (1 , n ), and since ~N (1 , n ) /n ≤
1, we get (4.5).The cases k = 2 and k > m = 1 and when m >
1. Then we will show that the equations for k = 2 and k > k ≥ m = 1. Analogously as we did when m = 1 and k = 1, consider the time t = n ( m + 1),i.e., when there are exactly n complete vertices. To account for what happens until when( n + 1) complete vertices appear, we need to recognize two steps of the process. Indeed, E ( ~N II-PA2 ,t +2 ) = E h ( ~N II-PA2 ,t + 1) (cid:16) ~N II-PA1 ,t + 1 t + 1 (cid:17) + ( ~N II-PA2 ,t −
1) 2 ~N II-PA2 ,t t + 1+ ~N II-PA2 ,t (cid:16) − ~N II-PA1 ,t + 1 + 2 ~N II-PA2 ,t t + 1 (cid:17)i = E ( ~N II-PA1 ,t ) + 1 t + 1 + (cid:16) − t + 1 (cid:17) E ( ~N II-PA2 ,t ) , (4.11)and for k > E ( ~N II-PA k,t +2 ) = E h ( ~N II-PA k,t + 1) (cid:16) ( k − ~N II-PA k − ,t t + 1 (cid:17) + ( ~N II-PA k,t − k ~N II-PA k,t t + 1+ ~N II-PA k,t (cid:16) − ( k − ~N II-PA k − ,t + k ~N II-PA k,t t + 1 (cid:17)i = ( k − E ( ~N II-PA k − ,t +1 ) t + 1 + (cid:16) − kt + 1 (cid:17) E ( ~N II-PA k,t +1 ) . (4.12)Note that in the last line of (4.11) and (4.12), we have replaced ~N II-PA k,t +1 with ~N II-PA k,t . In factif t = n ( m + 1), then at time t + 1 the process just adds deterministically a new vertex within-degree one, so when k ≥ ~N II-PA k,t +1 = ~N II-PA k,t , as well as ~N II-PA1 ,t +1 = ~N II-PA1 ,t + 1. Using thisobservation we can express (4.11) and (4.12) as a single equation holding for k ≥ m = 1.Using the notation ~N (1 , n ) it can be written as (4.6).Let m >
1. once more we need to consider when t is multiple of ( m + 1), and when it isnot. When t = n ( m + 1) we obtain again (4.11) and (4.12) for k = 2 and k >
2, respectively,while if t is not multiple of ( m + 1) and k ≥ E ( ~N II-PA k,t +1 ) = E h ( ~N II-PA k,t + 1) (cid:16) ( k − ~N II-PA k − ,t t (cid:17) + ( ~N II-PA k,t − k ~N II-PA k,t t + ~N II-PA k,t (cid:16) − ( k − ~N II-PA k − ,t + k ~N II-PA k,t t (cid:17)i = ( k − E ( ~N II-PA k − ,t ) t + (cid:16) − kt (cid:17) E ( ~N II-PA k,t ) . (4.13)Now, in order to get the corresponding equation for what happens in ( m +1) steps, i.e., duringthe time interval from when there are n vertices until when there are ( n + 1) vertices, it isnecessary to use (4.12) and (4.13) simultaneously. In the same manner as we did for k = 1,we take t = ( n + 1)( m + 1) −
1, so that at time t + 1 the process will have exactly ( n + 1)complete vertices. Since t is not multiple of ( m + 1), we need to begin using (4.13) ( m − m times, after some algebra we obtain that forany k ≥ E ( ~N II-PA k,t +1 ) = h m − X i =0 ( k − E ( ~N II-PA k − ,t − i ) t − i i − Y j =0 (cid:16) − kt − j (cid:17)i + h m − Y j =0 (cid:16) − kt − j (cid:17)i E ( ~N II-PA k,t − ( m − ) , (4.14) here the empty product (i.e., when i = 0) is equal to unity. Let now r = t − j , and since i ≤ m and m is fixed, then by Lemma 4.1 we have i − Y j =0 (cid:16) − kt − j (cid:17) = t Y r = t − ( i − (cid:16) − kr (cid:17) = 1 − kit + O (cid:16) k t (cid:17) . (4.15)Moreover, observe that | E ( ~N II-PA k − ,t − i ) − E ( ~N II-PA k − ,t − ( m − ) | ≤ m + 1 − i , since at each instant atmost one edge is added. Thus m − X i =0 E ( ~N II-PA k − ,t − i ) t − i = m − X i =0 E ( ~N II-PA k − ,t − ( m − ) t (cid:16) it − i (cid:17) + O (cid:16) t (cid:17) = m E ( ~N II-PA k − ,t − ( m − ) t + O (cid:16) t (cid:17) . (4.16)Then using (4.15) and (4.16) and noting that ( k − ~N II-PA k − ,t − i t − i ≤
1, we can write (4.14) as E ( ~N II-PA k,t +1 ) = ( k − m E ( ~N II-PA k − ,t − ( m − ) t + (cid:16) − kmt (cid:17) E ( ~N II-PA k,t − ( m − ) + O (cid:16) kt (cid:17) , (4.17)and using the notation ~N ( k, n ) we obtain (4.7).Theorem 4.2 gives the limit value to which ~N ( k, n ) /n converges when n goes to infinity.However, before proving the limit, we need to argue that such limit exists. Lemma 4.3.
