大学院共通授業科目トポロジー理工学特別講義Ｉネットワークトポロジー複雑ネットワークの統計的性質北海道大学工学研究科応用物理学専攻矢久保考介

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5 years ago
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1 大学院共通授業科目トポロジー理工学特別講義Ｉネットワークトポロジー複雑ネットワークの統計的性質北海道大学工学研究科応用物理学専攻矢久保考介

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7 4 WWW Web

5 type king yellow land race poem history apple MARCELLUS Good now, sit down, and tell me, he that knows, Why this same strict and most observant watch So nightly toils the subject of the land, And

8 5 type king yellow land race poem history apple MARCELLUS Good now, sit down, and tell me, he that knows, Why this same strict and most observant watch So nightly toils the subject of the land, And why such daily cast of brazen cannon, And foreign mart for implements of war; Why such impress of shipwrights, whose sore task Does not divide the Sunday from the week; What might be toward, that this sweaty haste Doth make the night jobjoint-labourer with the day: Who is't that can inform me? HORATIO That can I; At least, the whisper goes so. Our last king, Whose image even but now appear'd to us, Was, as you know, by Fortinbras of Norway, Thereto prick'd on by a most emulate manpride, Dared to the combat; blood in which our valiant Hamlet-- For so this side of our known world esteem'd him-- Did slay this Fortinbras; who by a seal'd compact, Well ratified by law and heraldry, Did forfeit, with his life, all those his lands Which he stood seized of, to the conqueror: Against the which, a moiety competent Was gaged by our king; which had return'd To the inheritance of Fortinbras, Had he been vanquisher; as, by the same covenant, And carriage of the article design'd, His fell to Hamlet. Now, sir, young Fortinbras, Of unimproved mettle hot and full, Hath in the skirts of Norway here and there Shark'd up a list of lawless resolutes, For food and diet, to some enterprise That hath a stomach in't; which is no other-- As it doth well appear unto our state-- But to recover of us, by strong hand And terms compulsatory, those foresaid lands So by his father lost: and this, I take it, Is the main motive of our preparations, The source of this our watch and the chief head Of this post-haste and romage in the land.

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11 1998

13 N n k i <k> 1 k = N d i i k i i 2n = k N j = 2 d ij l <d>

14 C i k i i i i k i : S i ki ( ki 1) Si = k C2 = i 2 E i i C i Ei 2Ei Ci = = S k ( k 1) C C = C i C i i i C=1 C=0

15 N l ln N C 0 C 1

16 k i (A) (B) WWW (C) (M) (NS)

17 ( ) γ P k k ( k >> 1) k =2.1 (WWW) =2.4 (Internet) =2.1 (NS)

19 l 1 l 2 N l D C l N N B l B D B l B N B l B D

20 l B N B N B l B D l B N = 6 B = 4

21 WWW N B l B D "Self-similarity of complex networks" C. Song, S. Havlin, and H. A. Makse Nature 433, 392 (2005).

22 N B l B D

23 N B l B D

24 N B exp( l / l B 0 )

26 Watts-Strogatz 1998 (1) N K (N>>K>>lnN) N=8 K=4 3( K C = 4( K (2) p 2) 1)

27 ) ( 1/ pkn f K N l d = u u u u u u f 4 tanh 4 4 ) ( = >> << = 1 ) / ln( 1 const. ) ( u u u u u f p=0 1) 4( 2) 3( (0) = K K C 3 ) 1 ( p 3 ) (1 1) 4( 2) 3( ) ( p K K p C =

28 K>>1 l C p=1 p=0 4 3 N K K N K N ln ln C l! = = ), ( 0 2 / 2 / 2 / 2 / )! 2 / ( 2) / ( ) (1 ) ( K k f n pk n K k n K n n K e n K k pk p p C k P

29 Barabasi-Albert 1999 (1) m 0 m( m 0 ) (2) ki Π(k i ) = k i i 3/10 2/6 2/10 2/6 3/10 2/6 2/10

30 Barabasi-Albert

31 l ln N ln ln N

32 Song el at. (Nature, 2005) N l D C N B l B l log N D B Yook el at. (PRL, 2005) disassortative Goh el at. (PRL, 2006) Song el at. (Nature Phys., 2006)?

33 (SOC) φ = 0

34 20.0 Average node-pair distance (l NP ) N=2000 N=4000 N=6000 N=8000 N= α =0.26 (= c ) M 0.5 P c (s) α/α c s

35 At = c 10 2 Box-covering 10 3 Cluster-growing 10 2 N B 10 1 N c l B D B = D C =1.77 l c

36 log N B D d log N B D d N B e l B / l 0 log l B log l B

37 SF SW D

38 J. Davidsen et al.: Phys. Rev. Lett. 99, (2002). M. Boguna et al.: Phys. Rev. E 68, (2003). k k d k γ = 1+ D d B k

40 Characterization of complex networks: A survey of measurements L. da F. Costa, F. A. Rodrigues, G. Travieso, and P. R. Villas Boas, Adv. Phys. 56, 167 (2007). The structure and function of complex networks M. E. J. Newman, SIAM Rev. 45, 167 (2003). "Evolution of Networks: From Biological Nets to the Internet and WWW S. N. Dorogovtsev and J. F. F. Mendes (Oxford Univ Pr., 2003). Statistical mechanics of complex networks R. Albert and A. L. Barabasi, Rev. Mod. Phys. 74, 47 (2002). Evolution of networks S. N. Dorogovtsev and J. F. F. Mendes, Adv. Phys. 51, 1079 (2002).

41 N p N (1) (2) (3)

k = The Last Samurai Tom Cruise [1] Oracle Ken Watanabe (I) has a Bacon number of 2. 1: 6(k 6) (small world p

k = The Last Samurai Tom Cruise [1] Oracle Ken Watanabe (I) has a Bacon number of 2. 1: 6(k 6) (small world p The size of the world It is a small world Araseki Hitoshi Can you believe that everyone is at most six steps away from any other person on the Earth? This phenomenon, which is called small world phenomenon,

大学院共通授業科目 トポロジー理工学特別講義 Ｉ ネットワークトポロジー 複雑ネットワークの統計的性質 北海道大学 工学研究科 応用物理学専攻 矢久保 考介

大学院共通授業科目トポロジー理工学特別講義Ｉネットワークトポロジー複雑ネットワークの統計的性質北海道大学工学研究科応用物理学専攻矢久保考介