Rate Equation Approach to Growing Networks

Transcription

1 Rate Equation Approach to Growing Networks Sidney Redner, Boston University Motivation: Citation distribution Basic Model for Citations: Barabási-Albert network Rate Equation Analysis: Degree and related distributions Global properties Finiteness & fluctuations Who is the leader? Protein Interaction Network: Cluster size distribution Lack of self averaging Outlook Paul Krapivsky (Boston University) Francois Leyvraz (CIC, Mexico) Geoff Rodgers (Brunel University, UK) Jeenu Kim & Byungnam Kahng (SNU)

2 Citation Distribution ISI: papers cites, n = paper cited 8907 times 64 papers >1000 times 282 papers >500 times 2103 papers >200 times papers <10 times papers 0 times! PRD: papers cites, n = P(n) PRD ISI slope n=number of cites J. Laherrere and D. Sornette, EPJB (1998) Redner, EPJB (1998)

3 Barabási-Albert Model nodes publications links citations Introduce nodes one at a time. 2. Attach to earlier node with k links at rate. H. A. Simon, Biometrica (1955) A. L. Barabási and R. Albert, Science (1999)

4 Rate Equation Approach KRL, PRL (2000), KR, PRE (2001) see also, Albert & Barabási, Rev. Mod. Phys. (2002) Dorogovtsev & Mendes, Adv. Phys. (2002) Basic Observable: N k Number of nodes with k links The degree distribution. Rate Equation: dn k dt = 1N k 1 N k A + δ k1. Attachment Rate: so that k γ A(t) = A j N j = j=1 j γ N j M γ (t). j=1

5 Moment equations: Ṁ 0 k Ṅk = 1; Ṁ 1 k kṅk = 2 These suggest: A(t) = j γ N j µ(γ)t N k (t) tn k. Rate eqs. Linear recursion relations Formal Solution: k (1 + µaj ) 1 n k = µ j=1 Asymptotics: k γ exp [ ( )] k µ 1 γ 2 1 γ 1 γ, 0 γ < 1; n k k ν, ν > 2, γ = 1; best seller 1 < γ < 2; bible γ > 2.

6 Heterogeneity Bianconi & Barabási, (2000); KR (2002). Each node has intrinsic attractiveness η and attachment rate (η). Rate equation: dn k (η) dt = 1(η)N k 1 (η) (η)n k (η) A + p 0 (η)δ k1. Solution for linear kernel (η) = ηk: n k (η) = µ p 0(η) η ( ) Γ(k) Γ 1 + µ η ( ). Γ k µ η Asymptotics for total degree distribution: Bounded support of p 0 (η): n k k (1+µ/ηmax) (lnk) ω, with µ determined by 1 = dη p 0 (η) ( µ η 1 ) 1. Unbounded support: condensation!

7 Age Distribution: KR, PRE (2001). N k (t, a) : # nodes of degree k, age a, time t. Rate Equation: ( t + ) N k = 1N k 1 N k a A + δ k1 δ(a). Age-Degree Distribution: = k : N k (t, a) = 1 a t ( 1 1 a t ) k 1 = 1 : ln(1 a/t) k 1 N k (t, a) = (1 a/t). (k 1)! Average Age: = k : a k t(1 12/k 2 ) = 1 : a k = t[1 (2/3) k ]. Average Degree: = k : k a (1 a/t) 1/2 = 1 : k a = ln(1 a/t).

8 Age-degree statistics: a k /t =1 =k k =1 =k k a a/t Message: = k, rich nodes must be old. = 1, rich nodes can be young.

9 Degree Correlations: C kl (t) number of nodes of degree k that attach to an ancestor node of degree l. Rate equation (linear kernel): k l dc kl dt = 1 { [(k 1)C k 1,l kc kl ] A } + [(l 1)C k,l 1 lc kl ] + (l 1)C l 1 δ k1. (i) (ii) (iii) (iv) (v) Asymptotic solution: (k, l 1 and k/l 1) c kl { 16 (l/k 5 ) l k, 4/(k 2 l 2 ) l k. c kl n k n l (k l) 3!

10 In- and Out-Components: out-component (cite tree from my paper) x in-component (cite tree to my paper) I s (t) No. of in-components with s nodes t s 2. O s (t) No. of out-components with s nodes (lnt)s s!.