Let ~N ( k, n ) be as in Lemma 4.2. Then, there exist values N ( k ) > and N ( k ) > such that ~N ( k, n ) n −→ N ( k ) a.s.,and ~mk ~N ( k, n )( m + 1) n −→ N ( k ) a.s.Proof. We make use of supermartingale’s convergence theorem (see [2], Theorem 35.5) andequations (4.4), (4.5), (4.6) and (4.7). Consider first (4.4) and (4.5) and observe that since ~N (1 , n ) / (( n + 1)( m + 1) − ≤ E ~N (1 , n + 1) ≤ E ~N (1 , n ) + 1 , (4.18)while for (4.6) and (4.7), E ~N ( k, n + 1) ≤ E ~N ( k, n ) + 1 + O (cid:16) kn (cid:17) . (4.19)Let H n be the filtration generated by the process { ~N ( k, n ) , ~N ( k − , n ) } n until time n , i.e., H n := σ ( N ( k, j ) , N ( k − , j ); 0 ≤ j ≤ n ). If k = 1, let Z (1 , n ) = ( ~N (1 , n ) − n ) /n , then by(4.18), E [ Z (1 , n + 1) | H n ] ≤ ~N (1 , n ) + 1 − ( n + 1) n + 1 ≤ ~N (1 , n ) + 1 − ( n + 1) n = Z (1 , n ) , (4.20)as N (1 , n ) is H n -measurable. Hence, { Z (1 , n ) } n is a supermartingale and in order to applysupermartingale convergence theorem to { Z (1 , n ) } n , it remains to prove thatsup n E ( | Z (1 , n ) | ) < ∞ . This is true as E ( | Z (1 , n ) | ) = E (cid:2) E (cid:0) | Z (1 , n ) | (cid:12)(cid:12) H n − (cid:1)(cid:3) ≤ n ( E ~N (1 , n −
1) + 1 + n ) < ∞ , (4.21) aving used that ~N ( k, n ) /n ≤
1, for any n ≥ k ≥
2, i.e., for (4.6) and (4.7), note first that if f ( n ) = O ( k/n ), then there exists M > | f ( n ) | = Mk/n + | δ | , where | δ | < k/n . Since k/n ≤
1, then there exists M such that | f ( n ) | ≤ M + 1. Thus, take Z ( k, n ) = [ ~N ( k, n ) − n ( c + 1)] /n , with c = M + 1, thenby (4.18) we also get that { Z ( k, n ) } n is a supermartingale, and similarly as we did above wealso show that sup n E ( | Z ( k, n ) | ) < ∞ . In this manner we have proved that Z ( k, n ) convergesalmost surely, thus ~N ( k, n ) /n converges almost surely. In perfect analogy we can prove that mkZ ( k, n ) / ( m + 1) converges almost surely, and thus obtain that mk ~N ( k, n ) /n ( m + 1) alsoconverges almost surely.In order to determine such a limit, we still need to prove the following lemma. Lemma 4.4.
Let p k := lim n −→∞ E ~N ( k, n ) /n . Then, p k = (1 + 1 /m )Γ( k )Γ(2 + 1 /m )Γ( k + 2 + 1 /m ) , k ≥ . (4.22) Proof.
