Distribution of Edge Load in Scale-free Trees

Distribution of Edge Load in Scal e-free T rees A ttila F ek ete ∗ and G´ ab or V atta y † Dep art ment of Physics of Complex Systems, E¨ otv¨ os University, P´ azm ´ any P. s ´ et´ any 1/A., H-1117 Budap est, Hungary Ljup co Ko carev ‡ Institute for Nonline ar Scien c es, University of California, 9500 Gilman Drive, L a Jo l la, CA 92093, San Die go, USA Abstract No de b et weenness has b een studied recently by a num b er of authors, b ut until no w less atten tion has b een paid to edge b et we enness. In this pap er, we present an exact analytic stud y of edge b et weenness in ev olving scale-free and non-scale-free trees. W e aim at the prob ab ility distribution of edge b et w eenn ess under th e condition that a lo cal p rop ert y , the in-degree of the “younger” no de of a r andomly selecte d edge, is kno wn . En route to the cond itional distribution of edge b et weenness the exa ct join t distribution of cluster size and in-degree, and its one dimensional marginal distributions h av e b een p resen ted in the pap er as w ell. F r om the derive d p robabilit y distributions t he expectation v alues of different quan tities ha v e b een calculated. Our results p ro vide an exact solution n ot only for infinite, but for fi nite net w ork s as wel l. P ACS n um ber s: 89.75.Da, 02.50 .Cw, 05.5 0.+q ∗ Electronic address: fekete@complex.elte.h u † Electronic a ddress: v attay@complex.elte.h u ‡ Electronic a ddress: lkocarev @ucsd.edu 1 I. INTR ODUCTION In recen t ye ars, the statistical prop erties o f complex netw orks ha ve b een extens iv ely in ve stigated by the phy sics comm unit y [1, 2 , 3, 4]. With the increasing computing p ow er of mo dern computers, analysis of lar ge-scale net w o rks and databases has b ecome p ossible. It has been sho wn that the degree statistics of m an y natural and artificial net w orks follow p o we r law . Examples for suc h net works v ary from so cial in terconnections and scien tific collab orations [5] to the w orld-wide w eb [6] and the In ternet [7, 8]. These ne t works are usually referred to as sc ale-fr e e net works , since the p o w er law distribution indicates that there is no characteristic scale in these systems. In the early 1960 s Erd˝ os a nd R´ en yi (ER) in tro duced ra ndom graphs that serv ed as the first mathematical mo del of complex netw orks [9]. In their mo del, the num b er of no des is fixed and connections are established ra ndomly , with p robabilit y p E R . Although the ER mo del leads to ric h theory , it fails to predict the p o wer law distributions observ ed in scale- free net works . Barab´ asi and Alb ert (BA) pro p osed a more suitable evolvin g mo del of these net works [10, 1 1]. The BA mo del is also based on the random graph theory , but inv olv es t w o k ey principles in addition: (a) gr owth , that is, t he size of the netw ork is increasing during dev elopmen t, and (b) pr efer ential attachment , that is, new netw ork elemen ts are connected to higher degree no des with higher probability . In the BA mo del eve ry new no de connects to the core netw ork with a fixed n umber of links m . The study of complex netw orks usually deals with the structural prop erties of net works , lik e degree distribution [12], shortest path distribution [13], degree–degree cor r elat io ns, clus- tering [14 ], etc. F or complex netw orks whic h inv olv e a tr a nsport mec hanism b etwe enn ess is the matter of imp orta nce. Roughly sp eaking, b et w eenness is the num b er of shortest paths passing through a certain net w o rk elemen t. F or example, in comm unication n et works in- formation flows b et w een remote hosts via intermediate stations and in the Internet data pac ke ts are t ransmitted through routers and cables. The exp ected traffic flo wing through a link or a router is prop ortional to the particular edge or no de b et w eenness , resp ectiv ely . News and rumors spread in so cial netw orks, and no de betw eenness measures the imp ort a nce or cen tr a lit y of an individual in so ciet y . No de b et w eenness has b een studied r ecen tly b y Goh, Kahng, a nd Kim [15 ], who argued that it follows p o wer law in scale-free net w o rks, and the exp onen t δ ≈ 2 . 2 is indepen- 2 den t from the degree distribution in a certain range. Szab´ o, Ala v a , and Kert ´ esz [13] used ro oted deterministic trees to mo del scale-free trees, and hav e found scaling exp onen t δ t = 2. The same scaling exp onen t has b een found exp erimen ta lly b y Goh, Kahng, and Kim for scale-free trees. The r ig orous pro of of the heuristic results of [13 ] has b een pro vided by Bollob´ as and Rior da n in [1 6]. Un til recen t ly , less att ention has b een paid to edge b et w eenness, ev en though edge b e- t wee nness is of ten essen tia l for estimating the load on links in complex net w orks. F or exam- ple, the edge b et we enness can measure the “imp ortance” of relationships in so cial net works , or it can measure the expected amount of data flo w on links in computer net w orks. The probabilit y distribution of edge b et w eenness giv es a rough statistical description of links and it characterizes the net w ork as a whole. Therefore, it is an imp ortant to ol for an ov erall description of links in complex net w orks. In some cases, a lo cal prop ert y of the net w ork is kno wn as w ell. F or instance, if the n um b er of friends of any individual can b e coun ted, then it is reasonable to ask the “ impo r t ance” of a relationship (i.e. an edge in a social net w ork) under the condition that the n umber of friends of t he related individuals is kno wn. In this case, the c onditional probabilit y distribution of edge b et w eenness provide s a muc h finer description of links than the t o tal distribution. In this pap er w e f o cus on ho w additional lo cal informat ion could b e used to describ e links. In particular, w e aim at deriving the probabilit y distribution of edge b et w eenness in ev olving scale-free trees, under the condition that the in-degree o f t he “y o unger” no de of an y randomly selected link is know n. F or the sake of simplicity we consider the in-degree of the “y ounger” no de only . Whether a no de is “y ounger” than another no de or not can b e defined uniquely in ev olving netw orks, since no des att a c h to the netw ork sequen tially . Note that the in- degree is considered instead of tota l degree f o r practical reasons only . The construction