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Network Science Class 5: BA model (Sept 15, 2014) Albert-László Barabási With Roberta Sinatra www.BarabasiLab.com Section 1 Introduction Section 1 Hubs represent the most striking difference between a random and a scale-free network. Their emergence in many real systems raises several fundamental questions: •Why does the random network model of Erdős and Rényi fail to reproduce the hubs and the power laws observed in many real networks? • Why do so different systems as the WWW or the cell converge to a similar scale-free architecture? Section 2 Growth and preferential attachment BA MODEL: Growth ER model: the number of nodes, N, is fixed (static models) networks expand through the addition of new nodes Barabási & Albert, Science 286, 509 (1999) BA MODEL: Preferential attachment ER model: links are added randomly to the network New nodes prefer to connect to the more connected nodes Barabási & Albert, Science 286, 509 (1999) Network Science: Evolving Network Models February 14, 2011 Section 2: Growth and Preferential Sttachment The random network model differs from real networks in two important characteristics: Growth: While the random network model assumes that the number of nodes is fixed (time invariant), real networks are the result of a growth process that continuously increases. Preferential Attachment: While nodes in random networks randomly choose their interaction partner, in real networks new nodes prefer to link to the more connected nodes. Barabási & Albert, Science 286, 509 (1999) Network Science: Evolving Network Models February 14, 2011 Section 3 The Barabási-Albert model Origin of SF networks: Growth and preferential attachment (1) Networks continuously expand by the addition of new nodes WWW : addition of new documents (2) New nodes prefer to link to highly connected nodes. GROWTH: add a new node with m links PREFERENTIAL ATTACHMENT: the probability that a node connects to a node with k links is proportional to k. WWW : linking to well known sites (ki ) ki jk j P(k) ~k-3 Barabási & Albert, Science 286, 509 (1999) Network Science: Evolving Network Models February 14, 2011 Section 4 Section 4 Linearized Chord Diagram Section 4 Degree dynamics All nodes follow the same growth law ¶ki ki µ P(ki ) = A ¶t å kj j Use: å ¶ki ki = ¶t 2t j k j = 2mt During a unit time (time step): Δk=m A=m k ¶k i ¶t = ki 2t æ t öb ki (t) = mç ÷ è ti ø b= ò m ¶ki = ki t ò ti ¶t 2t 1 2 β: dynamical exponent A.-L.Barabási, R. Albert and H. Jeong, Physica A 272, 173 (1999) Network Science: Evolving Network Models February 14, 2011 All nodes follow the same growth law k(t)~t ½ Degree (k) SF model: time (first mover advantage) Section 5.3 Section 5 Degree distribution Degree distribution æ t öb ki (t) = mç ÷ è ti ø b= 1 2 A node i can come with equal probability any time between ti=m0 and t, hence: 1 P(t i ) = m0 + t 1 P(t i < t ) = m0 + t t ò dt i 0 ¶P(ki (t) < k) 2m 2 t 1 -g \P(k) = = ~ k ¶k mo + t k 3 = t m0 + t γ=3 A.-L.Barabási, R. Albert and H. Jeong, Physica A 272, 173 (1999) Network Science: Evolving Network Models February 14, 2011 Degree distribution æ t öb ki (t) = mç ÷ è ti ø b= 1 2 2m 2 t 1 -g P(k) = ~ k mo + t k 3 γ=3 (i) The degree exponent is independent of m. (ii) As the power-law describes systems of rather different ages and sizes, it is expected that a correct model should provide a time-independent degree distribution. Indeed, asymptotically the degree distribution of the BA model is independent of time (and of the system size N) the network reaches a stationary scale-free state. (iii) The coefficient of the power-law distribution is proportional to m2. A.