Degree-distribution stability of scale-free networks

Degree-distribution stabilit y of scale-free net w orks Zhenting Hou 1 ∗ , Xiangxing K ong 1 ⋆ , Dingh ua Shi 1 , 2 † , and Guanro ng Chen 3 ‡ 1 Scho ol of Mathematics, Centr al South University, Changsha 410083, China 2 Dep artment of Mathematics, Shanghai University, Shang hai 200444, China 3 Dep artment of Ele ctr oni c Engine ering, City Univers ity of Hong Kong, Hong Kong, China (Dated: Nov ember 10, 2021) Based on the concept and techniques of first-passage probability in Ma rko v chain theory , this letter pro vid es a rigor ous pro of for the existence of the steady -state degree distribu tion of t he scale- free netw ork generated by the Barab´ asi-Alb ert (BA) mo del, and mathematically re-derives the exact analytic form ulas of the distribution. The approac h developed h ere is quite general, applicable to many other scale-free t yp es of complex net w orks. P A CS num bers: 89.75.Hc, 05.70.Ln, 87.23.Ge, 89.75.Da Intr o duction . The in tensive s tudy of complex netw or ks is per v ading all kinds of sciences to day , ranging from ph ysical to bio logical, ev en to so cia l sciences. Its impact on mo der n engineering and tec hnology is prominent and will b e far -reaching. Typical complex net works include the Internet, the W o rld Wide W eb, wir ed and wirele s s communication net works, p ow er gr ids, biological neur al net works, so cial rela tionship net works, scien tific co o p e r - ation and citation net works, and so on. Research on fundamen tal prop erties and dynamical features of such complex net works has beco me o verwhelming. In the inv estigation of v arious complex netw ork s , the degree distributions are alw ays the main co ncerns bec ause they characterize the f undamental topolog ical prop erties of t he underlying net works. Noticeably , for a ring -shap e regula r graph [1] of what- ever size, where every vertex is connected to its K nearest-neig h b oring vertices, all vertices have the sa me degree K . F or the well-known Erd¨ os-R´ en yi rando m graph mo del [2] with n vertices and m edg es, the degree distribution of v ertices is a pproximately Poisson with mean v alue 2 m/n . F or the small-world net work prop os ed by W atts and Strogatz [1] , the degree distribution of v er - tices also f ollows P oisson distr ibution approximately . A c ommon fea tur e of the ab ov e mo dels is that the de- gree distribution of vertices has a characteristic size h k i . In contrast, Barab´ asi and Alb ert [3] found that for many real-world complex netw o rks, e.g., the WWW, the fra c- tion P ( k ) of vertices with deg ree k is prop or tional over a large r ange to a “scale-fr ee” power-law tail: k − γ , where γ is a co nstant indep endent of the size of the netw o r k. Thu s, the fraction P ( k ) of vertices with degre e k is r e- ferred to as the degree distribution of a scale- free ne t- work. T o ex plain this phenomenon, they pro po sed the following net w ork - generating mec hanism [3] , known as the BA model: “ · · · starting with a small n umber ( m 0 ) of vertices, at every time step we add a new v ertex with m ( ≤ m 0 ) edges th at link the new vertex to m different vertices al- ready present in the system. T o incorp ora te preferential attachmen t, we ass ume that the pr o bability Π that a new vertex will b e connected to a v er tex dep ends o n the co n- nectivity k i of that vertex, s o that Π( k i ) = k i / P j k j . After t steps the mo del leads to a random net work with t + m 0 vertices a nd mt edges.” In [3], computer simulation show ed that for the BA mo del the degre e distr ibution of the netw o rk has a p ow er law fo r m with the exponent γ = 2 . 9 ± 0 . 1. In [4], a heuristic argument based o n the mea n-field theor y led to an analytic s olution P ( k ) ∼ 2 m 2 k − 3 , namely γ = 3. T o derive the following dynamic equation: ∂ k i ∂ t = m Π( k i ) = k i 2 t , k i ( i ) = m , it was ass umed [4] that the probability (can be interpreted as a co nt inuous r ate of c hang e of k i ) for an ex is ting v er- tex with degr ee k i to receive a new connection from the new vertex is exactly equal to m Π( k i ), which is simul- taneously pro po rtional to b oth t he degree k i ( t ) o f the existing vertex i and the n umber m of the new edg es that the new vertex brings in, a t time t . F or nota tional conv enience, this assumption will be simply refer red to as the “ m Π-h y po thesis” in this pape r . In all the consequent w ork s related