of the netw ork implies that t he in-degree is less tha n the total degree b y one for ev ery “younger” no de. T o obtain the desired conditional distribution w e calculate the exact joint distribution of cluster size and in-degree f o r a sp e cific link first. Then, the join t distribution of a r andomly sele cte d link is deriv ed, whic h is comparable with the edge ensem ble stat istics obtained from a netw ork realization. The exact marginal distributions of cluster size and in-degree follow next. After t ha t, w e giv e the distribution and mean of cluster size under the condition that in-degree is kno wn. F o r the sak e of completeness the conditional in-degree distribution is 3 presen ted as w ell. Finally , the distribution and mean o f edge b et we enness is derive d under the condition that the corresp onding in- degree is kno wn. Note that a ll of our analytic results are exact even for finite networks , whic h is v aluable since the size of the r eal netw orks are often m uch smaller than the v alid range of a symptotic formulae. Moreov er, exact r esults for unb ounde d networks are provided as well. As a mo del o f ev olving scale-free trees w e consider the BA mo del with parameter m = 1 , extended with initial attractiv eness [17, 18]. With the initial attractiv eness the scaling prop erties of the net work can b e finely tuned. Note, that in the limit of initia l attractiv eness to infinit y the preferen tial atta c hmen t disapp ears, and new no des are connected to the old ones with uniform probability . In this limit the netw ork lo oses its scale-free na ture and b ecomes similar to an ER netw ork with p E R = 2 / N . Therefore, scale-free and non-scale free net works can b e compared within one mo del. F or the sak e of simplicit y the infinite limit of initial a t tractiv eness is referred to as the “ ER limit” t hroughout this pap er. W e restrict our mo del to trees, that is to connected lo opless gra phs. The simplicit y of trees allows analytic results for edge b et w eenness, since the shortest paths in trees are unique b et w een any pair of no des. Although tr ees are sp ecial g raphs, a num b er of real netw orks can b e mo delled b y trees or b y tree-like gra phs with only a negligible num b er o f shortcuts. Imp ortan t examples of suc h net w orks are the Autonomous Systems in the In ternet [19]. The rest of this pap er is organized a s follo ws: In Section I I, a short introduction to the construction of BA trees is give n. Then, a master equation fo r the join t distribution of clus- ter size and in-degree of a sp ecific edge is deriv ed and solv ed in Section I I I and Section IV, resp ectiv ely . The total joint distribution of cluster size is calculated in Section V . The marginal and conditional distributions of cluster size and in-degree are deriv ed in Section VI and Section VI I, resp ectiv ely . In Section VI I I, the conditional distribution of edge b et w een- ness follows. Finally , w e conclude our work and discuss future directions in Section IX. I I. THE NET W ORK MODEL The concepts of graph theory are used thr o ughout this pap er. A graph consists o f vertic es (no des) and e dges (links). Edges a re ordered or un-ordered pairs of v ertices, dep ending o n whether an or dered or un-or dered graph is considered, resp ectiv ely . The or der of a gr a ph is the n umber of vertice s it holds, while the de gr e e o f a v ertex counts the n umber of edges 4 adjacen t to it. Path is a lso defined in the most natural w ay: it is a v ertex sequence, where an y tw o consecutiv e elemen ts form an edge. A path is called a simple p ath if none of t he v ertices in the path are rep eated. An y tw o ve rtices in a tr e e can b e connected by a unique simple path. The gr a ph is called connected if for any v ertex pair there exists a pa th whic h starts from one ve rtex and ends at the other. The construction of t he net work pr o c eeds in discrete time steps. Let us denote time with τ ∈ N , and the dev elop ed g raph with G τ = ( V τ , E τ ), where V τ and E τ denote the set of v ertices a nd the set of edges at time step τ , resp ectiv ely . Initially , a t τ = 0, the graph consists o nly of a single v ertex without any edges. Then, in ev ery time step, a new ve rtex is connected to the net w ork with a single edge. The edge is dir e cte d , whic h emphasize tha t the t wo sides of the edge are not symme tric. The newly connected no de, whic h is the source of the edge, is alw ays “y ounger” than the target no de. The term “ younger no de of a link” is used in this sense b elo w. Note that the initial vertex is different from all the others, since it has only incoming connections; w e r efer to it as t he r o ot vertex . The target of every new edge is selec ted randomly from the presen t vertice s of the graph. The probability that a new v ertex connects to an old one is prop ortional to the attractiv eness of the old vertex v , defined as A ( v ) = a + q , (1) where para meter a > 0 denotes the initial attractiv eness a nd q is the in-degree of ver- tex v . It has b een sho wn in [18] that the in-degree distribution is asymptotically P ( q ) ≃ (1 + a ) Γ(2 a +1) Γ( a ) ( q + a ) − (2+ a ) . W e will improv e this result and deriv e the exact in- degree distribution b elo w. Note that in the sp ecial case a = 0 the a ttractiv eness of ev ery no de is zero except of the ro o t v ertex. It follows that ev ery new v ertex is connected to the initial v ertex in this case, which corresp onds to a star t o po logy . The sp ecial case a = 1 practically returns the o riginal BA mo del. Indeed, except for the ro ot v ertex, the attrac- tiv eness of eve ry v ertex b ecomes equal to its degree if a = 1; this is exactly the definition of the attra ctiv eness in the BA mo del [1 0]. Finally , if a → ∞ , then preferen tial attachme n t disapp ears in the limit, and the mo del tends to a Poisson-t yp e graph, similar to a n ER graph. The attractive ness o f sub-graph S is the sum of the att r a ctiv eness of its elemen ts: A ( S ) = X v ′ ∈ S A ( v ′ ) . (2) 5 P S f r a g r e p l a c e m e n t s C e v Ro ot FIG. 1: (Color online) Sc h ematic illustration of the ev olving net work at time τ . V ertex v , connected to the netw ork at τ e , den ote s th e ro ot of cluster C . V ariables q and n = | C | − 1 denote the in-degree of vertex v and the num b er of no des in C without v (mark ed by circles), resp ectiv ely . W e refer to a connected sub-graph as a cluster . The a