Observe that for a function f ( k ), mf ( k )( n + 1)( m + 1) − mf ( k ) n ( m + 1) (cid:16) − m ( n + 1)( m + 1) − (cid:17) = mf ( k ) n ( m + 1) + O (cid:16) f ( k ) n (cid:17) . (4.23)By using (4.23) we can write the equations (4.4), (4.5), (4.6) and (4.7) as follows. For m = 1, k = 1, E ~N (1 , n + 1) = 1 + (cid:16) − n ( m + 1) (cid:17) E ~N (1 , n ) + O (cid:16) n (cid:17) , (4.24)for m > k = 1, E ~N (1 , n + 1) = 1 + (cid:16) − mn ( m + 1) (cid:17) E ~N (1 , n ) + O (cid:16) n (cid:17) , (4.25)for m = 1, k ≥ E ~N ( k, n + 1) = ( k − E ~N ( k − , n ) n ( m + 1) + (cid:16) − kn ( m + 1) (cid:17) E ~N ( k, n ) + O (cid:16) n (cid:17) , (4.26)and for m > k ≥ E ~N ( k, n + 1) = m ( k − E ~N ( k − , n ) n ( m + 1) + (cid:16) − mkn ( m + 1) (cid:17) E ~N ( k, n ) + O (cid:16) kn (cid:17) . (4.27)Looking at (4.24), (4.26), (4.25) and (4.27), we remark that they can be unified as E ~N ( k, n + 1) = g ( k − , n ) + (cid:16) − bn (cid:17) E ~N ( k, n ) + E n , (4.28)where b = km/ ( m + 1), g (0 , n ) = 1, g ( k − , n ) = ( mk/n ( m + 1)) E ~N ( k, n ) for k ≥
2, and E n = O (1 /n ) if m = 1 and of order O ( k/n ) if m >
1. We underline that k could be a functionof n and hence in general O ( k/n ) can be different of O (1 /n ).Note now that when the first complete vertex appears, it has in-degree equal to ( m +1), so ~N ( k,
1) = 0 for any k = ( m +1), and ~N ( m +1 ,
1) = 1. Iterating (4.28) we have, if k = ( m +1), E ~N ( k, n + 1) = n − X i =0 g ( k − , n − i ) i − Y j =0 (cid:16) − bn − j (cid:17) + n − X i =0 E n − i , (4.29)while, if k = m + 1, E ~N ( k, n + 1) = n − X i =0 g ( k − , n − i ) i − Y j =0 (cid:16) − bn − j (cid:17) + n − Y j =0 (cid:16) − bn − j (cid:17) + n − X i =0 E n − i . (4.30) o solve (4.30), let s = n − i and r = n − j so that i − Y j =0 (cid:16) − bn − j (cid:17) = n Y r = s +1 (cid:16) − br (cid:17) , then observe that if s < ⌊ b ⌋ , n Y r = s +1 (cid:16) − br (cid:17) = ⌊ b ⌋ Y r = s +1 (cid:16) − br (cid:17) n Y r = ⌊ b ⌋ +1 (cid:16) − br (cid:17) , which is equal either to 0 if b = ⌊ b ⌋ or to( − ⌊ b ⌋ ⌊ b ⌋ Y i =1 ( b − i )( ⌊ b ⌋ − i + 1) n Y r = ⌊ b ⌋ +1 (cid:16) − br (cid:17) , if b = ⌊ b ⌋ . Applying Lemma 4.1 (note that to apply this lemma is necessary to have b/r < r ≥ ⌊ b ⌋ + 1) we have n Y r = s +1 (cid:16) − br (cid:17) = , s < ⌊ b ⌋ , b = ⌊ b ⌋ ,O (cid:0) ⌊ b ⌋ n (cid:1) ⌊ b ⌋ , s < ⌊ b ⌋ , b = ⌊ b ⌋ , (cid:0) O (cid:0) n − ssn (cid:1)(cid:1)(cid:0) sn (cid:1) b , s ≥ ⌊ b ⌋ . Using this and (4.2), formula (4.30) can be written as E ~N ( k, n + 1) = n X s = ⌊ b ⌋ g ( k − , s ) (cid:16) sn (cid:17) b (cid:16) O (cid:16) n − ssn (cid:17)(cid:17) + O (cid:16) ⌊ b ⌋ n (cid:17) ⌊ b ⌋ + E = n X s = ⌊ b ⌋ g ( k − , s ) (cid:16) sn (cid:17) b + E , (4.31)where the error term E is of order O (ln n ) if m = 1 and of order O ( k ln n ) if m >
1. It is notdifficult to see that, following a similar procedure, we can get the same solution for (4.29).Now, by Lemma 4.3 we know that there exist some N ( k ) > N ( k ) >
0, suchthat ~N ( k, n ) /n −→ N ( k ) and mk ~N ( k, n ) / [ n ( m + 1)] −→ N ( k ) almost surely (observe thatin order to guarantee a.s. convergence, we will need to take k independent of n , hence wewill obtain N ( k ) and N ( k ) strictly positive). Thus, by the dominated convergence theorem p k := lim n −→ E ~N ( k, n ) /n , and for k ≥ g ( k −
1) := lim n −→ g ( k − , n ), exist.Note that g ( k −
1) = m ( k − p k − / ( m + 1), and let us write g ( k − , n ) = g ( k −
1) + O ( ε n ),where ε n −→ n −→ ∞ . Hence, n X s = ⌊ b ⌋ g ( k − , s ) (cid:16) sn (cid:17) b = g ( k − n b n X s = ⌊ b ⌋ s b + 1 n b n X s = ⌊ b ⌋ O ( ε s ) s b , (4.32)and using that P ns = ⌊ b ⌋ s b = n b +1 / ( b + 1) + o ( n b +1 ) (see 3.II of [19]), we obtain E ~N ( k, n + 1) n + 1 = (cid:26) g ( k − b +1 + O (cid:0) ln nn (cid:1) , m = 1 , g ( k − b +1 + O (cid:0) k ln nn (cid:1) , m > . (4.33)Observe that when m > k in order to determine thelimit of E ~N ( k, n + 1) / ( n + 1). Indeed it should satisfy that k ln n/n −→ n −→ ∞ , butthat is true since we are taking k fixed, i.e., independent of n . Thus, by (4.33), p k = lim n −→∞ E ~N ( k, n + 1) n + 1 = ( m +12 m +1 , k = 1 , m ( k − p k − m ( k +1)+1 , k > . (4.34)Solving (4.34) recursively we get (4.22). roof of Theorem 4.2. We follow the approach of Dorogovtsev, Mendes, and Samukhin [9],that uses master equations for the expected value of the number of vertices with in-degree k .To obtain the exact equations we need to consider two stages. For the first one we considerwhat happens in one step of the process, during which the number of vertices of in-degree k can be increased by counting also some vertices coming from those having previously in-degree ( k −
1) or in-degree ( k + 1), and then we consider what happens in ( m + 1) steps, thusobtaining the change of the vertices in-degree in an interval of time starting when the processhas n vertices, until it has ( n + 1) vertices. This part corresponds to finding the equations(4.4), (4.5), (4.6) and (4.7) given in Lemma 4.2. For the second stage, we iterate the previousequations with respect to n and obtain the limit of E ~N ( k, n ) /n as n −→ ∞ . This part wasproved in Lemma 4.4 determining (4.22).Finally, we use Azuma–Hoeffding inequality (2.1) to obtain (4.1). Let F t be the naturalfiltration generated by the process { ~N II-PA k,t } up to time t . Then, in the same way as itwas explained for to Simon model, Section 3.1, it is easy to show that for s ≤ t , Z II-PA s = E ( ~N II-PA k,t |F s ) is a martingale such that, | Z II-PA s − Z II-PA s − | ≤ Z II-PA t = ~N II-PA k,t and Z II-PA0 = E ~N II-PA k,t (at time t = 0 the random graph is the empty graph). Thus by (2.1) we get that forevery ǫ n ≫ / √ n , e.g. take ǫ n = p ln n/n , P (cid:16)(cid:12)(cid:12)(cid:12) ~N II-PA k,t n − E ~N II-PA k,t n (cid:12)(cid:12)(cid:12) ≥ ǫ n (cid:17) ≤ exp (cid:16) − ( nǫ n ) t (cid:17) −→ , as n goes to infinity. Here t = n ( m + 1) + i, for i = 0 , , . . . , m . Thus we obtain that for t = n ( m + 1), ~N II-PA k,t n −→ (1 + 1 /m )Γ( k )Γ(2 + 1 /m )Γ( k + 2 + 1 /m ) , in probability as n −→ ∞ . However, by Lemma 4.3 we actually have an almost sure conver-gence. Let M , M , . . . be independent and identically distributed random variables with E ( M i ) = m ,where m is a positive rational number and V ( M i ) = σ . Furthermore, let ( ˜ G nm ) n ≥ be therandom graph process defining the Price model as in Section 3.3 and take k = 1. If ~N Price k,n denotes the number of vertices with in-degree equal to k in ( ˜ G nm ) n ≥ , k ≥
1, then ~N Price k,n n −→ (1 + 1 /m )Γ( k )Γ(2 + 1 /m )Γ( k + 2 + 1 /m ) (4.35)almost surely as n −→ ∞ .A rigorous analysis of the previous result can be made using Chebyschev’s inequality andfollowing the same lines as in the proof of Theorem 4.2 for the II-PA model. Hence, we limitourselves to present a scheme of the proof.1. In the mathematical description of the Price model, we saw that ˜ G n +1 m is formed from ˜ G nm by adding a new vertex v n +1 with k directed loops, and from it a random number, M n +1 ,of directed edges to different old vertices. This happens with probabilities proportionalto their in-degrees as in (3.13). Conditioning on the number of vertices with in-degree k when there are n vertices, we obtain E ~N Price k ,n +1 = E (cid:2) E ( ~N Price k ,n +1 | ~N Price k ,n ) (cid:3) = E (cid:20) ~N Price k ,n − M n +1 k ~N Price k ,n nk + P ni =1 M i + ~N Price k ,n (cid:18) − M n +1 k ~N Price k ,n nk + P ni =1 M i (cid:19)(cid:21) = 1 + E (cid:20)(cid:18) − M n +1 k nk + P ni =1 M i (cid:19) ~N Price k ,n (cid:21) , (4.36) nd, for k > k , E ~N Price k,n +1 = E (cid:20) ( ~N Price k,n + 1) M n +1 ( k − ~N Price k − ,n nk + P ni =1 M i + ( ~N Price k,n − M n +1 k ~N Price k,n nk + P ni =1 m i + ~N Price k,n (cid:18) − M n +1 ( k − ~N Price k − ,n nk + P ni =1 M i − M n +1 k ~N Price k,n nk + P ni =1 M i (cid:19)(cid:21) = E (cid:20) M n +1 ( k − ~N Price k − ,n nk + P ni =1 M i + (cid:18) − M n +1 knk + P ni =1 M i (cid:19) ~N Price k,n (cid:21) . (4.37)2. Take k = 1, Y = n + P ni =1 M i and ǫ n = p ln n/n . By Chebyschev’s inequality, P [ | Y − n (1 + m ) | > nǫ n ] ≤ σ √ n ln n . (4.38)Let