to the BA mo del, this m Π-hypothesis plays a fundamental r o le. F or exa m- ple, Kr apivsky et al. [5] replaced the degr ee k i ( t ) of vertex i at time t by the total num b er N k ( t ) of deg ree- k vertices ov e r the whole netw ork a t time t , thereby o bta ining its rate equation dN k ( t ) dt = m ( k − 1) N k − 1 ( t ) − kN k ( t ) P k k N k ( t ) + δ km , where δ km accounts for new vertices bringing in new edges. In this study , the m Π-hypothesis w as a dopted in the der iv ations. Assuming tha t the stea dy -state de- gree distribution exists, using the law of large num b ers ( N k ( t ) t → P ( k ) a s t → ∞ ), they show ed that the differ- ence equation o f P ( k ) has an a nalytic solution P ( k ) = 4 k ( k + 1)( k + 2) for the BA mo del with m = 1. They a lso p ointed out that only the linea r pr eferential attac hment sc heme can lead to the scale-free structure but any nonlinear one will not. Dorogovtsev et al. [6] considered k i ( t ) as a random v ari- able and defined P ( k , i, t ) to b e the probability that v er- tex i has exa ctly k edges a t time t , where vertex i is the vertex that w as b eing a dded to the netw ork at time t = i , i = 1 , 2 , · · · . Moreover, they us e d the av erage of all vertex degrees as the netw ork degree: P ( k , t ) , 1 t P t i =1 P ( k, i, t ). They intro duced a mor e general attrac- tion model a nd allow ed m ultiple edges between vertices, 2 where each new vertex ha s an initial attra ction degree A . Sim ultaneously , m new directed edges co ming out from non-sp ecified v ertices are intro duced with the probability Π, ther efore k = A + q with q b eing th e in-degree of v er- tices. Consequently , when every new vertex is the so ur ce of the m new edg e s like in the BA mo del, the a ttrac- tion model makes more sens e than the BA mo del. They first arr ived at the mas ter e q uation of P ( k , i , t ) a nd then by summing all i ’s together they were a ble to der ive the following equa tion: P ( k , t + 1) = k − 1 2 t P ( k − 1 , t ) +  1 − k 2 t  P ( k , t ) + δ mk + O  P ( k , t ) t  . T o that end, by assuming the existence of P ( k ) [note that actually an additio nal assumption of lim t →∞ t [ P ( k , t + 1) − P ( k , t )] = 0 is als o needed], they obtained a difference equation for P ( k ). Fina lly , so lving the equation ga ve an analytic solution P ( k ) = 2 m ( m + 1) k ( k + 1)( k + 2) . Here, it should b e p ointed out th at if m ultiple edges are not allow ed, then t he m Π-hypothesis is still needed. As a s ide note, D oro g ovtsev et al. [7] also co nsidered the effect of accelerating gr owth, which is prop ortio na l to the power o f the time v ariable t at each time s tep. How ever, this destroys the scale-fre e feature and degr e e - distribution stability of the net work. Afterwards, Bollob´ as [8] made a general comment on the BA model: “F ro m a m athematical point of view, howev er, the de- scription ab ove, rep ea ted in many pap ers, do es not make sense. The first problem is getting star ted. The second problem is with the preferential attachmen t rule itself, and arises only f or m ≥ 2 . In order t o prov e r esults ab out the B A mo del, one must first decide on the details of the mo del itself. It tur ns o ut to b e c onv enient to allow m ultiple edges and lo o ps.” Consequently , he and his coauthors recommended a so-called LCD model, as follo ws: “W e start with the case m = 1. Consider a fixed se- quence of vertices v 1 , v 2 , · · · . W e shall inductively define a rando m graph pro cess { G t 1 } t ≥ 0 so that G t 1 is a directed graph on { v i : 1 ≤ i ≤ t } , as follows. Start with G 0 1 the “graph” with no vertices, or with G 1 1 the gr aph with one vertex and one lo o p. Given G t − 1 1 , for m G t 1 by a dding the vertex v t together with a single edge directed from v t to v i , where i is c hosen randomly with Π( i = s ) =    d G t − 1 1 ( v s ) 2 t − 1 , 1 ≤ s ≤ t − 1 1 2 t − 1 , s = t. F or m > 1 w e define the process { G t m } t ≥ 0 by running the pro cess { G t 1 } on a sequenc e v ′ 1 , v ′ 2 , · · · ; the gr aph G t m is for med from G mt 1 by iden tifying the vertices v ′ 1 , v ′ 2 , · · · , v ′ m to form v 1 , identifying v ′ m +1 , v ′ m +2 , · · · , v ′ 2 m to form v 2 , and s o on.” F or graph G n m , let # n m ( d ) b e the num b er of vertices of G n m with in-degree equal to d , i.e., with (total) degree m + d , and set a m,d = 2 m ( m + 1) ( d + m )( d + m + 1)( d + m + 2) . Bollob´ as et al. [9] rigoro usly proved the