ttractiv eness of cluster C can b e giv en easily: A ( C ) = (1 + a ) | C | − 1 , (3) where | C | denotes the size of the cluster. It is obv ious that the o v erall att r a ctiv eness of the net work at time step τ is A ( V τ ) = (1 + a ) ( τ + 1) − 1 . (4) I I I. MASTER EQUA TION FOR THE JOIN T DISTRIBUTION OF CLUSTER SIZE AND IN-DEGREE Let us consider the size of the net w ork N , an ar bitr ary edge e , whic h connected v ertex v to the graph a t time step τ e > 0, and let us denote b y C the cluster whic h has dev elop ed on v ertex v un til τ > τ e (Fig. 1). The calculation of b etw eennes s of the give n edge is straigh t f orw ard in trees, since the num b er of shortest paths going thro ug h the giv en edge, that is the b et wee nness of the edge, is ob viously L = | C | ( N − | C | ). Therefore, it is sufficien t to kno w t he size of the cluster o n the particular edge to get edge b et w eenness. The dev elopment of cluster C can b e regarded as a Mark ov pro cess. The states of the cluster a r e indexed by ( n, q ), where n = | C | − 1 denotes the num b er o f v ertices in cluster C without v . The in-degree of v ertex v is denoted b y q . T ransition probabilities can b e 6 obtained fro m the definition of preferential atta chmen t: W τ ,n,q = A ( C τ \ v ) A ( V τ ) = n − αq τ + 1 − α (5) W ′ τ ,q = A ( v ) A ( V τ ) = αq + 1 − α τ + 1 − α , (6) where α = 1 / (1 + a ) ∈ ]0 , 1] and W τ ,n,q denotes the transition probabilit y ( n, q ) → ( n + 1 , q ), and W ′ τ ,q denotes the transition probability ( n, q ) → ( n + 1 , q + 1), resp ectiv ely . The Master-equation, whic h describ es the Mark o v pro cess, follo ws f rom the fact that cluster C can dev elop to state ( n, q ) ob viously in three wa ys: a new v ertex can b e connected 1. to cluster C but not to v ertex v , and the cluster w as in state ( n − 1 , q ) , 2. to v ertex v , and the cluster w as in state ( n − 1 , q − 1), or 3. to the rest of the netw ork, and the cluster was in state ( n, q ). Therefore, the conditiona l probability P τ ( n, q | τ e ) that the dev elop ed cluster on edge e is in state ( n, q ) satisfies the followin g Master-equation: P τ ( n, q | τ e ) = W τ − 1 ,n − 1 ,q P τ − 1 ( n − 1 , q | τ e ) (7) + W ′ τ − 1 ,q − 1 P τ − 1 ( n − 1 , q − 1 | τ e ) +  1 − W τ − 1 ,n,q − W ′ τ − 1 ,q  P τ − 1 ( n, q | τ e ) , Since the pro cess start s with n = 0, q = 0 at τ = τ e , the initial condition of the ab o ve Master equation is P τ e ( n, q | τ e ) = δ n, 0 δ q , 0 , where δ i,j is the Kronec ker-delta sym b ol. IV. THE SOLUTI ON OF THE MASTER EQ UA TION After substituting t he ab ov e transition probabilities in t o (7), the fo llowing first order linear partia l difference equation is o btained: ( τ − α ) P τ ( n, q | τ e ) = ( n − 1 − α q ) P τ − 1 ( n − 1 , q | τ e ) + ( α q + 1 − 2 α ) P τ − 1 ( n − 1 , q − 1 | τ e ) + ( τ − n − 1) P τ − 1 ( n, q | τ e ) , (8) 7 Let us seek a pa rticular solution of (8) in pro duct form: f ( τ ) g ( n ) h ( q ) . The following equation is obtained a f ter substituting the prob e f unction in to (8): ( τ − α ) f ( τ ) f ( τ − 1) − τ = ( n − 1 − α q ) g ( n − 1) g ( n ) − n − 1 + ( αq + 1 − 2 α ) g ( n − 1) g ( n ) h ( q − 1) h ( q ) . The ab ov e partia l difference equation can b e separated in to a system of three ordinary difference equations. The solutions of the separated equations ar e: f ( τ ) = Γ( τ + λ 1 ) Γ( τ − α + 1) , (9) g ( n ) = Γ( n + λ 2 ) Γ( n + λ 1 + 1 ) , (10) h ( q ) = Γ( q + 1 /α − 1 ) Γ( q + λ 2 /α + 1) , (11) where λ 1 and λ 2 are separation parameters. The solutio n of (7), whic h fulfils the initial conditions, is constructed from the linear com bination of the ab o v e pa rticular solutions: P τ ( n, q | τ e ) = X λ 1 ,λ 2 C λ 1 ,λ 2 f ( τ ) g ( n ) h ( q ) , (12) where C λ 1 ,λ 2 co efficien ts are indep enden t of τ , n and q . T o obtain co efficien ts C λ 1 ,λ 2 , the initial condition of (7) is expanded on the bases of g ( n ) and h ( q ). The detailed calculation is presen t ed in App endix A. The solution of (7) is P τ ( n, q | τ e ) = Γ( τ − τ e + 1 ) Γ( τ e ) Γ( n + 1) Γ( τ − n ) Γ( τ − τ e − n + 1) Γ( τ e + 1 − α ) Γ( τ + 1 − α ) Γ( q + 1 / α − 1) Γ(1 /α − 1) Φ α ( n, q ) (13) where Φ α ( n, q ) = P q k =0 ( − 1) k k !( q − k )! ( − αk ) n and ( x ) n ≡ Γ( n + x ) / Γ( x ) denotes P o chhamme r’s sym b ol. Note tha t P τ ( n, q | τ e ) 6 = 0 iff 0 ≤ q ≤ n ≤ τ − τ e . The conditions 0 ≤ q and n ≤ τ − τ e are obv ious, since 1 / Γ( k ) = 0 b y definition if k is a negativ e integer or zero. F urthermore, the condition q < n can b e easily seen if Φ α ( n, q ) is tra nsfor med in to the following equiv alen t fo r m: Φ α ( n, q ) = 1 q ! d n dz n z n − 1 (1 − z − α ) q   z =1 . This result coincides with t he fact that the size of a cluster n cannot b e less tha n the corresp onding num b er of in-degrees q . V. JOINT DISTRIBUTIO N O F CLUSTER SI ZE AND IN -DEGREE Equation (13) provide s the conditional probability that a particular edge whic h was con- nected to the net work at τ e is in state ( n, q ) at τ > τ e . In a fully dev elop ed netw ork, ho w ev er, 8 the time when a particular edge is connected to the net work is usually not known. Moreo ve r, the dev elopmen t of a n individual link is usually not as imp ortan t as the pro p erties of the finally deve lop ed link ensem ble. Therefore, w e are mor e inte rested in the total probabilit y P τ ( n, q ), that is t he probabilit y that a randomly selected edge is in state ( n, q ) at τ , tha n the conditional probability (13). T he total probability can b e calculated with the help of the tota l probability theorem: P τ ( n, q ) = τ X τ e =1 P τ ( n, q | τ e ) P τ ( τ e ) , (14) where P τ ( τ e ) is the probabilit y that a randomly selected edge w as included into the net work at τ e . According to the construction of the net w o rk one edge is added to the netw ork at ev ery t ime step, therefore P τ ( τ e ) = 1 /τ . The follo wing formula can b e obtained after the ab o ve summation ha s b een carried out: P τ ( n, q ) = τ + 1 − α τ (1 /α − 1) q (2 − α ) n +1 Φ α ( n, q ) , (15) where 0 < α ≤ 1. In star top ology , that is when α = 1, the join t distribution P τ ( n, q ) eviden tly degenerates to P τ ( n, q ) = δ n, 0 δ q , 0 . The ER limit of join t distribution can b e obtained via the α → 0 limit of (15) (see App endix B for details): lim α → 0 P τ ( n, q ) = τ + 1 τ n − 1 X k = q − 1 ( − 1) k + n − 1  k q − 1  S ( k ) n − 1 Γ( n + 3) (16) where 0 < q ≤ n < τ a nd S ( m ) n denote the