X be another random variable such that E ( X/Y ) is bounded, and 0 ≤ E ( X ) ≤ ( k − mn . In addition, define the event E n := { n (1 + m − ǫ n ) ≤ Y ≤ n (1 + m + ǫ n ) } .Conditioning on E n and applying (4.38), E ( X/Y ) ≈ E ( X/Y | Y ∈ E n ) + O (cid:18) √ n ln n (cid:19) , (4.39)because E ( X/Y ) is bounded. Furthermore, note that E ( X ) n (1 + m ) (cid:16) − ǫ n m + ǫ n (cid:17) ≤ E (cid:16) XY (cid:12)(cid:12)(cid:12) Y ∈ E n (cid:17) ≤ E ( X ) n (1 + m ) (cid:16) − ǫ n m − ǫ n (cid:17) , thus E ( X/Y | Y ∈ E n ) = E ( X ) n (1+ m ) + O ( p ln n/n ). Replacing this in (4.39) we have E ( X/Y ) ≈ E ( X ) n (1 + m ) + O ( p ln n/n ) . (4.40)Using (4.40) in (4.36) with X = M n +1 ~N Price1 ,n and in (4.37) with X = M n +1 ( k − ~N Price k − ,n and X = M n +1 k ~N Price k,n , respectively, we get from (4.36) and (4.37) that E ~N Price k ,n +1 ≈ (cid:18) − mn (1 + m ) (cid:19) E ~N Price k ,n + O ( p ln n/n ) , (4.41)and, for k > E ~N Price k,n +1 ≈ m ( k − E ~N Price k − ,n n (1 + m ) + (cid:18) − mkn (1 + m ) (cid:19) E ~N Price k,n + O ( p ln n/n ) . (4.42)3. Note now that (4.41) and (4.42) are almost the same as (4.5) and (4.7) for the II-PAmodel, respectively. In order to derive (4.35) we then proceed as in the proof of Theorem4.2. More specifically, to ensure the existence of the limit value of ~N Price k,n /n as n → ∞ ,we use supermartingale’s convergence theorem (see [2], Theorem 35.5), in analogy toLemma 4.3. Then we find thatlim n →∞ E ~N Price k,n n = (1 + 1 /m )Γ( k )Γ(2 + 1 /m )Γ( k + 2 + 1 /m ) , k ≥ , (4.43)as in Lemma 4.4. Finally by Azuma–Hoeffding inequality (2.1) we obtain ~N Price k,n n → (1 + 1 /m )Γ( k )Γ(2 + 1 /m )Γ( k + 2 + 1 /m ) , k ≥ , in probability as n → ∞ . By Lemma 4.3 the result follows almost surely.Notice that the Price model is by definition equivalent to the II-PA if M i = m = 1 almostsurely. Moreover, Price and II-PA models have the same limit in-degree distribution when E ( M i ) = m . .2 Relation between Simon and Yule models Bearing in mind the construction of Yule model as explained in Section 3.5, we underlinethat the inter-event times of in-links appearance and those related to creation of new verticesare exponentially distributed. In order to relate Yule and Simon models we investigate herethe inter-event times characterizing Simon model showing that a suitable rescaling in thelimit leads to exponential random variables. The idea is to identify two different processeswhich conditionally describe Simon model, and clarify how these are related with the twoYule processes which define a Yule model.The next theorem together with Remarks 4.6 and 4.7 allows us to recognize the first processinside a Simon model behaving asymptotically as a Yule process with parameter (1 − α ), whileTheorem 4.4 and Remark 4.8 determine the second process which behaves asymptotically asa Yule process with parameter equal to one. The first process models how the vertices getnew in-links, thus at each moment a new vertex appears, a process starts. On the other hand,the second process is related to how the vertices appear.Let ( G tα ) t ≥ be the random graph process associated to Simon model of parameter α , asdescribed in Subsection 3.1, and let { ~d ( v i , t ) } t ≥ t i be the in-degree process associated to thevertex v i , which appears at time t i , i.e., t i = min { t : ~d ( v i , t ) = 1 } (note that ~d ( v i , t i ) = 1 asthe vertex appears together with a directed loop in the model).Our first focus is on the study of the distributions of the waiting times between the instantin which each vertex has in-degree k , till that in which it has in-degree k + 1. Formally, westudy the distribution of the random variables W ik = t ik − t ik − , k ≥
1, where t ij = min { t : ~d ( v i , t ) = j + 1 } for j = 0 , , , . . . . Theorem 4.3.
Let z = ln (cid:16) xt ik − − (cid:17) , k ≥ , x > . It holds (cid:12)(cid:12)(cid:12) P ( W ik ≤ x ) − P ( Z ik ≤ z ) (cid:12)(cid:12)(cid:12) < O (cid:16) t ik − (cid:17) , (4.44) where Z ik is an exponential random variable of parameter (1 − α ) k . Remark 4.6.