following res ult: lim n →∞ E [# n m ( d )] /n = a m,d . Then, based on the ma rtingale theory , they prov ed that # n m ( d ) /n con verges to a m,d in probability . It has b een o bserved that mo st real-world a nd simu- lated net works follow certain rules to add or r e mov e their vertices a nd edges, whic h are not entirely ra ndo m. Mor e impo rtantly , at each time step, these rules ar e applied only to the previously formed netw ork, therefor e the pro- cess has prominent Marko vian pro p er ties. Shi et a l. [10] established a close relationship b etw een the BA mo del and Mar kov chains. According to the evolution of the BA mo del, the degree k i ( t ) of vertex i a t time t constitutes a nonhomogeneo us Marko v chain as time evolves. Thu s, all vertices together for m a family of Ma rko v chains. Co n- sequently , based on the Marko v chain theo ry , sta r ting from an initia l distribution and iteratively mu ltiplying the state-transition proba bilit y matrices, the fina l net- work degree distribution can b e ea sily obtained. Lately , Shi et al. [11] developed an evolving netw ork model b y using an an ti-prefere ntial attac hment mechanism, whic h can ge nerate scale- free netw o rks with pow er-law exp o- nent s v a r ying b etw een 1 ∼ 4. There are se veral mo dified and genera lized BA mo de ls in the literature, including such as the lo cal-world BA mo del [12] , which will not be listed and r eviewed here. All in all, the BA model indee d is a breakthr o ugh dis- cov ery with significan t impact on netw ork science to day . Therefore, it is quite impo rtant to support the mo del with a r igorous mathematical founda tion. It is clear from the a bove discussions that tw o key ques- tions need to be ca r efully answered for the BA mo del: 1 ) F or the c ase o f m ≥ 2, ca n one find a scheme of adding new edges fro m t he new v ertex to the exis ting ones that has a proba bilit y pr ecisely e q ual t o m Π? This is the key of the BA mo deling. 2 ) Do es the s teady-state de- gree distribution of the netw ork exist a nd, if so, what is it? This is the key to the v alidity o f the mean-field, rate-equa tio n, master -equation, and Marko v- chain ap- proaches. The present paper will give complete a nswers to these t wo q uestions. De gr e e-distribution stability . T o start, conside r the first question. Recall that Holme and Kim [13] prop osed a scheme for new edge connection: When a new vertex comes into the net work, the first edg e co nnec ts to an ex- isting vertex with the preferential attachmen t probabil- it y Π. After that, t he res t m − 1 edges r andomly co nnec t with proba bilit y p to the v ertices in the neig h b orho o d o f the vertex that the fir st edge was co nnected to, or con- nect with probability 1 − p to those vertices that the fir st edge did not connect to. Here, co nsider this appr oach with p = 1 in the fo llowing scena rio: When a new v ertex 3 comes into the net work, the first edge co nnects to an ex- isting vertex with the sp ecified prefer ent ial attachmen t probability Π, sa me as above. Y et, the rest m − 1 edges simult aneous ly co nnec t to m − 1 vertices randomly c ho- sen from inside the neig h b orho o d o f the vertex that the first e dge was connected to. By random s a mpling the- ory this is equiv alent to the a b ove Holme-Kim scheme which c ontin ually co nnec ts the edg es to m − 1 vertices randomly chosen from ins ide the neighborho o d without allowing multiple edges. F or t his specia l scheme, the f ol- lowing res ult c a n be rigoro usly prov ed. Prop ositio n. F or the BA mo del with the ab ov e sp ecia l attachmen t scheme, if vertex i ha s degr ee k i ( t ) at time t , then the probability that v ertex i receives a new edge from the new vertex a t time t + 1 is exactly eq ua l to m Π( k i ). Pr o of. Let P i ( t +1) b e the pr obability of vertex i receiv ing a new edge from vertex t + 1 at time t + 1. Then, P i ( t + 1) = k i ( t ) P j k j ( t ) + X l ∈ O i ( t ) k l ( t ) P j k j ( t ) C m − 2 k l ( t ) − 1 C m − 1 k l ( t ) = k i ( t ) P j k j ( t ) + X l ∈ O i ( t ) m − 1 P j k j ( t ) = m k i ( t ) P j k j ( t ) , where C m − 2 k l ( t ) − 1 C m − 1 k l ( t ) = ( k l ( t ) − 1)! / [( m − 2)!( k l ( t ) − m + 1)!] k l ( t )! / [( m − 1)!