䝛䝑䝖䝽䞊䜽䛸䛿 䠊䠊䠊኱Ꮫ㝔ඹ㏻ᤵᴗ⛉┠䛂䝖䝫䝻䝆䞊⌮ᕤᏛ≉ูㅮ⩏ 䠥 䠖⛉Ꮫ䛸䝖䝫䝻䝆䞊䛃䝛䝑䝖䝽䞊䜽䝖䝫䝻䝆䞊ᙐᩃȍȃȈȯȸǯƷወᚘႎࣱឋ䝜䞊䝗໭ᾏ㐨኱Ꮫ ᕤᏛ◊✲㝔 ᛂ⏝≀⌮Ꮫ㒊㛛▮ஂಖ ⪃௓䝸䞁䜽ࣀ࣮ࢻ㸸 ሙᡤ䚸ே䚸≀䚸ゝⴥ䚸ᴫᛕ䚸䞉䞉䞉ࣜࣥࢡ㸸 䝜䞊䝗䛸䝜䞊䝗䛾㛵ಀ䝛䝑䝖䝽䞊䜽䛾౛䠍䠖 ᆅୗ㕲䝛䝑䝖䝽䞊䜽ᵝ䚻䛺䝛䝑䝖䝽䞊䜽䝜䞊䝗䠖㥐䝸䞁䜽䠖㊰⥺

䝛䝑䝖䝽䞊䜽䝜䞊䝗䝸䞁䜽䜲䞁䝍䞊䝛䝑䝖䝁䞁䝢䝳䞊䝍㏻ಙᅇ⥺㼃㼃㼃㼃㼑㼎䝨䞊䝆䝸䞁䜽┦஫స⏝㟁Ꮚ䜽䞊䝻䞁┦஫స⏝ேⓗ䝁䝭䝳䝙䜿䞊䝅䝵䞁 ே཭ே㛵ಀ䚸ᩛᑐ㛵ಀ

஺㏻⥙

✵㊰⥺

ᫎ⏬ತඃ䝛䝑䝖䝽䞊䜽ᫎ⏬ತඃඹ₇㛵ಀㄽᩥඹⴭ◊✲⪅ඹⴭ㛵ಀㄽᩥᘬ⏝ㄽᩥᘬ⏝㛵ಀ༢ㄒ䝛䝑䝖䝽䞊䜽༢ㄒᩥ୰䛻㞄᥋䠄ḟ㞄᥋䠅䛧䛶఩⨨䛩䜛䝛䝑䝖䝽䞊䜽䛾౛䠎䠖 䝁䞁䝢䝳䞊䝍䞉䝛䝑䝖䝽䞊䜽䝜䞊䝗䠖䝁䞁䝢䝳䞊䝍䝸䞁䜽䠖ᅇ⥺䝛䝑䝖䝽䞊䜽䛾౛4䠖 WWW
䝜䞊䝗䠖Web䝨䞊䝆䝸䞁䜽䠖䝝䜲䝟䞊䝸䞁䜽䝛䝑䝖䝽䞊䜽䛾౛3䠖 䜲䞁䝍䞊䝛䝑䝖䝜䞊䝗䠖䝁䞁䝢䝳䞊䝍䝸䞁䜽䠖≀⌮ᅇ⥺䝛䝑䝖䝽䞊䜽䛾౛5䠖㓝ẕ䛻䛚䛡䜛䝍䞁䝟䜽㉁㛫┦஫స⏝䝛䝑䝖䝽䞊䜽䝜䞊䝗䠖⺮ⓑ㉁䝸䞁䜽䠖┦஫స⏝䝛䝑䝖䝽䞊䜽䛾౛6䠖 ⛉Ꮫㄽᩥඹⴭ㛵ಀ䝛䝑䝖䝽䞊䜽䝛䝑䝖䝽䞊䜽䛻㛵䛩䜛⌮ㄽⓗᢅ䛔䠙䝜䞊䝗䠖◊✲⪅䝸䞁䜽䠖ඹⴭ㛵ಀ䜾䝷䝣⌮ㄽ䠄䠍䠔ୡ⣖䛾䜸䜲䝷䞊䜢๰ጞ⪅䛸䛩䜛ᩘᏛ䠖䝖䝫䝻䝆䞊䛾㉳※䠅⭾኱䛺䝜䞊䝗ᩘ䜢䜒䛴」㞧䛺䝛䝑䝖䝽䞊䜽」㞧䝛䝑䝖䝽䞊䜽⛉Ꮫ䠄」㞧䝛䝑䝖䝽䞊䜽䛾⤫ィⓗᛶ㉁䜢ゎ᫂䛩䜛䠅
1998䡚䝛䝑䝖䝽䞊䜽䛾୍⯡ⓗᛶ㉁䝛䝑䝖䝽䞊䜽䜢≉ᚩ௜䛡䜛ᵝ䚻䛺㔞䖃䝛䝑䝖䝽䞊䜽ᵓ㐀䛻䝴䞊䜽䝸䝑䝗㊥㞳䛿ᐃ⩏䛥䜜䛶䛔䛺䛔ྠ䛨ᵓ㐀䖃 ⥲䝜䞊䝗ᩘ N䝛䝑䝖䝽䞊䜽䛾䝃䜲䝈䖃 ⥲䝸䞁䜽ᩘ n䖃 䝜䞊䝗䛾ḟᩘ ki
䝜䞊䝗 i 䛛䜙ฟ䛶䛔䜛䝸䞁䜽ᩘ䖃 ᖹᆒḟᩘ <k>k䖃㐃⤖ᛶ䛜ኚ䜟䜜䜀ᵓ㐀䛿ኚ䜟䜛