11 Degree Distribution of Finite Networks: Dorogovtsev et al PRE ( 01), Moreira et al cond-mat , KR ( 02). n k N=10 2 N=10 3 N=10 4 N=10 5 N=10 6 =k degree k N=10 2 N=10 3 N=10 4 N=10 5 n k =k+λ λ= degree k

12 Scaling Near the Extreme: F(k/N 1/2 ) 2 1 N=10 2 N=10 3 N=10 4 N=10 5 N=10 6 =k k/n 1/2 F(k/N 1/(2+λ) ) N=10 2 N=10 3 N=10 4 N=10 5 =k+λ λ= k/n 1/(2+λ)

13 Who is the most popular node? leader index J =1 =k time Using J = t a kmax, results for k max & a kmax : For = k, leader among the oldest! J 1.9. P , P , P , For = 1, leader index J(t) t ψ, with ψ = 1 + ln(2/3)/ ln

14 How many lead changes occur? # lead changes =1 =k time Basic feature: Number of lead changes up to time t is proportional to lnt.

15 Protein Interaction Network addition β/ν new node duplication 1- δ Duplication: A new node duplicates a random pre-existing node by connecting to each of its neighbors with probability 1 δ. Addition: A new node links to any previous node with probability β/n. Uetz et al., Nature (2000). Wagner, PNAS (1994). Vazquez et al., cond-mat/ (2001). Solé et al., Adv. Complex Systems (2002).

16 Rate Equation (equiprob. node selection): K 3 R (2002) dn k dn = 1N k 1 N k N + G k. Attachment Rate: = (1 δ)k }{{} duplication + β }{{} addition Source : G k = a+b=k s=a (1 δ) γ 1 n k. ( ) s n s (1 δ) a δ s a a }{{} duplication: a links β b b! e β }{{} addition: b links n k = N k /N.

17 Average node degree: In each event, number of links evolves as dl dn = β + (1 δ) 2L N, Combining with D(N) = 2L(N)/N, the average node degree D is D(N) = finite δ > 1/2, β lnn δ = 1/2, const. N 1 2δ δ < 1/2. The degree distribution (for δ < 1/2): The rate equation is a recursion. Substituting n k k γ determines γ via: γ = δ (1 δ)γ 2 γ δ

18 Addition Only: A non-local percolation. Rate equation for cluster size distr. C s dc s dn = β sc s β n + n }{{ N } n! e β n=0 s 1 s n j=1 loss by linking s j C sj N } {{ } gain by n body merging sum s 1 1,..., s n 1, with s s n + 1 = s. Generating function: Define g(z) = 1 sc se sz, with c s = C s /N g = βg + (1 + βg ) e z+β(g 1)., Basic results: s = g (0) = 1 2β 1 4β 2β 2 ; c s As τ, τ = 1 + G(β) e π/ 4β β ;

19 Mean cluster size s, percolation prob. G s G(x10) β Cluster size distribution: β < 1 4 : non-universal power law τ rapidly decreasing in β; τ 3 as β 1 4. β = 1 4 : logarithmic correction c s 8s 3 (lns) 2.

20 Duplication Only: Rate equation for complete duplication dn k dn = (k 1)[N k 1 N k ]. N Solution: N k = 2 ( ) 1 2 k 1 N irrelevant! Instead: Strong sample-specific fluctuations No self-averaging. Asymptotic behavior: K 1,1 evolves to K n,m. Each state {n, m} (with n + m = N) occurs with uniform probability.

21 K n,m K n+1,m prob. n n+m K n,m+1 prob. m n+m n sites degree m m sites degree n

22 Failure of scaling: If isolated sites created, they evolve independently and with strong sample-specific fluctuations. Simple example: Start with K 1,1,0 ( ). After N sites, with complete duplication: P(N 0, N) = 2 N 1 N 0 (N 1)(N 2) 2/N 2/N 2 P(N 0,N) N 0 N N 0 = N 3, while (N 0) mp 1. Incomplete duplication: any isolated sites created will evolve independently of N k>0!

23 Outlook The Rate Equation! Simple yet powerful tool. Basic Messages Degree distribution easily computable: Power law not generic or robust. Stretched exponential is robust. Heterogeneity, age distribution, correlations, global features, extremes, etc. Fluctuations unresolved. Other Growth Mechanisms: Biological processes. New percolation process. Self averaging can fail. Errors for robust behavior.