( k l ( t ) − m + 1)!] = m − 1 k l ( t ) , which is the pr obability of cho osing v ertex i , among the m − 1 vertices that were r andomly chosen from inside the neighborho o d O l ( t ) of v er tex l , to perform simult aneous connections. The Prop ositio n answers the first question pos ted ab ov e a nd shows that the s pec ial Holme-Kim preferen- tial attachmen t scheme is o ne w ay to implement the m Π- hypothesis. In order to prove the degree-distr ibution stability of the general BA netw ork, the BA m o del is spe c ified first. Start with a co mplete graph with m 0 vertices, which has a total degr e e N 0 = m 0 ( m 0 − 1), and denote these ver- tices by − m 0 , · · · , − 1, re s pec tiv ely . In all the following deriv ations , the m Π-h yp othesis will be a ssumed. The general BA netw orks will b e further discus s ed in the la st section below. F ollowing Doro govtsev et al. [6] , consider the degree k i ( t ) as a r andom v aria ble, and let P ( k , i , t ) = P { k i ( t ) = k } b e the probability of vertex i having degr ee k at tim e t , and mo reov er let the netw ork degree distribution b e the a verage ov er all its vertices at time t , namely , P ( k , t ) , 1 t + m 0 t X i = − m 0 ,i 6 =0 P ( k , i, t ) . Recall that k i ( t ) is a ra ndom v a r iable for any fixed t a nd it is a nonhomo geneous Mar ko v chain for v ar i- able t [10] . Under t he m Π-hypothesis, the state-tr a nsition probability o f this Marko v chain is given by P { k i ( t + 1) = l | k i ( t ) = k } =      1 − k 2 t + N 0 m , l = k k 2 t + N 0 m , l = k + 1 0 , otherwise , (1) where k = 1 , 2 , · · · , m + t − i , and i = 1 , 2 , · · · . The existence of the steady-state degree distribution for this specified BA netw ork ca n b e pro ved in three steps as follows. Detailed deriv ations a re s upplied in the Ap- pendix of the paper. 1. Co nsider the first-pa ssage proba bilit y of the Markov chain: f ( k , i, t ) = P { k i ( t ) = k , k i ( l ) 6 = k , l = 1 , 2 , · · · , t − 1 } . Then, the rela tio nship b etw een the first-pa s sage pro ba- bilit y and the vertex degrees is established. Lemma 1. Under the m Π- h yp othesis, for the BA mo del with k > m , f ( k , i, s ) = P ( k − 1 , i, s − 1 ) k − 1 2( s − 1 ) + N 0 m , (2) P ( k , i, t ) = t X s = i + k − m f ( k , i, s ) t − 1 Y j = s 1 − k 2 j + N 0 m ! . (3) 2. Under the m Π-hypothesis , using the sta te-transition probability of the Mar ko v chain, o ne fir st finds the ex- pression of P ( m, t ), as follows: P ( m, t ) = t − 1 Y i =1 1 − m 2 i + N 0 m ! i + m 0 i + 1 + m 0 × 2 6 6 6 4 P ( m, 1) + t − 1 X l =1 1 l +1+ m 0 l Q j =1 „ 1 − m 2 j + N 0 m « j + m 0 j +1+ m 0 3 7 7 7 5 = 1 t + m 0 t − 1 Y i =1 1 − m 2 i + N 0 m ! × " (1 + m 0 ) P ( m, 1) + t − 1 X l =1 l Y j =1 1 − m 2 j + N 0 m ! − 1 # . Then, one can sho w the existence of the limit lim t →∞ P ( m, t ) b y using the following classic al Sto lz- Ces´ aro Theorem in Calculus. Stolz-Ces´ aro Theorem [14] . In sequence { x n y n } , assume that { y n } is a mo notone increa s ing sequence with y n → ∞ . If the limit lim n →∞ x n +1 − x n y n +1 − y n = l exis ts, where − ∞ ≤ l ≤ + ∞ , then lim n →∞ x n y n = l . Lemma 2 . Under the m Π-h yp othesis, for the BA mo del, the limit lim t →∞ P ( m, t ) exists and is indep endent of the initial net work: P ( m ) , lim t →∞ P ( m, t ) = 2 m + 2 > 0 . (4) 3. Under the m Π-hypo thesis, similar ly , one finds the expression of P ( k , t ) using the first-pas s age probability of the Markov chain, and then shows the existence of the limit lim t →∞ P ( k , t ) by using the Stolz- Ces´ aro Theor em, if the lim it lim t →∞ P ( k − 1 , t ) exists. Lemma 3. Under the m Π- h yp othesis, for the BA mo del with k > m , if the limit lim t →∞ P ( k − 1 , t ) exists then the limit lim t →∞ P ( k , t ) also exists: P ( k ) , lim t →∞ P ( k , t ) = k − 1 k + 2 P ( k − 1) > 0 . (5) 4 Finally , by mathematica l induction, it follows from Lemmas 2 and 3 that the steady-sta te degree distribu- tion of the s pecifie d BA netw ork exists. T o this end, by solving the difference equation (5) iter atively , o ne arrives at the following conclusion. Theorem 1. Under the m Π-hypo thesis, for the BA mo del with k ≥ m , the steady-state degree distribution exists, indep endent of the initial net work, and is given by P ( k ) = 2 m ( m + 1) k ( k + 1)( k + 2) ∼ 2 m 2 k − 3 > 0 . (6) Clearly , this degree distribution formula is consistent with the for mu la obtained b y Dorog ovtsev et al. [6] and Bollob´ as et al. [9] , which allow mu ltiple edges and loops. Discussion . Bollob´ a s [8] once discusse d the BA de s crip- tion (the m Π-hypo thesis) of prefer e ntial a ttachmen t in detail. His re s ult gives a range of models fitting the BA description