Stirling n um b ers o f the first kind. Note, that for the sp ecial case n = q = 0 the ER limit is lim α → 0 P τ (0 , 0) = τ +1 2 τ . The a bov e f orm ulae hav e b een v erified b y extensiv e n umerical sim ulatio ns. The join t empirical cluster size and in-degree distribution has b een compared with the analytic formula (15) for α = 1 / 2 in Fig 2. Subfigures 2( a ) and 2 (b) represen t inte rsections of the join t distribution with cutting planes of fixed in-degrees and cluster sizes, resp ectiv ely . The figures confirm that the empirical distributions, obtained as relativ e frequencies of links with cluster size n and in-degree q in 100 net w ork realizations, are in complete a g reemen t with the deriv ed analytic results. Equation (15) is the fundamen tal r esult of this section. The deriv ed distribution is exact for an y finite v alue of τ , that is for any finite BA trees. This result is precious f or mo deling 9 P S f r a g r e p l a c e m e n t s Cluster size, n S i m u l a t i o n A n a l y t i c R e l a t i v e d i ffe r e n c e I n - d e g r e e , q q = 1 q = 10 q = 20 n = 5 n = 1 0 n = 2 0 n = 4 0 P τ ( n, q ) 5 1 5 2 0 2 5 3 0 1 1 10 100 1 0 0 0 10 − 1 10 − 2 10 − 3 10 − 4 10 − 5 10 − 6 10 − 7 1 0 − 8 1 0 − 1 0 1 0 − 1 2 1 0 − 1 4 (a)Joint distribution of cluster size and in-degr e e a s the function of cluster size. P S f r a g r e p l a c e m e n t s C l u s t e r s i z e , n S i m u l a t i o n A n a l y t i c R e l a t i v e d i ffe r e n c e In-degree, q q = 1 q = 1 0 q = 2 0 n = 5 n = 10 n = 20 n = 40 P τ ( n, q ) 5 1 5 2 0 2 5 3 0 1 1 10 100 1 0 0 0 10 − 1 10 − 2 10 − 3 10 − 4 10 − 5 10 − 6 10 − 7 10 − 8 1 0 − 1 0 1 0 − 1 2 1 0 − 1 4 (b)Joint distribution o f cluster size and in-degree a s the function of in-degree. FIG. 2: (Color online) Join t empirical d istribution of cluster size and in-degree at α = 1 / 2 (sym- b ols), and analytic form ula (15 ) (solid lines) are compared on doub le-logarithmic plot. Sim ulation results hav e b een obtained from 100 realizations of 10 5 size net works. a n umber of real net works , where the size of the net work is small, compared to the relev ant range of cluster size or in-degree. If the size of the net work is m uch larger than t he relev an t range of cluster size or in-degree, then it is pra ctical to consider the netw ork as infinitely large, tha t is to tak e the τ → ∞ limit. F or the ab o ve join t distributions (15) a nd (16) the τ → ∞ limit is eviden t, since the τ dep enden t prefactor s obv iously tend t o 1 if the size of the netw orks gro w b ey ond ev ery limit. 10 VI. MAR GINAL DISTRIBUTIONS OF C LUSTER SIZ E AN D IN-DEGREE W e ha ve deriv ed the join t probability distribution of the cluster size and the in- degree in the previous section. In man y cases it is sufficien t to kno w the probability distribution of only one random v ariable, since the infor mat ion on the other v a riable is either unav ailable or not needed. It is also p ossible tha t the o ne dimensional distribution is needed esp ecially , for example, for the calculation of a conditional distribution in Section VI I. The one dimensional (marginal) distributions P τ ( n ) and P τ ( q ) can b e obtained f rom join t distribution P τ ( n, q ) as follow s: P τ ( n ) = n X q =0 P τ ( n, q ) , P τ ( q ) = τ − 1 X n = q P τ ( n, q ) . After substituting (15) into the ab ov e formulae the following expressions are obtained: P τ ( n ) = τ + 1 − α τ 1 − α ( n + 1 − α ) ( n + 2 − α ) . (17) if 0 ≤ n < τ a nd P τ ( n ) = 0 if n ≥ τ . F urthermore, P τ ( q ) = τ + 1 − α τ 1 α (1 /α − 1) 1 /α ( q + 1 / α − 1) 1 /α +1 − τ + 1 − α τ (1 /α − 1) q (2 − α ) τ q X k =0 ( − 1) k k ! ( q − k )! ( − αk ) τ αk + 2 − α . (18) if 0 ≤ q < τ and P τ ( q ) = 0 otherwise. Rice’s metho d [20] has b een applied to ev aluate the first term of P τ ( q ) in closed form. The ER limit of the marginal cluster size distribution can ob viously b e obtained from (17) at α = 0. F urthermore, the ER limit of the marginal in-degree distribution can b e deriv ed analogo usly t o the limit of the join t distribution, sho wn in App endix B: lim α → 0 P τ ( q ) = τ + 1 τ 1 2 q +1 + τ + 1 τ 1 Γ( τ + 2)Γ( q ) d q − 1 dα q − 1 (1 + α ) τ − 1 2 − α      α =0 . (19) If the size of the netw ork gr ows b ey ond ev ery limit, that is if τ → ∞ , then the marginal distributions b ecome mu c h simpler: P ∞ ( n ) = 1 − α ( n + 1 − α ) ( n + 2 − α ) (20) P ∞ ( q ) = 1 α (1 /α − 1) 1 /α ( q + 1 / α − 1) 1 /α +1 (21) lim α → 0 P ∞ ( q ) = 2 − q − 1 . (22) 11 P S f r a g r e p l a c e m e n t s Cluster size, n I n - d e g r e e , q S i m u l a t i o n A n a l y t i c F c τ ( n ) F c τ ( q ) α = 0 α = 1 / 3 α = 1 / 2 α = 2 / 3 0 5 1 0 1 5 2 0 2 5 1 1 10 10 2 10 3 10 4 10 5 10 6 10 − 1 10 − 2 10 − 3 10 − 4 10 − 5 10 − 6 10 − 7 1 0 − 8 1 0 − 9 1 0 − 1 0 FIG. 3: (Color online) Figure sh o ws comparison of empirical CCDFs of clus ter size distr ibutions (p oin ts) with analytic formula (24) (lines) on logarithmic plots, at α = 0, and 1 / 2. Emp irical distributions ha ve b een obtained from 10 realizations of N = 10 6 size netw orks. The asymptotic behavior of the cluster size and in-degree distributions differ significan tly . The ta il of the cluster size distribution follows p o w er law with exp onen t 2 either in BA o r ER net work, indep enden tly of α . Ho wev er, w e learned t ha t the ta il o f the in-degree distribution follo ws p o w er law with exp onen t 1 / α + 1 = 2 + a in BA net w orks, and it f a lls exp onen tially in ER to po logy , whic h agree with the we ll know n results of previous w orks [9]. It is w orth noting that the mean cluster size div erges logar it hmically as the size of the net work tends to infinit y: E τ { n } = P τ − 1 n =0 n P τ ( n ) = (1 − α ) ln τ + O (1). The exp ectation v alue o f the in-degree, ho w ev er, obvious ly remains finite: E τ { q } = τ τ +1 < 1, and E ∞ { q } = 1 if the size of the net w ork is infinite. Moreo v er, the standard error of the in-degree can b e also given exactly when the size of the netw ork g r o ws b ey ond ev ery limit: E ∞  ( q − 1) 2  = 2 | 1 − 2 α | . (23) This result implies that the fluctuatio ns o f the in- degree div erge in a b oundless net w ork, if α = 1 / 2, that is in the classical BA mo del. Our a nalytic results hav e b een v erified with computer sim ulations. Since cumulativ e distributions are more suitable to be compared with sim ulations tha n o rdinary distributions, w e matched the correspo nding complemen tary cumulativ e