Theorem 4.3 states that for any t ∗ large enough but fixed, (cid:12)(cid:12)(cid:12) P ( W jk ≤ x ) − P ( Z jk ≤ z ) (cid:12)(cid:12)(cid:12) < O (cid:16) t ∗ (cid:17) , (4.45) ∀ j ≥ min { i : t i ≥ t ∗ } , and for k ≥ . This means that from a fixed but large time t ∗ , all thewaiting times W jk are approximately exponential random variables, with an error term smallerthan O (1 /t ∗ ) .Proof of Theorem 4.3. By the preferential attachment probabilities (3.1) of ( G tα ) t ≥ , for x ≥
1, we have P [ W ik = x ] = (cid:16) ~d ( v i , t ik − )(1 − α ) t ik − + x − (cid:17)(cid:16) − ~d ( v i , t ik − )(1 − α ) t ik − + x − (cid:17) . . . (cid:16) − ~d ( v i , t ik − )(1 − α ) t ik − (cid:17) = (cid:16) k (1 − α ) t ik − + x − (cid:17)(cid:16) − k (1 − α ) t ik − + x − (cid:17) . . . (cid:16) − k (1 − α ) t ik − (cid:17) = (cid:16) k (1 − α ) t ik − + x − (cid:17) t ik − + x − Y r = t ik − (cid:16) − k (1 − α ) r (cid:17) , (4.46)as t ik − > k . Then k (1 − α ) /r <
1, so we can apply Lemma 4.1 to the product to obtain t ik − + x − Y r = t ik − (cid:16) − k (1 − α ) r (cid:17) = (cid:16) t ik − − t ik − + x − (cid:17) k (1 − α ) (cid:16) O (cid:16) x ( t ik − + x − t ik − − (cid:17)(cid:17) . (4.47) hus, using (4.46), (4.47), the Euler–Maclaurin formula, n X j =1 j s = 1 n s − − s Z n ⌊ y ⌋ y s +1 dy, with s ∈ R \ { } (see [19]) and the fact that ⌊ y ⌋ ≤ y , we arrive at P [ W ik ≤ x ] = x X w =1 (cid:16) k (1 − α ) t ik − + w − (cid:17)(cid:16) t ik − − t ik − + w − (cid:17) k (1 − α ) (cid:16) O (cid:16) w ( t ik − ) + wt ik − (cid:17)(cid:17) = k (1 − α )( t ik − − k (1 − α ) x X w =1 (cid:16) t ik − + w − (cid:17) k (1 − α )+1 (cid:16) O (cid:16) w ( t ik − ) + wt ik − (cid:17)(cid:17) < (cid:16) O (cid:16) t ik − (cid:17)(cid:17) k (1 − α )( t ik − − k (1 − α ) t ik − + x − X j = t ik − (cid:16) j (cid:17) k (1 − α )+1 = (cid:16) O (cid:16) t ik − (cid:17)(cid:17) k (1 − α )( t ik − − k (1 − α ) × (cid:16) t ik − + x − k (1 − α ) − t ik − ) k (1 − α ) + ( k (1 − α ) + 1) Z t ik − + x − t ik − ⌊ y ⌋ y k (1 − α )+2 dy (cid:17) < (cid:16) O (cid:16) t ik − (cid:17)(cid:17) ( t ik − − k (1 − α ) h t ik − − k (1 − α ) − t ik − + x − k (1 − α ) i = (cid:16) O (cid:16) t ik − (cid:17)(cid:17)h − exp h − k (1 − α ) ln (cid:16) xt ik − − (cid:17)ii . (4.48)Thus, we get (cid:12)(cid:12)(cid:12) P ( W ik ≤ x ) − (cid:2) − exp (cid:0) − k (1 − α ) ln(1 + x/ ( t ik − − (cid:1)(cid:3) (cid:12)(cid:12)(cid:12) < O (cid:16) t ik − (cid:17) . Then, by taking z = ln(1 + x/ ( t ik − − (cid:12)(cid:12)(cid:12) P ( W ik ≤ x ) − P ( Z ik ≤ z ) (cid:12)(cid:12)(cid:12) < O (cid:16) t ik − (cid:17) , where Z ik is a random variable exponentially distributed with parameter (1 − α ) k . Remark 4.7.