with v ery different pro p er ties. When m ≥ 2 , as a new vertex comes in, it is no pro blem for its first edge to pre fer entially connect to an existing v er tex. But what ab out the other m − 1 new edges? This q uestion was not car e fully a ddressed b efore. Clearly , after the first edge ha s been connected from the new vertex to an existing vertex, the pr eferential attachmen t pro bability Π is no long er the same if later o pe rations do no t al- low m ultiple edg es and lo ops. It is a ls o clear that when m ≥ 2, the proba bilit y of vertex i receiv ing a new edge is always greater than Π. But what is it? On the other hand, it is a lso p ossible that the pr obability of vertex i receiving a new e dg e dep ends o n other vertex deg rees. Barab´ a s i alwa ys emphasizes the m Π-hypothesis but did not discuss this “how” question either. Th us, tw o ques- tions ar ise: 1) F or the BA mo del, or for any other BA-lik e mo del, how to prov e the degr ee-distribution stability if the m Π-hypo thesis holds only a pproximately? 2) Is there a preferent ial a ttachmen t scheme for m ≥ 2 suc h tha t the probability of vertex i r eceiving a new edge is indep en- dent of other v ertex degrees? T o answer these tw o questions, a new preferential at- tachmen t sc heme is prop osed and discussed in [15], where a new v ertex will be simultaneously connected to m dif - ferent vertices and it is a ssumed that the preferent ial a t- tachmen t probability Π is pro po rtional to the sum of the degrees k i 1 , · · · , k i m of those vertices. They show ed that the pr o bability that the existing v ertex i received an edge from the new vertex is indep endent of other vertex de- grees, namely , Π t +1 m ( k i ( t )) = m 0 + t − m m 0 + t − 1 k i ( t ) 2 mt + N 0 + m − 1 m 0 + t − 1 , where m 0 is the n umber o f vertices and N 0 is the total degree in the initial netw o rk. Consequently , under the ( a t k i ( t ) + b t )(1 + o (1) k i ( t ) ,t )-hypothesis and some mild conditions, they prov ed the degree- dis tribution stability of Bara b´ asi-Alb ert t yp e netw o rks. Esp ecially , the p ower- law ex p o ne nt of the netw ork deg r ee distributio n in this new preferen tial attac hment scheme is γ = 2 m + 1. Finally , it sho uld b e emphasized that the theory and scheme develop ed in this pa per has great generality [16] , in the sense that it ca n b e applied to ma ny B A-like mod- ified and genera lized models, such as the LCD mo del of Bollob´ as et al. [9] , the attraction mo del o f Do rogovtsev et al. [6] , the lo ca l-world BA-like mo del of Li and Chen [12] , and the evolving net work model of Shi et a l. [11] , etc. W e summarize the res ults and findings in this pa per as follows: (1) Our proving method differs from the one based on martingale theory , and can be applied to man y other scale- free types of complex netw orks; (2) W e do not need to change the BA mo del, e.g ., to allow mul- tiple edges and lo o ps; (3) W e provide a sp ecial Ho lme- Kim preferential attachment scheme such that the “ m Π- hypothesis” holds. This re search w as supp orted by the National Natural Science F oundation under Gr ant No. 1 06712 12, and by the NSF C-HKRGC Joint Research Pr o jects under Grant N-CityU 107 /07. ∗ Email a ddress: zthou@ csu.edu.cn ⋆ Email a ddress: kongxiangx ing 2008@ 163.com † Email addr ess: shidh20 01@26 3.net ‡ Email addr ess: gchen@ee.cityu.edu.hk [1] W atts D. J . and Strogatz S. H., Natur e 393 , 1998, 440- 442 [2] Erd¨ os P . and R´ enyi A., Public ations Mathematic ae 6 , 1959, 290-297 [3] Barab´ asi A.-L. and Alb ert R., Scienc e 286 , 1999, 509 -512 [4] Barab´ a si A.-L., Alb ert R. and Jeong H., Physic a A 272 , 173 (1999). [5] Krapivsky P . L., Redner S. and Leyvraz F., Phys. R ev. L ett. 85 , 2000, 4629-4632 [6] Dorogo vtsev S. N., Mendes J. F. F. and S am ukhin A. N., Phys. R ev. L ett. 85 , 20 00, 4633-4636 [7] Dorogo vtsev S. N . and Mendes J. F. F., Phys. R ev. 