ネットワーク科学 蔓延する利己主義とScale-Free則

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1. Scale-Free 2 2019 ( ) Scale-Free 1 / 26
2. 1. Best Business Books ’02 ( ) Scale-Free 2 / 26
3. , , , . , , , , . . . ⇒ !! ( ) Scale-Free 3 / 26
4. Many Real Networks , R.Albert,A.-L.Barab´ asi, Rev.Med.Phys. 74, 2002. P(k) ∼ k−γ, 2 < γ < 3. , , , , , AS , WWW, , , , , ⇒ ( ) Scale-Free 4 / 26
5. Technological Nets (Infra.) http://www.nd.edu/∼networks/ppt/SanDiego.ppt/ ( ) Scale-Free 5 / 26
6. Heterogeneous Structure Internet connectivity with selected backbone ISPs Nature 406 (CAIDA) 2000 Java Class Components (JDK1.2) Europhysics Letters 2002 ( ) Scale-Free 6 / 26
7. Social Nets http://www.nd.edu/∼networks/ppt/SanDiego.ppt/ ( ) Scale-Free 7 / 26
8. Bio-Chemical Nets ( ) Scale-Free 8 / 26
9. Metabolic Pathways , , , 2002 ATP( ) ( ) Scale-Free 9 / 26
10. Surfing the p53 Network p53 ⇒ K.W. Kohn, Mol.Bio.Cell 10, 1999 & B. Vogelstein et al., Nature 408, 2000 ( ) Scale-Free 10 / 26
11. Food Web , (C.A. Simenstand et al., Science 200, 1978.) ( ) Scale-Free 11 / 26
12. 2. → , . . . , , , ” (rich get richer)” . , ! ( ) Scale-Free 12 / 26
13. GN BA 1 2 3 4 5 6 7 8 9 10 1 1 2 4 5 6 7 8 : : NEW 9 BA model Krapivsky-Redner’s GN model : 10 NEW : 3 P.L. Krapivsky, S.Redner, F.Leyvraz, Phy.Rev.Lett. 85, 2000, A.-L.Barab´ asi, R.Albert, H.Jeong, Physica A 272, 1999 ( ) Scale-Free 13 / 26
14. ti i t k p(k, ti , t + 1) = k − 1 2t p(k − 1, ti , t) + 1 − k 2t p(k, ti , t), t+1 ti =1 p(k, ti , t + 1) = k − 1 2t t ti =1 p(k − 1, ti , t) + 1 − k 2t t ti =1 p(k, ti , t). p(k) = limt→∞ ti p(k, ti , t)/t ≈ t ti =1 p(k, ti , t)/t , (t + 1)p(k) = k − 1 2t tp(k − 1) + 1 − k 2t tp(k), p(k) = k − 1 k + 2 p(k − 1) = 4 k(k + 1)(k + 2) ∼ k−3. ( ) Scale-Free 14 / 26
15. P(k) Step 0: 0 N0 Step 1: t = 1, 2, 3, . . . , m i Πi ∝ kν i . Step 2: N Step 1 ← → ν = 0 0 < ν < 1 ν = 1 ν > 1 P(k) Ak sublinear linear superlinear BA ( ) Scale-Free 15 / 26
16. : t k Nk (t) k Ak /A(t) dNk (t) dt = Ak−1Nk−1(t) − AkNk (t) A(t) + δk,1, Ak = k A(t) = j≥1 jNj (t) = 2t , N1 = 2t/3 N2 = t/6, . . . Nk = t × pk pk = k − 1 k + 2 × pk−1 = 4 k(k + 1)(k + 2) ≈ k−3, m = 1 BA ( ) Scale-Free 16 / 26
17. Growing Exponential Network(GEN) , ν = 0 Aj = 1 t A(t) = j Nj (t) = t Nk (t + 1) − Nk (t) = Nk−1(t) − Nk (t) t + δk,1. P(k, t) = Nk (t)/t ∂tP(k, t) ∂t = P(k − 1, t) − P(k, t) + δk,1. t → ∞ ∂P(k,t) ∂t = 0 2P(k) − P(k − 1) = δk,1 P(k) = 2−k S.N.Dorogovtsev, J.F.F.Mendes, Evolution of Networks, Oxford University Press, 2003 ( ) Scale-Free 17 / 26