distribution functions (CCDF) against sim ulation data. The CCDF o f cluster size, F c τ ( n ) = P τ − 1 n ′ = n P τ ( n ′ ) can b e calculated 12 P S f r a g r e p l a c e m e n t s C l u s t e r s i z e , n In-degree, q In-degree, q S i m u l a t i o n A n a l y t i c F c τ ( n ) F c τ ( q ) F c τ ( q ) α = 0 α = 0 α = 1 / 3 α = 1 / 2 α = 2 / 3 0 5 1 0 15 20 25 1 1 1 10 10 10 2 10 3 10 4 10 5 10 − 1 10 − 1 10 − 2 10 − 2 10 − 3 10 − 3 10 − 4 10 − 4 10 − 5 10 − 5 10 − 6 10 − 6 10 − 7 10 − 7 1 0 − 8 1 0 − 9 1 0 − 1 0 FIG. 4: (Color online) Figure sh o ws comparison of empirical CC DFs of in -d egree d istributions (p oin ts) with analytic formula (25) (lines) on logarithmic plots, at α = 0, 1 / 3, 1 / 2, and 2 / 3. Empirical d istributions ha v e b een obtained f rom 10 realizations of N = 10 6 size n et w orks. Inset: Comparison at α = 0 on semi-logarithmic plot. straigh t f orw ardly: F c τ ( n ) = τ + 1 − α τ 1 − α n + 1 − α − 1 − α τ , (24) where 0 ≤ n < τ and 0 ≤ α ≤ 1 . The CCDF of in-degree, F c τ ( q ) = P τ − 1 q ′ = q P τ ( q ′ ) is more complex, how ev er: F c τ ( q ) = τ + 1 − α τ (1 /α − 1) 1 /α ( q + 1 / α − 1) 1 /α − 1 − α τ + τ + 1 − α τ (1 /α − 1) q (2 − α ) τ q − 2 X k =0 ( − 1) k k ! ( q − 2 − k )! (1 − α − αk ) τ − 1 ( k + 1 /α ) ( k + 2 /α ) (25) where 0 ≤ q < τ and 0 < α ≤ 1. If the size of the net w ork gro ws b ey ond eve ry limit, then the CCDFs are the following: F c ∞ ( n ) = 1 − α n + 1 − α , F c ∞ ( q ) = (1 /α − 1) 1 /α ( q + 1 / α − 1) 1 /α , (26) where 0 ≤ n , 0 ≤ q and 0 < α < 1. Comparison of analytic CCDF of cluster size (2 4) and empirical distributions are shown in Fig ure 3 for α = 0, 1 / 3, 1 / 2, and 2 / 3 . Exp erimental data has b een collected from 1 0 realizations of 10 6 no de net works. Figure 3 show s that sim ulations fully confirm our analytic result. 13 On F igure 4 analytic formula (25) a nd the empirical CCDFs of in-degree, obtained f r o m the same 10 6 no de realizations, are compared. Note the precise matc h o f the sim ulat io n and the theoretical distribution on almost the whole dat a range. Some small discrepancy can b e observ ed ar o und the lo w probability ev ents . This deviation is caused b y t he a ggregation of errors on the cum ulativ e distribution when some rare ev en t o ccurs in a finite net w ork. VI I. CONDITI ONAL PR OBABILI TIES AND E XPECT A TION V ALUES In the previous sections exact jo in t and marginal distributions of cluster size and in- degree ha ve b een ana lyzed for b oth finite and infinite netw orks. All these distributions prov ide general statistics of the netw ork. In this section we pro ceed f ur t her, and w e in v estigate the scenario when the “y ounger” in-degree of a ra ndomly selected link is kno wn. W e ask the cluster size distribution under this condition, that is the conditional distribution P τ ( n | q ). The results of the previous sections a re referred to b elo w to obtain the conditional probability distribution, and ev en tually the conditional exp ectation of cluster size. F o r the sak e of completeness , the conditional distribution and exp ectation o f in-degree are given as well at the end of this section. The conditional cluster size distribution can b e giv en by the quotien t of the join t and the marginal in-degree distributions by definition: P τ ( n | q ) = P τ ( n, q ) P τ ( q ) . (27) The exact conditio na l distribution for an y finite net w ork can b e obtained aft er substituting (15) and (18) in to the ab ov e expression. F or a b oundless net w ork the conditio na l distribution tak es the simpler f o rm: P ∞ ( n | q ) = α (2 /α − 1) q +1 (2 − α ) n +1 Φ α ( n, q ) , (28) where 0 ≤ q ≤ n . If n ≫ 1, t hen P ∞ ( n | q ) ∼ α (2 /α − 1) q +1 /n 3 + O (1 /n 4 ), that is the conditional cluster size distribution f a lls faster than the ordinary cluster size distribution. It follow s that the mean of the conditional cluster size distribution will not div erge like the mean of the ordinary distribution. What is the exp ected size of a cluster under the conditio n that the in-degree of its ro ot is know n? F or practical reasons, we do not calculate E τ { n | q } directly , but we calculate 14 E τ { n + 2 − α | q } = E τ { n | q } + 2 − α instead: E τ { n + 2 − α | q } = 1 P τ ( q ) τ − 1 X n = q ( n + 2 − α ) P τ ( n, q ) . (29) Since ( n + 2 − α ) P τ ( n, q ) = τ +1 − α τ (1 /α − 1) q (2 − α ) n Φ α ( n, q ), the ab ov e summation can b e g iven sim- ilarly to the marg ina l distribution P τ ( q ) in (18): τ − 1 X n = q ( n + 2 − α ) P τ ( n, q ) = τ + 1 − α τ 1 /α − 1 q + 1 / α − 1 − τ + 1 − α τ (1 /α − 1) q (2 − α ) τ − 1 q X k =0 ( − 1) k k ! ( q − k )! ( − αk ) τ αk + 1 − α After replacing the a bov e sum in E τ { n | q } , the following equation can b e obtained: E τ { n + 2 − α | q } = ( 1 − α ) ( q + 1 / α ) 1 /α (1 /α − 1) 1 /α G τ ( q ) , (30) where G τ ( q ) = 1 − (1 /α − 1) q +1 (1 − α ) τ q X k =0 ( − 1) k k ! ( q − k )! ( − αk ) τ k + 1 /α − 1 1 − (2 /α − 1) q +1 (2 − α ) τ q X k =0 ( − 1) k k ! ( q − k )! ( − αk ) τ k + 2 /α − 1 . (31) The iden tity lim τ →∞ G τ ( q ) ≡ 1 implies that G τ ( q ) in v olv es the finite scale effects, and the factors preceding G τ ( q ) give the asymptotic form of E τ { n + 2 − α | q } : E ∞ { n + 2 − α | q } = ( 1 − α ) ( q + 1 / α ) 1 /α (1 /α − 1) 1 /α . (32) It can b e seen that the exp ectation of cluster size, under the condition that the in- degree is kno wn, is finite in a n un b ounded net w ork. It stands in contrast to the unconditional cluster size, discuss ed in the previous section, whic h div erges loga rithmically as the size of the netw ork gr ows b ey ond ev ery limit. In the ER limit, the exp ected conditional cluster size b ecomes lim α → 0 E ∞ { n + 2 | q } = 2 q +1 . (33) The f undamental difference b et w een the scale-free and non-scale-free netw orks can b e ob- serv ed again. In t he scale-free case the exp ected conditio nal cluster size a symptotically gro ws with the in-degree to the p ow er of 1 /α , while in the later case it gr ows exp onen tially . 