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App endix to “Degree-d istribut ion s tabilit y of scale-free ne t works” Zhen ting Hou 1 Xiangxing Kong 1 Dingh ua Shi 1 , 2 Guanrong Chen 3 1 School of Mathematics, Cen tral South Univ ersity , Changsha 410083 , China 2 Department of Mathematics , Shanghai Universit y , Shangh ai 200 444, China 3 Department of Electronic Engineering, Cit y Universi ty of Hong K ong, Hong Kong, China No v em b er 10, 2021 T o provide a rigorous proof of the degree-distr ibutio n stability o f the scale- free netw o r k genera ted by the BA mo del, some parameter s ar e sp ecified as follows: (i) start with a complete g r aph with m 0 vertices, which ha s the total degree N 0 = m 0 ( m 0 − 1), and denote these vertexes by − m 0 , · · · , − 1, resp ec- tively; (ii) as sume that at e a ch time step t , the probability of the new vertex connecting to a n existing v er tex i is exactly equal to m Π( k i ( t )). Here, in (ii), the prefer e n tial attachmen t proba bilit y is simultaneously pro- po rtional to bo th th e degree k i ( t ) of the existing vertex i and the num b er m of new edge s that the new vertex brings in, at tim e t . F or notatio nal conv enience, this assumption will b e referr ed to as the “ m Π-h y po thesis” below. Observe that the degree k i ( t ) of vertex i at time t is a random v ar iable [6] . Let P ( k , i , t ) = P { k i ( t ) = k } denote the probability of v er tex i ha ving degree k at time t , and define the degree distr ibutio n of the whole netw o rk by the average v alue of probabilities of vertex degrees P ( k , t ) , 1 t + m 0 t X i = − m 0 ,i 6 =0 P ( k , i, t ) . (1) Observe also that the degr ee k i ( t ) a s a pro cess in time t is an nonhomoge- neous Markov c hain [10] . Thus, for k = 1 , 2 , · · · , t + i − m , the sta te transitio n probabilities of t he Markov chain, under the m Π-hypothesis, are giv en b y P { k i ( t + 1) = l | k i ( t ) = k } =        1 − k 2 t + N 0 m , l = k k 2 t + N 0 m , l = k + 1 0 , otherwise . (2) 1 1 The BA mo del with m = 1 Denote the fir st-passag e pro bability of the Markov chain by f ( k , i, t ) = P { k i ( t ) = k , k i ( l ) 6 = k , l = 1 , 2 , · · · , t − 1 } . First, the relationship betw een the first-passa ge proba bilit y and the vertex degr ees is esta blished. Lemma 1 F or k > 1, f ( k , i, s ) = P ( k − 1 , i, s − 1) k − 1 2( s − 1) + N 0 , (3) P ( k , i, t ) = t X s = i + k − 1 f ( k , i, s ) t − 1 Y j = s  1 − k 2 j + N 0  . (4) Pro of Fir st, consider Eq. (3). The degree of a vertex is alwa ys nondecrea s- ing, a nd increa s ing at most by 1 each time, accor ding to the c o nstruction rule of the BA mo del. Th us, it follo ws from the Markovian pro per ties that f ( k , i, s ) = P { k i ( s ) = k , k i ( l ) 6 = k , l = 1 , 2 , · · · , s − 1 } = P { k i ( s ) = k , k i ( s − 1) = k − 1 , k i ( l ) 6 = k , l = 1 , 2 , · · · , s − 2 } = P { k i ( s ) = k , k i ( s − 1) = k − 1 } = P { k i ( s − 1) = k − 1 } P { k i ( s ) = k | k i ( s − 1) = k − 1 } = P ( k − 1 , i, s − 1) k − 1 2( s − 1 ) + N 0 . Second, observe that the earliest time for the degree of vertex i to r each k is at s tep k + i − 1, and the latest time to do so is at step t . After this vertex degree becomes k , it will no t increase a ny more. T hus, Eq. (4) is prov ed. Lemma 2 (Stolz-Ces´ aro Theor em) In sequence { x n y n } , assume that { y n } is a mono tone increasing sequence with y n → ∞ . If the limit lim n →∞ x n +1 − x n y n +1 − y n = l exists, where −∞ ≤ l ≤ + ∞ , then lim n →∞ x n y n = l . Pro of This is a classical r esult, see [14]. Lemma 3 F or the proba bilit y P ( k , t ) defined in Eq. (1), the limit lim t →∞ P (1 , t ) exists and is indep endent o f the init ial netw o rk; mo reov er, P (1) , lim t →∞ P (1 , t ) = 2 3 > 0 . (5) Pro of F rom the construction o f t he BA model or Eq. (2), it follows that P (1 , i, t + 1 ) =  1 − 1 2 t + N 0  P (1 , i, t ) . Since P (1 , t + 1 , t + 1) = 1, one has P (1 , t + 1 ) = 1 t + 1 + m 0 t +1 X i = − m 0 ,i 6 =0 P (1 , i, t + 1 ) = t + m 0 t + 1 + m 0  1 − 1 2 t + N 0  P (1 , t ) + 1 t + 1 + m 0 . 2 Then, b y iteration, P (1 , t ) = t − 1 Y i =1  1 − 1 2 i + N 0  i + m 0 i + 1 + m 0      P (1 , 1) + t − 1 X l =1 1 l +1+ m 0 l Q j =1  1 − 1 2 j + N 0  j + m 0 j +1+ m 0      = 1 t + m 0 t − 1 Y i =1  1 − 1 2 i + N 0    (1 + m 0 ) P (1 , 1) + t − 1 X l =1 l Y j =1  1 − 1 2 j + N 0  − 1   . Next, let x n , (1 + m 0 ) P (1 , 1) + n − 1 P l =1 l Q j =1  1 − 1 2 j + N 0  − 1 and y n , ( n + m 0 ) n − 1 Y i =1  1 − 1 2 i + N 0  − 1 > 0 . Thu s, it follo ws that x n +1 − x n = n Y j =1  1 − 1 2 j + N 0  − 1 and y n +1 − y n = 3 n + N 0 + m 0 2 n + N 0 n Y i =1  1 − 1 2 i + N 0  − 1 > 0 . Since y n > 0 a nd y n +1 − y n > 0, { y n } is a strictly monotone increas ing no nneg- ative sequence, hence y n → ∞ . Mo reov er, x n +1 − x n y n +1 − y n = 2 n + N 0 3 n + N 0 + m 0 → 2 3 ( n → ∞ ) . F rom Lemma 2, one has P (1) , lim t →∞ P (1 , t ) = lim n →∞ x n y n = lim n →∞ x n +1 − x n y n +1 − y n = 2 3 > 0 . This completes the pr o of. Lemma 4 F or k > 1, if