-L.Barabási, R. Albert and H. Jeong, Physica A 272, 173 (1999) Network Science: Evolving Network Models February 14, 2011 The mean field theory offers the correct scaling, BUT it provides the wrong coefficient of the degree distribution. So assymptotically it is correct (k ∞), but not correct in details (particularly for small k). To fix it, we need to calculate P(k) exactly, which we will do next using a rate equation based approach. Network Science: Evolving Network Models February 14, 2011 MFT - Degree Distribution: Rate Equation < N(k,t) >= tP(K,t) Number of nodes with degree k at time t. Since at each timestep we add one node, we have N=t (total number of nodes =number of timesteps) P(k) = k å j kj = k 2mt 2m: each node adds m links, but each link contributed to the degree of 2 nodes Total number of k-nodes Number of links added to degree k nodes after the arrival of a new node: Nr. of degree k-1 nodes that acquire a new link, becoming degree k Nr. of degree k nodes that acquire a new link, becoming degree k+1 k -1 P(k -1,t) 2 Preferential attachment k P(k, t) 2 (N +1)P(k,t +1) = NP(k,t) + # k-nodes at time t+1 k k ´ NP(k,t) ´ m = P(k,t) 2mt 2 # k-nodes at time t k -1 k P(k -1,t) - P(k,t) 2 2 Gain of knodes via k-1 k Loss of knodes via k k+1 New node adds m new links to other nodes MFT - Degree Distribution: Rate Equation k -1 k (N +1)P(k,t +1) = NP(k,t) + P(k -1,t) - P(k,t) 2 2 # k-nodes at time t+1 # k-nodes at time t Gain of knodes via k-1 k Loss of knodes via k k+1 We do not have k=0,1,...,m-1 nodes in the network (each node arrives with degree m) We need a separate equation for degree m modes (N +1)P(m,t +1) = NP(m,t) + 1 # m-nodes at time t+1 # mnodes at time t - Add one m-degeree node m P(m,t) 2 Loss of an m-node via m m+1 Network Science: Evolving Network Models February 14, 2011 MFT - Degree Distribution: Rate Equation k -1 k P(k -1,t) - P(k,t) 2 2 m (N +1)P(m,t +1) = NP(m,t) + 1 - P(m,t) 2 (N +1)P(k,t +1) = NP(k,t) + k>m We assume that there is a stationary state in the N=t∞ limit, when P(k,∞)=P(k) (N +1)P(k,t +1) - NP(k,t) ®NP(k,¥) + P(k,¥) - NP(k,¥) = P(k,¥) = P(k) (N +1)P(m,t +1) - NP(m,t) ®P(m) k -1 k P(k -1) - P(k) 2 2 m P(m) = 1 - P(m) 2 P(k) = k -1 P(k -1) k +2 2 P(m) = 2+m P(k) = k>m Network Science: Evolving Network Models February 14, 2011 MFT - Degree Distribution: Rate Equation k -1 P(k) = P(k -1) k +2 k P(k +1) = P(k) k +2 2 m+2 m 2m P(m +1) = P(m) = m+3 (m + 2)(m + 3) P(m) = P(m + 2) = m +1 2m(m +1) P(m +1) = m+4 (m + 2)(m + 3)(m + 4) P(m + 3) = m+2 2m(m +1) P(m + 2) = m+5 (m + 3)(m + 4)(m + 5) ... 2m(m +1) P(k) = k(k +1)(k + 2) Krapivsky, Redner, Leyvraz, PRL 2000 Dorogovtsev, Mendes, Samukhin, PRL 2000 Bollobas et al, Random Struc. Alg. 2001 m+3 k P(k) ~ k -3 for large k Network Science: Evolving Network Models February 14, 2011 MFT - Degree Distribution: A Pretty Caveat Start from eq. P(k) = k -1 k P(k -1) - P(k) 2 2 2P(k) = (k -1)P(k -1) - kP(k) = -P(k -1) - k[P(k) - P(k -1)] P(k) - P(k -1) ¶P(k) 2P(k) = -P(k -1) - k = -P(k -1) - k k - (k -1) ¶k P(k) = - Its solution is: 1 ¶[kP(k)] 2 ¶k P(k) ~ k -3 Dorogovtsev and Mendes, 2003 Network Science: Evolving Network Models February 14, 2011 Degree distribution æ t öb ki (t) = mç ÷ è ti ø 1 b= 2 2m(m +1) P(k) = k(k +1)(k + 2) P(k) ~ k -3 γ=3 for large k (i) The degree exponent is independent of m. (ii) As the power-law describes systems of rather different ages and sizes, it is expected that a correct model should provide a time-independent degree distribution. Indeed, asymptotically the degree distribution of the BA model is independent of time (and of the system size N) the network reaches a stationary scale-free state. (iii) The coefficient of the power-law distribution is proportional to m2. Network Science: Evolving Network Models February 14, 2011 NUMERICAL SIMULATION OF THE BA MODEL P(k) = 2m(m +1) k(k +1)(k + 2) Section 6 absence of growth and preferential attachment MODEL A growth preferential attachment Π(ki) : uniform ¶k i m = AP(k i ) = ¶t m 0 + t -1 ki (t) = m ln( m 0 + t -1 )+m m + t i -1 e k P(k) = exp(- ) ~ e - k m m MODEL B growth k i t A (ki ) k i (t ) 1 N 2 ( N 1) N ( N 2) preferential attachment ki N N 1 2t N t Ct 2 ( N 1 ) ~ 1 N 2 t N pk : power law (initially) Gaussian Fully Connected Do we need both growth and preferential attachment? YEP. Network Science: Evolving Network Models February 14, 2011 Section 7 Measuring preferential attachment Section 7 Measuring preferential attachment k i t (ki ) ~ ki t Plot the change in the degree Δk during a fixed time Δt for nodes with degree k. To reduce noise, plot the integral of Π(k) over k: (k ) (K ) K k No pref. attach: κ~k Linear pref. attach: κ~k2 (Jeong, Neda, A.-L. B, Europhys Letter 2003; cond-mat/0104131) Section 7 Measuring preferential attachment citation network Internet Plots shows the integral of Π(k) over k: (k ) (K ) K k No pref. attach: κ~k neurosci actor collab collab. Linear pref. attach: κ~k2 (k ) A k , 1 Network Science: Evolving Network Models February 14, 2011 Section 8 Nonlinear preferenatial attachment Section 8 Nonlinear preferential attachment α=0: Reduces to Model A discussed in Section 5.4. The degree distribution follows the simple exponential function. α=1: Barabási-Albert model, a scale-free network with degree exponent 3. 0<α<1: Sublinear preferential attachment. New nodes favor the more connected nodes over the less connected nodes. Yet, for the bias is not sufficient to generate a scale-free degree distribution. Instead, in this regime the degrees follow the stretched exponential distribution: Section 8 Nonlinear preferential attachment α=0: Reduces to Model A discussed in Section 5.4. The degree distribution follows the simple exponential function. α=1: Barabási-Albert model, a scale-free network with degree exponent 3. α>1: Superlinear preferential attachment. The tendency to link to highly connected nodes is enhanced, accelerating the “rich-gets-richer” process. The consequence of this is most obvious for , when the model predicts a winner-takes-all phenomenon: almost all nodes connect to a single or a few super-hubs. Section 8 Nonlinear preferential attachment The growth of the hubs. The nature of preferential attachment affects the degree of the largest node. While in a scale-free network the biggest hub grows as (green curve), for sublinear preferential attachment this dependence becomes logarithmic (red curve). For superlinear preferential attachment the biggest hub grows linearly with time, always grabbing a finite fraction of all links (blue curve)). The symbols are provided by a numerical simulation; the dotted lines represent the analytical predictions. Section 9 The origins of preferential attachment hen ce i t i s i n her en t ly local an d r an dom . Un li ke t he Bar abási -Alber t m odel, i t 9 lacks a bui lt -i n Section EXISTING NETWORK (k) f un ct i on . Yet n ext w e show t hat i t gen er Link selection model at es pr ef er en t i al at t achm en t . p Link selection model -- perhaps the simplest example of a local ofn ode generating . t he Weor st arrandom t by w r i t i nmechanism g t he pr obabi li tcapable y qk t hat at t he en d of a r an preferential attachment. dom ly chosen li n k has degr ee k as Growth: at each time stepqk weCkp add a new node to the (5.26) k network. Equat i on (5.26) capt ur es t w o ef f ect s: TARGET u CHOOSE TARGET Link selection: we select a link at random and connect the • node The hi gher t he degr ee of a n ode, t he hi gher t he chan ce t hat i t i s lonew to one of nodes at the two ends of the selected Figure 5.14 cat ed at t he en d of t he chosen li n k. link. Copying Model • The m or e degr ee-k n odes ar e i n t he n et w or k (i .e., t he hi gher i s p k), To show that this simple mechanism generates linear t he m or e li kely t hat a degr