15 On Fig ure 5 the exact analytic fo rm ula (30) is compared with sim ulation results at α = 0, 1 / 3, 1 / 2 , a nd 2 / 3. The simulations clearly justify our analytic solution. Let us in v estigate shortly the opp osite scenario, tha t is when the cluster size is kno wn and the statistics of the in-degree under this condition is sough t. The conditional distribution can b e obtained fr o m the com bination o f Eqs. (15), (17) and t he definition P τ ( q | n ) = P τ ( n, q ) P τ ( n ) . (34) The conditional expectation of in-degree can b e a cquired b y the same t echniq ue as the conditional exp ectation of cluster size. Let us calculate E τ { q + 1 / α − 1 | n } = E τ { q | n } + 1 /α − 1 instead of E τ { q | n } directly: E τ { q + 1 / α − 1 | n } = 1 P τ ( n ) n X q =0 ( q + 1 / α − 1) P τ ( n, q ) = Γ(2 − α ) α ( n + 1 − α ) α , (35) where 0 ≤ n < τ . Note, that the conditiona l exp ectation of in-degree is indep enden t of τ , that is of the size o f the netw ork. In t he ER limit the exp ectation of the in-degree b ecomes lim α → 0 E τ { q | n } = Ψ ( n + 1) + γ , (36) where Ψ( x ) = d dx ln Γ( x ) denotes the digamma function, and γ = − Ψ(1) ≈ 0 . 5 772 is the Euler–Masc heroni constan t. Asymptotically the exp ectation of the in-degree in a scale- free tree gro ws with the cluster size to the p ow er of α , while in a ER tree it grows o nly logarithmically , since Ψ( n + 1) = log n + O (1 /n ). Therefore, conditional in-degree and conditional cluster size are m utually in v erses asymptotic al ly . Figure 6 sho ws the analytic solution (35) and sim ulation data at α = 0, 1 / 3, 1 / 2, a nd 2 / 3 parameter v alues. Sim ula tion data has b een collected from 100 realizations of 10 5 size net works . VI I I. CONDITION AL DISTRIBUTION OF EDGE BET WEENNESS Using the results of the previous sections, w e are finally ready to answ er the problem whic h motiv ated our w ork, that is the distribution of the edge b et we enness under the condition that the in-degree of the “y ounger” no de of the link is kno wn. It has b een noted at the b eginning of Section I I I t hat the edge b et w eenness can b e expressed with cluster size: L = ( n + 1) ( τ − n ) . (37) 16 P S f r a g r e p l a c e m e n t s In-degree, q E τ { n | q } α = 0 α = 1 / 3 α = 1 / 2 α = 2 / 3 0 1 1 10 10 10 2 10 2 10 3 10 3 10 4 10 4 10 5 1 0 6 FIG. 5: (Color online) Figure shows the a verag e clus ter size as the function of the in-degree q , obtained from 100 realizati ons of 10 5 size net works. Simulat ion data has b een collect ed at α = 0, 1 / 3, 1 / 2, and 2 / 3 parameter v alues. An alytical result (30) of conditional exp ectation E τ { n | q } is sho w n with contin uous lines. P S f r a g r e p l a c e m e n t s Cluster size, n E τ { q | n } α = 0 α = 1 / 3 α = 1 / 2 α = 2 / 3 0 1 1 10 10 10 2 10 2 10 3 10 3 10 4 10 5 1 0 6 FIG. 6: (Color online) Figure sh o ws th e a v erage in-degree as the fun ctio n of the cluster size n , obtained from 100 realizati ons of 10 5 size net works. Simulat ion data has b een collect ed at α = 0, 1 / 3, 1 / 2, and 2 / 3 parameter v alues. An alytical result (35) of conditional exp ectation E τ { q | n } is sho w n with contin uous lines. Therefore, conditional edge b et w eenness can b e give n formally by the following transforma- tion o f random v ariable n : P τ ( L | q ) = τ − 1 X n =0 δ L, ( n +1)( τ − n ) P τ ( n | q ) . (38) 17 Ob viously , P τ ( L | q ) is non- zero only at those v alues of L , where (37) has integer solution for n . If n L = τ − 1 2 − s ( τ + 1 ) 2 4 − L (39) is suc h an integer solutio n of the quadratic equation (37), and L 6 = ( τ + 1) 2 / 4, then P τ ( L | q ) = P τ ( n L | q ) + P τ ( τ − 1 − n L | q ) . (40) If L = ( τ + 1) 2 / 4 is inte ger, then P τ ( L | q ) = P τ ( n L | q ) . The conditional exp ectation o f edge b et w eenness can b e obtained from (3 7 ): E τ { L | q } = τ E τ { n + 1 | q } − E τ { ( n + 1) n | q } . (41) Therefore, for the exact calculation of E τ { L | q } the first and the second momen t of the conditional cluster size distribution are required. The first moment, that is the mean, has b een deriv ed in the previous section. In order to calculate the second momen t let us use the tec hnique we ha ve dev elop ed in the previous sections. Let us consider: E τ { ( n + 2 − α ) ( n + 1 − α ) | q } = τ + 1 − α τ (1 /α − 1) q P τ ( q ) τ − 1 X n = q Φ α ( n, q ) (2 − α ) n − 1 . (42) W e shall b e cautious when the summation for n is ev aluated. The k = 1 term in Φ α ( n, q ) = P q k =0 ( − 1) k k !( q − k )! ( − αk ) n m ust b e treated separately to av oid a div ergen t term: τ − 1 X n = q Φ α ( n, q ) (2 − α ) n − 1 = 1 − α ( q − 1)! [ α Ψ( τ − α ) − α Ψ(1 − α ) − Ψ( q ) − γ ] − 1 α 1 (2 − α ) τ − 2 q X k =2 ( − 1) k k ! ( q − k )! ( − αk ) τ k − 1 The exact form ula for E τ { L | q } can b e obtained stra ig h tforwardly , after (30) and the ab ov e expressions hav e b een substituted in t o (41). Let us consider the scenario when the size of the net work tends to infinit y . Equation (37) implies that edge b et w eenness dive rges as τ → ∞ , therefore L should b e rescaled for an infinite net w ork. F ro m the asymptotics of the digamma function Ψ( τ − α ) = ln τ + O (1 /τ ) it follows that E τ { ( n + 2 − α ) ( n + 1 − α ) | q } grow s only logarithmically , slow er than the linear growth of τ E τ { n + 2 − α | q } . Therefore, edge betw eenness asymptotically g ro ws linearly as the size of the netw ork g ro ws beyond ev ery limit. Let us rescale edge b et w eenness Λ τ = L ( τ ) τ + 1 (43) 18 and let us consider the limit Λ = lim τ →∞ Λ τ = n Λ + 1. The CCDF of the rescaled edge b etw eennes s can b e giv en by F c ∞ (Λ | q ) = lim τ →∞ P τ − 1 − n Λ τ n = n Λ τ P τ ( n | q ) = 1 P ∞ ( q ) P ∞ n =Λ − 1 P ∞ ( n, q ). When the summation has b een carried out, the following equation is obtained: F c ∞ (Λ | q ) = (2 /α − 1) q +1 (2 − α ) Λ − 1 q X k =0 ( − 1) k k ! ( q − k )! ( − αk ) Λ − 1 k + 2 /α − 1 , (44) where q + 1 ≤ Λ. If 1 < q ≪ Λ, t hen only the first term of the sum should b e ta k en in to accoun t, and it is easy to see tha t F c ∞ (Λ | q ) = α 2 (1 − α ) 2Γ(2 /α − 1 ) q 2 /α Λ 2 + O  1 / Λ 2+ α  . (45) It can b e seen that the scaling expo nent − 2 is indep enden t of α . The ab o v e asymptotic form ula has b een obtained for infinite net works . The same p ow er law scaling can b e observ ed in finite size net works as (4 5) if Λ τ ≪ τ . Ho w ev er, F c τ (Λ τ | q ) ≡ 0 if Λ τ > τ in finite net w o rks, therefore asymptotic formula (45) eviden t ly b ecomes in v alid if Λ τ ≈ τ . It is obvious that as the size of the netw ork g ro ws la rger and larg er, asymptotic form ula (44) b ecomes mor e and more accurate. One can a sk ho w fast the con v ergence is. F rom elemen tary estimations of F c τ (Λ τ | q ) one can show that for fixed Λ τ : F c τ (Λ τ | q ) = F c ∞ (Λ τ | q ) − (1 − F c ∞ (Λ τ | q )) α 2 (1 − α ) 2 1 τ 2 + O  1 /τ 2+ α  , (46) that is corrections to the a symptotic formula decrease with τ − 2 for large τ . On F igure 7 comparison of analytic form ula (44) with sim ulation results is presen ted fo r q = 1 and q = 2. The empirical CCDF of rescaled edge b etw eenness , under the condition that in-degree q is know n, is shown for 10 4 , 1 0 5 , and 10 6 size net w orks, at α = 1 / 2 parameter v alue. T he empirical CCDFs of rescaled edge b et wee nness eviden tly collapse to the same curv e for differen t size net w orks, a nd they coincide precisely with our analytic result. The exp ectation o f the rescaled edge b et wee nness under the condition that in-degree q is kno wn can b e given by E ∞ { Λ | q } = E ∞ { n Λ + 1 | q } . Using (3 2) and (3 3) w e get E ∞ { Λ | q } = (1 − α ) ( q + 1 / α ) 1 /α (1 /α − 1) 1 /α − 1 + α , (47) lim α → 0 E ∞ { Λ | q } = 2 q +1 − 1 . (48) One can see that E ∞ { Λ | q } ∼ q 1 /α for q ≫ 1 if α > 0 and E ∞ { Λ | q } ∼ e q for q ≫ 1 if α → 0. 