the limit lim t →∞ P ( k − 1 , t ) exists, then the limit lim t →∞ P ( k , t ) also exists and, mor eov er, P ( k ) , lim t →∞ P ( k , t ) = k − 1 k + 2 P ( k − 1) > 0 . (6) 3 Pro of Fir st, observ e that P ( k , t ) = 1 t + m 0 t X i = − m 0 i 6 =0 P ( k , i, t ) = 1 t + m 0 − 1 X i = − m 0 P ( k , i, t ) + t t + m 0 1 t t X i =1 P ( k , i, t ) . Next, denote P ( k , t ) , 1 t t P i =1 P ( k , i, t ). O ne only needs to prov e that the limit lim t →∞ P ( k , t ) exists, which will imply that the limit lim t →∞ P ( k , t ) = lim t →∞ P ( k , t ) exists. T o s how that the limit of P ( k , t ) exis ts as t → ∞ , o bserve that P ( k , i, t ) = 0 when i > t + 1 − k , since in this case even if this vertex i increases its degree by 1 eac h time, it cannot reach degree k . Then, it follows from Lemma 1 that P ( k , t ) = 1 t t +1 − k X i =1 P ( k , i, t ) = 1 t t +1 − k X i =1 t X s = i + k − 1 f ( k , i, s ) t − 1 Y j = s  1 − k 2 j + N 0  = 1 t t +1 − k X i =1 t X s = i + k − 1 P ( k − 1 , i, s − 1) k − 1 2( s − 1) + N 0 t − 1 Y j = s  1 − k 2 j + N 0  = 1 t t X s = k s +1 − k X i =1 P ( k − 1 , i, s − 1) k − 1 2( s − 1) + N 0 t − 1 Y j = s  1 − k 2 j + N 0  = 1 t t X s = k s − 1 X i =1 P ( k − 1 , i, s − 1) k − 1 2( s − 1) + N 0 t − 1 Y j = s  1 − k 2 j + N 0  = 1 t t X s = k P ( k − 1 , s − 1) ( s − 1)( k − 1 ) 2( s − 1 ) + N 0 t − 1 Y j = s  1 − k 2 j + N 0  = 1 t t − 1 Y i = k  1 − k 2 i + N 0  " P ( k − 1 , k − 1) ( k − 1) 2 2( k − 1) + N 0 + t − 1 X l = k P ( k − 1 , l ) l ( k − 1) 2 l + N 0 l Y j = k  1 − k 2 j + N 0  − 1 # . Next, let x n , P ( k − 1 , k − 1 ) ( k − 1) 2 2( k − 1) + N 0 + n − 1 X l = k P ( k − 1 , l ) l ( k − 1) 2 l + N 0 l Y j = k  1 − k 2 j + N 0  − 1 4 and y n , n n − 1 Y i = k  1 − k 2 i + N 0  − 1 > 0 → ∞ . Obviously , x n +1 − x n = P ( k − 1 , n ) n ( k − 1) 2 n + N 0 n Y j = k  1 − k 2 j + N 0  − 1 , and since y n +1 − y n =  ( n + 1) − n  1 − k 2 n + N 0  n Y i = k  1 − k N 0 + 2 i  − 1 = ( k + 2) n + N 0 2 n + N 0 n Y i = k  1 − k 2 i + N 0  − 1 > 0 , one has that { y n } is a strictly mo notone increasing nonnegative sequence, hence y n → ∞ . Also , b y assumption, x n +1 − x n y n +1 − y n = ( k − 1) n ( k + 2) n + N 0 P ( k − 1 , n ) → k − 1 k + 2 P ( k − 1) ( n → ∞ ) . Thu s, it follows fr o m Lemma 2 that lim t →∞ P ( k , t ) = lim n →∞ x n y n = lim n →∞ x n +1 − x n y n +1 − y n = k − 1 k + 2 P ( k − 1) > 0 , therefore, lim t →∞ P ( k , t ) exists, and Eq. (6) is t hus proved. Theorem 1 The steady-state degree dis tribution of the BA mo del with m = 1 exists, and is given b y P ( k ) = 4 k ( k + 1)( k + 2) ∼ 4 k − 3 > 0 . (7) Pro of By mathematical induction, it follows from Lemmas 3 a nd 4 that the steady-state degree distribution of the BA mo del with m = 1 exists. Then, solving Eq. (6) iteratively , o ne obtains P ( k ) = k − 1 k + 2 P ( k − 1) = k − 1 k + 2 k − 2 k + 1 k − 3 k P ( k − 3) . By con tinu ing the proces s till k = 3 + 1 , one finally obtains P ( k ) = 4 k ( k + 1)( k + 2) ∼ 4 k − 3 > 0 . F rom Theorem 1, o ne can see that the degre e distribution formula o f K rapivsky et al. [5] is exact, although the mathematical pro of there w as not as rigorous as that given ab ove. 5 2 The BA mo del with m ≥ 2 Lemma 5 Under the m Π-h y po thesis, the BA mo del when k > m satisfies f ( k , i, s ) = P ( k − 1 , i, s − 1) k − 1 2( s − 1) + N 0 m , (8) P ( k , i, t ) = t X s = i + k − m f ( k , i, s ) t − 1 Y j = s 1 − k 2 j + N 0 m ! . (9) Pro of First, consider Eq. (8). The de g ree of vertex is nondecr e a sing, and increasing at most by 1 e a ch time, accor ding to the construction of the BA mo del. Thus, it follows f rom the Marko vian prop erties that f ( k , i, s ) = P { k i ( s ) = k , k i ( l ) 6 = k , l = 1 , 2 , · · · , s − 1 } = P { k i ( s ) = k , k i ( s − 1) = k − 1 , k i ( l ) 6 = k , l = 1 , 2 , · · · , s − 2 } = P { k i ( s ) = k , k i ( s − 1) = k − 1 } = P { k i ( s − 1) = k − 1 } P { k i ( s ) = k | k i ( s − 1) = k − 1 } = P ( k − 1 , i , s − 1) k − 1 2( s − 1) + N 0 m . Second, observe that the earliest time for the degree of vertex i to r each k is at step k + i − m , a nd the latest time to do so is a t step t . After this vertex degree becomes k , it will no t increase a ny more. T hus, Eq. (9) is prov ed. Lemma 6 Under the m Π-hypo thes is, in the BA model the limit lim t →∞ P ( m, t ) exists and is indep endent o f the init ial netw o rk; mo reov er, P ( m ) , lim t →∞ P ( m, t ) = 2 m + 2 > 0 . (10) Pro of F rom the construction o f t he BA model or (2), it follows that P ( m, i, t + 1 ) = 1 − m 2 t + N 0 m ! P ( m, i, t ) . Since P ( m, t + 1 , t + 1 ) = 1, one has P ( m, t + 1) = 1 t + 1 + m 0 t +1 X i = − m 0 ,i 6 =0 P ( m, i, t + 1 ) = t + m 0 t + 1 + m 0 1 − m 2 t + N 0 m ! P ( m, t ) + 1 t + 1 + m 0 . Then, iterative ca lculation yields P ( m, t ) = t − 1 Y i =1 1 − m 2 i + N 0 m ! i + m 0 i + 1 + m 0      P ( m, 1) + t − 1 X l =1 1 l +1+ m 0 l Q j =1  1 − m 2 j + N 0 m  j + m 0 j +1+ m 0      6 = 1 t + m 0 t − 1 Y i =1 1 − m 2 i + N 0 m !   (1 + m 0 ) P ( m, 1) + t − 1 X l =1 l Y j =1 1 − m 2 j + N 0 m ! − 1   . Next, let x n , (1 + m 0 ) P ( m, 1) + n − 1 X l =1 l Y j =1 1 − m 2 j + N 0 m ! − 1 and y n , ( n + m 0 ) n − 1 Y i =1 1 − m 2 i + N 0 m ! − 1 > 0 . It follo ws that x n +1 − x n = n Y j =1 1 − m 2 j + N 0 m ! − 1 and y n +1 − y n = ( m + 2) n + N 0 m + mm 0 2 n + N 0 m n Y i =1 1 − m 2 i + N 0 m ! − 1 > 0 . Consequently , x n +1 − x n y n +1 − y n = 2 n + N 0 m ( m + 2) n + N 0 m + mm 0 → 2 m + 2 ( n → ∞ ) . It follo ws from Lemma 2 that P ( m ) = lim t →∞ P ( m, t ) = lim n →∞ x n y n = lim n →∞ x n +1 − x n y n +1 − y n = 2 m + 2 > 0 . This completes the pr o of. Lemma 7 Under the m Π-hypothesis, in the BA mo del with k > m , if lim t →∞ P ( k − 1 , t ) exists, then lim t →∞ P ( k , t ) exists and, moreov er, P ( k ) , lim t →∞ P ( k , t ) = k − 1 k + 2 P ( k − 1) > 0 . (11) Pro of Fir st, observ e that P ( k , t ) = 1 t + m 0 t X i = − m 0 ,i 6 =0 P ( k , i, t ) = 1 t + m 0 − 1 X i = − m 0 P ( k , i, t )+ t t + m 0 1 t t X i =1 P ( k , i, t ) . Denote P ( k , t ) , 1 t t P i =1 P ( k , i, t ). O ne only needs to prove that the limit lim t →∞ P ( k , t ) exists, whic h will imply that the limit lim t →∞ P ( k , t ) = lim t →∞ P ( k , t ) exists. 7 T o s how that the limit of P ( k , t ) exists as t → ∞ , observe that P ( k , i, t ) = 0 when i > t + m − k . Therefor e, it follows fro m Lemma 5 that P ( k , t ) = 1 t t + m − k X i =1 P ( k , i, t ) = 1 t t + m − k X i =1 t X s = i + k − m f ( k , i, s ) t − 1 Y j = s 1 − k 2 j + N 0 m ! = 1 t t + m − k X i =1 t X s = i + k − m P ( k − 1 , i, s − 1) k − 1 2( s − 1) + N 0 m t − 1 Y j = s 1 − k 2 j + N 0 m ! = 1 t t X s = k − m +1 s + m − k X i =1 P ( k − 1 , i, s − 1) k − 1 2( s − 1 ) + N 0 m t − 1 Y j = s 1 − k 2 j + N 0 m ! = 1 t t X s = k − m +1 s − 1 X i =1 P ( k − 1 , i, s − 1) k − 1 2( s − 1) + N 0 m t − 1 Y j = s 1 − k 2 j + N 0 m ! = 1 t t X s = k − m +1 P ( k − 1 , s − 1) ( s − 1)( k − 1 ) 2( s − 1) + N 0 m t − 1 Y j = s 1 − k 2 j + N 0 m ! = 1 t t − 1 Y i = k − m +1 1 − k 2 i + N 0 m ! " P ( k − 1 , k − m ) ( k − 1)( k − m ) 2( k − m ) + N 0 m + t − 1 X l = k − m +1 P ( k − 1 , l ) l ( k − 1) 2 l + N 0 m l Y j = k − m +1 1 − k 2 j + N 0 m ! − 1 # Next, let x n , P ( k − 1 , k − m ) ( k − 1)( k − m ) 2( k − m ) + N 0 m + n − 1 X l = k − m +1 P ( k − 1 , l ) l ( k − 1) 2 l + N 0 m l Y j = k − m +1 1 − k 2 j + N 0 m ! − 1 and y n , n n − 1 Y i = k − m +1 1 − k 2 i + N 0 m ! − 1 > 0 . It follo ws that x n +1 − x n = P ( k − 1 , n ) n ( k − 1) 2 n + N 0 m n Y j = k − m +1 1 − k 2 j + N 0 m ! − 1 and y n +1 − y n = " ( n + 1) − n 1 − k 2 n + N 0 m !# n Y i = k − m +1 1 − k 2 i + N 0 m ! − 1 8 = ( k + 2) n + N 0 m 2 n + N 0 m n Y i = k − m +1 1 − k 2 i + N 0 m ! − 1 > 0 . By assumption, x n +1 − x n y n +1 − y n = ( k − 1) n ( k + 2) n + N 0 m P ( k − 1 , n ) → k − 1 k + 2 P ( k − 1) ( n → ∞ ) . It then f ollows fro m Lemma 2 that lim t →∞ P ( k , t ) = lim n →∞ x n y n = lim n →∞ x n +1 − x n y n +1 − y n = k − 1 k + 2 P ( k − 1) > 0 . Thu s, lim t →∞ P ( k , t ) exists and Eq. (11) is pro ved. Theorem 2 Under the m Π-hypothes is , the steady-sta te degr ee dis tr ibution of the BA mo del with m ≥ 2 exists, and is given by P ( k ) = 2 m ( m + 1) k ( k + 1)( k + 2) ∼ 2 m 2 k − 3 > 0 . (12) Pro of By induction, it follows from Lemmas 6 and 7 that the steady-state degree distr ibutio n of the BA mo del with m ≥ 2 exists. Equation (11) follows from iteration P ( k ) = k − 1 k + 2 P ( k − 1) = k − 1 k + 2 k − 2 k + 1 k − 3 k P ( k − 3) , till k = 3 + m . Thus, o ne obtains P ( k ) = 2 m ( m + 1) k ( k + 1)( k + 2) ∼ 2 m 2 k − 3 > 0 . One can s ee that this degree distr ibution formula is consistent with the formula obta ined by Dorog ovtsev et al. [6] and Bo llob´ as et al. [9] , which allow m ultiple edges and lo o ps. Finally , the authors thank Pro fes s or Yirong Liu for many helpful discussions. Z. Hou 1 : zthou@csu.edu.cn X. Kong 1 : k ongxia ng xing200 8@163.com D. Shi 1 , 2 : shidh2001@2 63.net G. Chen 3 : gc hen@ee.cityu.edu.hk 9