Note that the knowledge of { W ik } , k ≥ , is equivalent to the knowledge of ~d ( v i , t ) , as ~d ( v i , t ) := min { k : P kb =1 W ib > ( t − t i ) } . Thus, due to Theorem 4.3, the process { ~d ( v j , t ) } t ≥ t j , ∀ j ≥ min { i : t i ≥ t } , behaves asymptotically as a Yule process with parameter (1 − α ) . Let us now consider the growth of the vertices in Simon model, where at each instant oftime t , a new vertex is created with a fixed probability α . This fact can be re-thought froma different perspective as follows. Remember that in Simon model the number of vertices attime t is a random variable V ( t ), distributed Binomially, Bin( t, α ), and that at each instantof time, one and only one vertex can appear. Think for a moment that we know the numberof vertices at time t , then, conditionally on that, at time t + 1 choose uniformly at randoman existing vertex, i.e., with probability 1 /V ( t ) select one vertex, and with probability α duplicate it. Note that, as time increases, each existing vertex may give birth to a newvertex with probability α/V ( t ). In this way we have that a new vertex appears with constantprobability α ; since there are V ( t ) vertices, then the probability of the birth of a new vertexis V ( t )( α/V ( t )) = α .Now fix a time, take for example t i , the time when the i th vertex appears, so V ( t i ) = i .For each of the existing vertices at time t i , say v j , 1 ≤ j ≤ i , define the birth process { D j ( t ) } t ≥ t i of all its descendants as follows. Start at time t i with one vertex, v j . Since attime t +1 each existing vertex in Simon model may give birth to a new vertex with probability α/V ( t ), then if at time t the number of vertices descendent of v j (i.e., itself + its children + ts grandchildren + etc.) is k , the probability that a new descendent of v j appears at time t + 1 is kα/V ( t ). Formally, let D j ( t ) be the total number of descendent of v j at time t with D j ( t i ) = 1 (itself), then if D j ( t ) = k , k ≥
1, the probability that a new descendent of v j appears at time t + 1 is kα/V ( t ).Observe that since at each time we are selecting one and only one vertex in Simon modelto duplicate, the probability of either no duplications or more than two at each instant of time t is zero. Clearly this is different from the case in which we had taken independent processes { D j ( t ) } t ≥ t i , therefore, they are dependent. However, by definition, these processes, are equalin distribution, i.e., P ( D j ( t ) ≤ d ) is the same for each 1 ≤ j ≤ i .We will see in the following theorem that the processes { D j ( t ) } t ≥ t i , 1 ≤ j ≤ i , convergein distribution to Yule processes with parameter 1, i.e., if at time t , D j ( t ) = k , k ≥ v j , converges in distribution to anexponential random variable with parameter k . Thus, starting with i vertices, we will see thatfrom t i , the process of appearance of new vertices in Simon model approximates i dependentbut identically distributed Yule processes with parameter 1. If the interest is to study theasymptotic characteristics of a uniformly chosen random vertex in Simon model, we could dothat first by choosing uniformly at random a Yule process with parameter 1, and then, bychoosing uniformly at random an individual belonging to it.Formally, let ( G tα ) t ≥ be the random graph process corresponding to Simon model (de-scribed in Subsection 3.1) and, as above, let t i be the time when the i th vertex appears. Then,for each vertex in this process up to time t i , say v j , 1 ≤ j ≤ i , let Y jk be the random variables Y jk := ℓ jk − ℓ jk − , for k = 1 , , . . . , where ℓ j = t i , and ℓ jk is the minimum t when there areexactly k + 1 descendants of v j in { D j ( t ) } t ≥ t i , k ≥
1. Hence Y jk represents the waiting timebetween the appearance of the k th and the ( k + 1)th vertex in { D j ( t ) } t ≥ t i . Theorem 4.4.
Let z = ln(1+ y/ ( ℓ jk − − , k ≥ , y > , and < ε t < such that tε t −→ ∞ as t −→ ∞ . Then, (cid:12)(cid:12)(cid:12) P ( Y jk ≤ y ) − P ( Z jk ≤ z ) (cid:12)(cid:12)(cid:12) < O (cid:16) ℓ jk − ε ℓ jk − (cid:17) , (4.49) where Z jk is an exponentially distributed random variable of parameter k . Remark 4.8.
Since t i = ℓ j and ℓ jk > ℓ j , k ≥ , the previous theorem states that for any t ∗ = t i large enough but fixed, (cid:12)(cid:12)(cid:12) P ( Y jk ≤ y ) − P ( Z jk ≤ z ) (cid:12)(cid:12)(cid:12) < O (cid:16) t ∗ ε t ∗ ) (cid:17) . (4.50) In words it means that from a fixed but large time t ∗ , all the waiting times Y jk are ap-proximately exponential random variables of parameter k , with an error term smaller than O (cid:0) / ( t ∗ ε t ∗ ) ) (cid:1) . Thus, for t ∗ large enough we start to see a process which is very close to aYule process with parameter k .Proof of Theorem 4.4. Let us define the Bernoulli random variables X jk,ℓ , ℓ ≥