18. Fitness Model G. Bianconi et al., PRL 86, 2001. Bose-Einstein , (MS ) ( ) Scale-Free 18 / 26
19. Duplication-Divergence Model X duplication deletion new link add new hub ramdom selection hub ! ⇒ , , ! R.V.Sol´ e et al., SantaFe Inst. Working Paper, 01-08-041, 2001 ( ) Scale-Free 19 / 26
20. 3. Optimal Topology SF SW ← 0 < λ < 1 → Random (tree) - Pref. (SF) - Forced (star, clique) min E(λ) = λd + (1 − λ)ρ, d def = i<j Dij nC2 /Dmax , ρ def = i<j aij nC2 , k pk H def = − n−1 k=1 pk log pk R.F.iCancho, R.V.Sol´ e, SantaFe Inst. Working Paper 01-11-068, 2001 ( ) Scale-Free 20 / 26
21. Hub Evolution , L W min E = λL + (1 − λ)W , Metropolis SA λ ≈ 0 λ ≈ 1 , (a) regular → (g)∼(j) SF → (l) random ⇒ SF ! N.Mathias, V.Gopal, Phy.Rev. E 63, 2001 ( ) Scale-Free 21 / 26
22. 4. Kmax P(k) = Ck−γ, C 1 = ∞ Kmin P(k)dk = C γ − 1 K1−γ min , 2 ≤ γ ≤ 3 C = (γ − 1)Kγ−1 min N Kmax 1 N = ∞ Kmax P(k)dk = C γ − 1 K1−γ max , Kmax = KminN 1 γ−1 , BA γ = 3 O( √ N) ( ) Scale-Free 22 / 26
23. Kmax α > 0 P(k) = Ce−αk 1 = C ∞ Kmin e−αkdk = C α e−αKmin , C = αeαKmin 1 N = C ∞ Kmax e−αkdk = C α e−αKmax , Kmax = 1 α (αKmin + log N), O(log N). ( ) Scale-Free 23 / 26
24. 5. Degree-degree correlation ¯ knn (k) def = k′ k′P(k′|k) = N i=1 δ(ki , k) j∈∂i kj kNP(k) . Assortative , Disassortative , · ⇒ , ! M.E.J.Newman, PRL 89, 2002 & PRE 67, 2003 ( ) Scale-Free 24 / 26
25. Assortativity Pearson Assortativity: r def = 4M e (kek′ e ) − [ e (ke + k′ e )]2 2M e (k2 e + k′2 e ) − [ e (ke + k′ e )]2 , , ke k′ e e = (i, j) i j , −1 ≤ r ≤ 1 . r > 0 . r , r def = S1Se − S2 2 S1S3 − S2 2 , . Se def = i,j Aij ki kj , S1 def = i ki , S2 def = i k2 i , S3 def = i k3 i . , , 3 . M.E.J.Newman, Networks -Introduction, Oxford Univ. Press, 2010. ( ) Scale-Free 25 / 26
26. ki µ cov(ki , kj ) , r . µ def = ij Aij xi ij Aij = i ki xi i ki = 1 2M i ki xi , cov(xi , xj ) def = ij Aij (xi − µ)(xj − µ) ij Aij = 1 2M ij Aij (xi xj − µxi − µxj + µ2) = 1 2M ij Aij xi xj − µ2 = 1 2M ij Aij xi xj − 1 (2M)2 i,j ki kj xi xj , r def = cov(ki , kj ) cov(ki , ki ) cov(ki , ki ) = ij (Aij − ki kj /2M)ki kj ij (ki δij − ki kj /2M)ki kj . M.E.J.Newman, Networks -Introduction, Oxford Univ. Press, 2010. ( ) Scale-Free 26 / 26