19 P S f r a g r e p l a c e m e n t s Rescaled edge b etw eenness, Λ F c ∞ (Λ | q ) N = 10 4 N = 10 5 N = 10 6 q = 1 q = 2 1 1 10 − 1 10 − 2 10 − 3 10 − 4 10 − 5 10 − 6 10 10 2 10 3 1 0 4 1 0 5 1 0 6 FIG. 7: (Color onlin e) Figure sho w s CCDF of edge b et weenness und er the condition that the in-degree q is kno wn. Emp ir ical CCDF h as b een obtained from 100 realizations of N = 10 4 and N = 10 5 , and 10 realizations of N = 10 6 size net w orks at α = 1 / 2 p arameter v alue. Con tinuous lines sh o w analytic result (44). Analytic results (47) and (48), a nd sim ulatio n da ta are sho wn in Figure 8 at α = 1 / 2 and α = 0 parameter v alues. Numerical data has b een collected from the same 10 4 , 10 5 , and 10 6 size net works as ab o v e. As the size of the net work gro ws, a larger and la rger ra ng e of the rescaled empirical data collapses to the same analytic curv e. On the high degree region some discrepancy can b e observ ed due to the finite scale effects. Finally , let us note that the precise unconditional distribution of edge b et w eenness P τ ( L ) = P τ − 1 n =0 δ L, ( n +1)( τ − n ) P τ ( n ) can b e o btained from (17) as w ell. F urthermore, CCDF of the unconditional b et wee nness F c τ ( L ) = P τ − n L − 1 n = n L P τ ( n ) can b e deriv ed in closed form: F c τ ( L ) = τ + 1 − α τ (1 − α ) ( τ − 2 n L ) ( n L + 1 − α ) ( τ − n L + 1 − α ) . (49) F or the sak e of simplicit y w e hav e assumed during our calculations t ha t in-degrees of the “y ounger” no des are pro vided. How ev er, it is p ossible that eve n tho ug h b oth tw o in-degrees of ev ery link a re kno wn, w e cannot distinguish them from each other, that is w e cannot tell whic h is the “y o unger” no de. How could w e extend our results to this scenario? Let us consider a new edge when it is connected to the net work. The in-degree of the new no de is ob viously 0. The in-degree of the other no de, whic h the new no de is connected to, is equal to o r larger tha n one. Due to preferen t ia l att a c hmen t the larger the in-degree is the faster it gro ws. Ev en if preferen tial attach men t is absen t, the grow th rate of eve ry in-degree is 20 P S f r a g r e p l a c e m e n t s In-degree, q In-degree, q E ∞ { Λ | q } N = 10 4 N = 10 5 N = 10 6 α = 0 α = 1 / 2 0 1 1 1 5 10 10 10 10 15 20 10 2 10 2 10 2 10 3 10 3 10 3 10 4 10 4 10 5 10 5 10 6 10 6 FIG. 8: (Color on lin e) Figure sh o ws av erage edge b et w eenness u nder the condition that the in - degree q is kno wn as the function of q on log-lo g plot. Numerical d ata has b een collected from 100 realizatio ns of N = 10 4 and N = 10 5 , and 10 realizatio ns of N = 10 6 size net works at α = 1 / 2 parameter v alue. In set sh o ws the same scenario at α = 0 parameter v alue on semi-logarithmic plot. Con tinuous lines sho w analytic results (47 ) and (48) . the same. Therefore, it is exp ected that the initial deficit in t he in-degree of the “younger” no de gro ws o r r emains at the same lev el during the ev olution of the net work. It follows that it is a reasonable approximation t o substitute the in-degree of the “younger” no de q with q min = min( q 1 , q 2 ) in our fo rm ulae. IX. CONCLUSIO NS A t ypical net w ork construction problem is to design netw ork infrastructure without w a st- ing precious resources at places where not needed. An appropriate design strategy is if net work resources are allo cated prop ortionally to the exp ected traffic. In a mean field appro ximation the exp ected traffic is prop ortio na l to the num b er of shortest paths going through a certain net w ork elemen t, that is the b et w eenness. The precise calculation of all the b et w eenness require complete informatio n on the net w ork structure. In real life, how ev er, the num b er of shortest paths is often imp ossible to tell b ecause the structure of the net work is not fully kno wn. One of the practical results o f this pap er is that the exp ectation of edge b et w eenness can b e estimated precisely when a limited lo cal information on netw ork structure—the in- degree of the “younger” no de—is av ailable. 21 Another difficult y of net w ork design is that the size of real net works is finite. Moreo v er, the size of real netw orks is often so small that asymptotic fo rm ulas can b e applied only with unacceptable error. The other imp ortant nov elt y of our r esults is that the deriv ed formulas are exact ev en for finite netw orks, whic h allows b etter design of realistic finite size netw orks. V arious statistical prop erties of ev olving random trees hav e b een in ves tigated in t his pap er. W e hav e fo cused on t he cluster size, the in-degree and the edge b et we enness. W e ha ve considered the m = 1 case of the BA mo del extended with initial attractiv eness f o r mo deling random trees. Initial attractiv eness allows fine tuning of the scaling parameter. Moreo ve r, in the limit of the tuning parameter α → 0 the applied mo del tends to a non- scale-free structure, whic h is in man y a spects similar to t he classical ER mo del. Therefore, w e were able to in ves tigate b oth the scale-free and the no n- scale-free scenario within t he same framew ork. First, the ev olutio n o f cluster size and in-degree of a specific edge ha ve b een mo deled as a biv ariate Mark ov pro cess. The master equation, asso ciated with the Marko v pro cess, has led us to a linear partial difference equation. An exact analytic solution of the master equation, whic h satisfies the initial conditions as w ell, has b een found. The solution provides the join t probabilit y