1, with P ( X jk,ℓ =1) = kα/V ( ℓ jk − + ℓ ) = 1 − P ( X jk,ℓ = 0), so, {X jk,ℓ = 1 } denotes the event that any of the k descendant of v j in { D j ( t ) } t ≥ t i gives birth to a new one at time ℓ jk − + ℓ . Note that the event {Y jk = y } is equivalent to the event {X jk, = 0 , X jk, = 0 , . . . , X jk,y = 1 } . Now define the events E t := { t ( α − ε t ) ≤ V ( t ) ≤ t ( α + ε t ) } . By Chebyschev’s inequality we have P ( E ct ) ≤ α (1 − α ) /tε t ,so P ( E t ) −→ tε t −→ ∞ as t −→ ∞ . Then observe that P ( X jk,ℓ = x ) ∼ P ( X jk,ℓ = x | E ℓ jk − + ℓ − ) + O (cid:16) ℓ jk − + ℓ − ε ℓ jk − + ℓ − (cid:17) , and P ( Y jk = y ) ∼ h P ( X jk,y = 1 | E ℓ jk − + y − ) y − Y ℓ =1 P ( X jk,ℓ = 0 | E ℓ jk − + ℓ − ) i + O (cid:16) ℓ jk − ) ε ℓ jk − (cid:17) . (4.51) ssuming that ε ℓ jk − + x − > ε ℓ jk − + x − > · · · > ε ℓ jk − , we obtain that the right side of (4.51)is bounded above by αk ( ℓ jk − + y − α − ε ℓ jk − + y − ) − αk ( ℓ jk − + y − α + ε ℓ jk − + y − ) × · · · × − αkℓ jk − ( α + ε ℓ jk − ) + O (cid:16) ℓ jk − ε ℓ jk − (cid:17) = αk ( ℓ jk − + y − α − ε ℓ jk − + y − ) ℓ jk − + y − Y r = ℓ jk − (cid:18) − αkr ( α + ε r ) (cid:19) + O (cid:16) ℓ jk − ε ℓ jk − (cid:17) , (4.52)and bounded below by αk ( ℓ jk − + y − α + ε ℓ jk − + y − ) − αk ( ℓ jk − + y − α − ε ℓ jk − + y − ) × · · · × − αkℓ jk − ( α − ε ℓ jk − ) + O (cid:16) ℓ jk − ε ℓ jk − (cid:17) = αk ( ℓ jk − + y − α + ε ℓ jk − + y − ) ℓ jk − + y − Y r = ℓ jk − (cid:18) − αkr ( α − ε r ) (cid:19) + O (cid:16) ℓ jk − ε ℓ jk − (cid:17) . (4.53)Thus, in a similar manner as we did in the proof of Theorem 4.3, by using Lemma 4.1 andEuler–Maclaurin formula to (4.52) and (4.53), we find that (cid:12)(cid:12)(cid:12) P ( Y jk ≤ y ) − [1 − exp( − k ln(1 + y/ ( ℓ jk − − (cid:12)(cid:12)(cid:12) < O (cid:16) ℓ jk − ε ℓ jk − (cid:17) . Then, taking z = ln(1 + y/ ( ℓ jk − − (cid:12)(cid:12)(cid:12) P ( Y jk ≤ y ) − P ( Z jk ≤ z ) (cid:12)(cid:12)(cid:12) < O (cid:16) ℓ jk − ε ℓ jk − (cid:17) , where Z jk is an exponentially distributed random variable with parameter k , which proves thethesis. To compare the Barab´asi–Albert and Simon models, we considered a third model that wecalled here the II-PA model, first introduced in [16] with a different name. Then we gave acommon description of the three models by introducing three different random graph processesrelated to them. This representation allowed us to clarify in which sense the three modelscan be related. For each fixed time, if m = 1, we proved that the Barab´asi–Albert andthe II-PA models have exactly the same preferential attachment probabilities (Theorem 4.1).Furthermore, since in the first model the preferential attachment is meant with respect to thewhole degree of each vertex while in the second case it is meant with respect only to the in-degree, the conclusion is that, for a uniformly selected random vertex, the degree distributionin the Barab´asi–Albert model equals the in-degree distribution in the Simon model. Notethat m = 1 is the only case in which this is true.Since the direct comparison between Barab´asi–Albert and Simon model is not possiblewe first compared II-PA model with Barab´asi–Albert model (Theorem 4.1), and then II-PA odel with Simon model (Theorem 4.2). We underline that, even if the introduction of II-PAmodel was functional to the study of the connections between the Barab´asi–Albert and Simonmodels, this hybrid model is interesting in itself.Regarding the connections between Simon and II-PA models, Theorem 4.2 shows thatwhen time goes to infinity, the II-PA model has the same limiting in-degree distribution asthat of the Simon model with parameter α = 1 / ( m + 1), for any m ≥
1. The proof uses theAzuma–Hoeffding concentration inequality and the supermartingale’s convergence theorem.Combining Theorem 4.1 and 4.2, we conclude that, in the limit, the Simon model has thesame in-degree distribution as that of the Barab´asi–Albert model, for α = 1 / m = 1.The existing relations between the three models are summarized in Figure 3.On the other hand, Yule model is defined in continuous time. In Section 4.2 we givea mathematical explanation of the reason why, when time goes to infinity the distributionof the size of a genus selected uniformly at random in the Yule model coincide with the in-degree distribution of Simon model. More precisely, we recognize which are the two differentprocesses that describe Simon model and how they are related with a Yule model. Theorem4.3 and Theorem 4.4 show that, as time flows, these two different processes approximatesthe behavior of a continuous time process that in fact corresponds to a Yule model withparameters (1 − α, Simon ( α )in-degree II-PA (m)in-degree B–A (m)degreePrice ( M i )in-degree t → ∞ , α = 1 / ( m + 1) t → ∞ , m = 1, α = 1 / t ≥ m = 1 n → ∞ , E ( M i ) = m n ≥ M i = m = 1, a.s. Figure 3:
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