ee k n ode i s at t he en d of t he li n k. preferential attachment, we write the probability that the node at the of a randomly link In (5.26) C canend be calculat ed usi n g t hechosen n or m ali zat i on has con didegree t i on q =k1,as k obt ai n i n g C=1/ k . Hen ce t he pr obabi li t y t o f i n d a degr ee-k n ode at t he en d of a r an dom ly chosen li n k i s qk = kpk , k (5.27) 1-p u u CHOOSE ONE OF THE OUTGOING LINKS OF TAR The m ai n st eps of t he copyi ng m odel. node connect s wi t h pr obabi li t y p t o a r an chosen t ar get node u, or wi t h pr obabi li t y one of t he nodes t he t ar get u poi nt s t o. I wor ds, wi t h pr obabi lt y 1-p t he new node c li nk of i t s t ar get u. Section 9 Copying model (a) Random Connection: with probability p the new node links to u. (b) Copying: with probability we randomly choose an outgoing link of node u and connect the new node to the selected link's target. Hence the new node “copies” one of the links of an earlier node (a) the probability of selecting a node is 1/N. (b) is equivalent with selecting a node linked to a randomly selected link. The probability of selecting a degree-k node through the copying process of step (b) is k/2L for undirected networks. The likelihood that the new node will connect to a degree-k node follows preferential attachment Social networks: Copy your friend’s friends. Citation Networks: Copy references from papers we read. Protein interaction networks: gene duplication, Section 9 Optimization model Section 9 Optimization model Section 9 Optimization model Section 9 Optimization model Section 9 Section 10 Diameter and clustering coefficient Section 10 Diameter Bollobas, Riordan, 2002 Section 10 Clustering coefficient Reminder: for a random graph we have: Crand = <k> ~ N -1 N What is the functional form of C(N)? m (ln N) 2 C= 8 N Konstantin Klemm, Victor M. Eguiluz, Growing scale-free networks with small-world behavior, Phys. Rev. E 65, 057102 (2002), cond-mat/0107607 CLUSTERING COEFFICIENT OF THE BA MODEL C= Nr(◃ ) k(k -1) 2 1 2 C= 2 6 Denote the probability to have a link between node i and j with P(i,j) The probability that three nodes i,j,l form a triangle is P(i,j)P(i,l)P(j,l) The expected number of triangles in which a node l with degree kl participates is thus: Nrl (◃ ) = N N i=1 j =1 ò di ò djP(i, j)P(i,l)P( j,l) We need to calculate P(i,j). Network Science: Evolving Network Models February 14, 2011 CLUSTERING COEFFICIENT OF THE BA MODEL Calculate P(i,j). Node j arrives at time tj=j and the probability that it will link to node i with degree ki already in the network is determined by preferential attachment: æ t ö1/ 2 æ j ö1/ 2 ki (t) = mç ÷ = mç ÷ èiø è ti ø N N m (ln N) 2 C= 8 N N N ò di ò dj (ij) i=1 ki ( j) j åk =m ki ( j) 2mj l l =1 1 m P(i, j) = (ij) 2 2 Where we used that the arrival time of node j is tj=j and the arrival time of node is ti=i m3 Nrl (◃ ) = ò di ò djP(i, j)P(i,l)P( j,l) = 8 i=1 j =1 m3 (ln N) 2 C = 8l kl (k l -1) /2 P(i, j) = mP(k i ( j)) = m - 1 2 j =1 (il) - 1 2 ( jl) - 1 2 m3 = 8l æ N ö1/ 2 Which is the degree of node l kl (t) = mç ÷ è lø at current time, at time t=N There is a factor of two difference... Where does it come from? N di ò i i=1 N dj m 3 ò j = 8l (ln N)2 j =1 Let us approximate: kl (kl -1) » kl 2 = m 2 N l Network Science: Evolving Network Models February 14, 2011 Section 10 Clustering coefficient Reminder: for a random graph we have: Crand = <k> ~ N -1 N What is the functional form of C(N)? m (ln N) 2 C= 8 N Konstantin Klemm, Victor M. Eguiluz, Growing scale-free networks with small-world behavior, Phys. Rev. E 65, 057102 (2002), cond-mat/0107607 Section 11: Summary The network grows, but the degree distribution is stationary. Section 11: Summary The network grows, but the degree distribution is stationary. Section 11: Summary