distribution of cluster size and in-degree fo r a sp ecific edge. Using the ab ov e results w e ha v e derive d the j o in t probability distribution of cluster size and in-degree for a randomly selec ted edge. It is of more practical imp ortance than the join t distribution for a sp ecific edge b ecause, in con trast t o t he former distribution, it provid es the statistical description of the whole net w ork. W e also deriv ed t he join t distribution in the ER limit. Note that the obt a ined form ulae are exact for ev en finite size net w orks. In addition, t he formulae for un b ounded netw orks ha v e b een presen ted as w ell. W e ha ve con tinu ed our analysis with the one dimensional marginal distributions. W e hav e sho wn some fundamen t al differences in the scaling prop erties of t he margina l cluster size and in- degree distributions. The nov elt y of our results here, compared to previous r esults in the literat ur e, is that w e ha v e fo und exact analytic formulae not only for the la rge, but also fo r the small cluster size and in-degree region. Although the margina l distributions hav e their own imp ortance, w e ha v e deriv ed them in order to obtain conditional probabilit y distributions. F rom the combination of the joint and the margina l distributions w e hav e giv en the conditional distributions of cluster size and in-degree. W e hav e also presen t ed conditional exp ectations of cluster size and in-degree 22 for b oth finite and un b ounded net w orks. W e hav e found that asymptotically t he conditional cluster size grows with in-degree to the p o wer of 1 /α and the conditiona l in-degree grows with cluster size to the p ow er of α , resp ective ly . The ER limit has b een discusse d as we ll. W e ha ve sho wn that the conditional cluster size gro ws exp onen tially and the conditional in-degree grows logarithmically when α → 0. Finally , b y applying the transformation of random v ariables w e hav e deriv ed the distri- bution of edge b et wee nness under the condition that the corresp onding in-degree is known. W e ha v e fo und that the conditional expectatio n o f edge b et wee nness grows linearly with the size of t he net work. F or the analysis of un b ounded net w orks w e hav e defined the rescaled edge b et ween ness Λ , and deriv ed its distribution and exp ectation under the condition that in-degree q is provided. Our analytic results hav e b een verifie d at differen t net w or k sizes and parameter v alues b y extensiv e n umerical sim ula tions. W e hav e demonstrated that n umerical sim ulations fully confirm our analytic results. F or the future, w e hop e that the metho ds w e ha v e dev elop ed in this pap er allow us to describe cluster size and edge b et w eenness in more general scenarios. F or example, when not only the “younger”, but b oth tw o in-degrees of links are considered. Ac kno wledgemen t The authors thank the pa r t ia l supp o rt of the National Science F oundation (OTKA T37903), the Nationa l Office for Researc h and T ec hnology (NKFP 02/032/ 2 004 and NAP 2005/ K CKHA005 ) and the EU IST FET Complexity EVER GRO W In tegra ted Pro ject. The autho rs also thank M´ at ´ e Mar´ odi for the fruitful discussions and his helpful commen ts. APPENDIX A: EXP ANSI ON OF THE KR ONE CKER-DEL T A FUNCT ION W e hav e se en that the general solution of Eq. (7) is P τ ( n, q | τ e ) = P λ 1 ,λ 2 C λ 1 ,λ 2 f ( τ ) g ( n ) h ( q ), and the initial condition is P τ e ( n, q | τ e ) = δ n, 0 δ q , 0 , where δ n,m =      1 , if n = m, 0 , if n 6 = m (A1) 23 is the Kronec k er-delta function, and n and m a r e integers. Co efficien ts C λ 1 ,λ 2 are calculated in this section. First w e show that δ n, 0 = n X k =0 ( − 1) k k ! 1 Γ( n − k + 1) . (A2) Note that w e can consider m = 0 without any loss o f generalit y , since δ n,m ≡ δ n − m, 0 . If n < 0, then the summand in (A2) is zero b y definition, indeed. If n > 0, t hen n X k =0 ( − 1) k k ! 1 Γ( n − k + 1) = 1 n ! n X k =0  n k  ( − 1) k = 0 (A3) follo ws from the binomial theorem. Fina lly , for n = 0, 0 X k =0 ( − 1) k k ! 1 Γ( − k + 1 ) = ( − 1) 0 0! 1 Γ(1) = 1 . (A4) Co efficien ts C λ 1 ,λ 2 can b e obtained from the term by term comparison of P τ e ( n, q | τ e ) = P λ 1 ,λ 2 C λ 1 ,λ 2 f ( τ e ) g ( n ) h ( q ) with the expansion of the initial condition δ n, 0 δ q , 0 , shown ab o v e. One can easily confirm with the help of identit y f ( n ) δ n, 0 ≡ f (0) δ n, 0 that the same terms app ear on b oth sides, if λ 1 = − k 1 , and λ 2 = − αk 2 , and co efficien ts C k 1 ,k 2 are the f o llo wing: C k 1 ,k 2 = ( − 1) k 1 + k 2 k 1 ! k 2 ! Γ( τ e + 1 − α ) Γ( τ e − k 1 ) 1 Γ( − αk 2 ) 1 Γ(1 /α − 1) . (A5) Finally , to obtain (15) the summation fo r k 1 can b e carried out explicitly: n X k 1 =0 ( − 1) k 1 k 1 !Γ( n − k 1 + 1 ) Γ( τ − k 1 ) Γ( τ e − k 1 ) = Γ( τ − τ e + 1 ) Γ( n + 1)Γ( τ e ) Γ( τ − n ) Γ( τ − τ e − n + 1) APPENDIX B: THE α → 0 LIMIT OF JOINT DISTRIBUTION P τ ( n, q ) In this section w e prov e that the ER limit of the jo int probability P τ ( n, q ) is (16). Theorem L et us c onsider P τ ( n, q ) as define d in (15), wher e 0 < q < n < τ ar e i n te gers. Then the fol lowing limit holds: lim α → 0 P τ ( n, q ) = τ + 1 τ Γ( n + 3) n − 1 X k = q − 1 ( − 1) n − 1 − k S ( k ) n − 1  k q − 1  . (B1) 24 Pr o of. First, let us note that Φ α ( n, q ) in (15) can b e rewritten in the fo llo wing equiv a lent form: Φ α ( n, q ) = α P q − 1 k =0 ( − 1) k (1 − α − αk ) n − 1 k !( q − 1 − k )! . Next, Pochhamme r’s sym b ol (1 / α − 1) q is re- placed with its a symptotic form: (1 /α − 1) q = 1 /α q (1 + O ( α )). After the o bvious limits ha ve b een ev aluated the following equation is obtained: lim α → 0 P τ ( n, q ) = τ + 1 τ Γ( n + 3) lim α → 0 P q − 1 k =0 ( − 1) k (1 − α − αk ) n − 1 k !( q − 1 − k )! α q − 1 . (B2) The ab ov e limit, b y definition, can b e substituted with q − 1 order differential at α = 0, if all the low er order deriv ates of the sum a re zero at α = 0. Indeed, lim α → 0 P q − 1 k =0 ( − 1) k (1 − α − αk ) n − 1 k !( q − 1 − k )! α q − 1 = 1 m ! d m dα m q − 1 X k =0 ( − 1) k (1 − α − αk ) n − 1 k ! ( q − 1 − k )!      α =0 = 1 m ! d m (1 + α ) n − 1 dα m      α =0 q − 1 X k =0 ( − 1) k ( − k − 1) m k ! ( q − 1 − k )! , where the sum is 0 if m < q − 1 and 1 if m = q − 1 . Therefore, the limit can b e tr ansformed to lim α → 0 P τ ( n, q ) = τ + 1 τ Γ( n + 3) 1 ( q − 1)! d q − 1 (1 + α ) n − 1 dα q − 1     α =0 . 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