1
¦ ki
N i

2nk N

䖃 䝜䞊䝗㛫㊥㞳 d␗䛺䜛ᵓ㐀ji䝜䞊䝗ᑐ䜢⤖䜆䝸䞁䜽ᩘ䛾᭱ᑠ್d ij

2㐃⥆ኚᙧ䛧䛶䜒⧅䛞᭰䛘䛜䛺䛡䜜䜀ྠ䛨ᵓ㐀䖃 䝛䝑䝖䝽䞊䜽ᚄ l䝛䝑䝖䝽䞊䜽ෆ䛾඲䛶䛾䝜䞊䝗㛫㊥㞳䛾᭱኱್䝖䝫䝻䝆䞊䖃 ᖹᆒ䝜䞊䝗㛫㊥㞳 <d>䝛䝑䝖䝽䞊䜽ෆ䛾඲䛶䛾䝜䞊䝗㛫㊥㞳䛾ᖹᆒ್䖃 䜽䝷䝇䝍䞊ಀᩘ Cከ䛟䛾⌧ᐇ䝛䝑䝖䝽䞊䜽䛻ඹ㏻䛾ᛶ㉁䝜䞊䝗 i 䛾ḟᩘki䠍䠊䝇䝰䞊䝹䝽䞊䝹䝗ᛶ䝜䞊䝗 i 䛾㏆᥋䝜䞊䝗ᩘkii㏆᥋䝜䞊䝗䛜஫䛔䛻䝸䞁䜽䛩䜛⤌䜏ྜ䜟䛫䛾ᩘ : SiSiiki

C2ki (ki  1)
2㏆᥋䝜䞊䝗䛜ᐇ㝿䛻䝸䞁䜽䛧䛶䛔䜛ᩘ 䠖 Ei

䝜䞊䝗 i 䛾䜽䝷䝇䝍䞊ಀᩘ 䠖 Ci
CiEiSi䝛䝑䝖䝽䞊䜽䝃䜲䝈N䛻ẚ䜉䛶ᖹᆒ䝜䞊䝗㛫㊥㞳䛜ᅽಽⓗ䛻ᑠ䛥䛔
㧗䛔䜽䝷䝇䝍䞊ᛶl v ln N

᏶඲䜾䝷䝣
C=1

2 Eiki (ki  1)䝛䝑䝖䝽䞊䜽䛾䜽䝷䝇䝍䞊ಀᩘ 䠖 CC Ci

C |0䝒䝸䞊ᵓ㐀
C=0C䛿䝛䝑䝖䝽䞊䜽䛜䛹䛾⛬ᗘᐦ䛻⧅䛜䛳䛶䛔䜛䛛䜢⾲䛩㔞䠎䠊䝇䜿䞊䝹䝣䝸䞊ᛶC |1ྵܱƷȍȃȈȯȸǯƷșǭЎࠋḟᩘ ki 䛾ศᕸ㛵ᩘ䛜䝧䜻㛵ᩘP ( k ) v k J(k !! 1)ǹǱȸȫȕȪȸࣱ 䠄≉ᚩⓗ䛺k䛜Ꮡᅾ䛧䛺䛔䠅䃒㻩㻞㻚㻝㻌㻌㻌㻌㻔㼃㼃㼃㻕䃒㻩㻞㻚㻠㻌㻌㻌㻌㻔㻵㼚㼠㼑㼞㼚㼑㼠㻕䃒㻩㻞㻚㻝㻌㻌㻌㻌㻔㻺㻿㻕䞉䞉䞉(A) ತඃ䛾ඹ₇㛵ಀ䚸(B) WWW䚸(C) 㟁ຊ⥙ȏȖನᡯㄽᩥ䛾ඹⴭ㛵ಀ䚹(M)ᩘᏛ䚸(NS)⚄⤒⛉Ꮫ䝧䜻ศᕸᕧ኱䛺䝸䞁䜽ᩘ䜢ᣢ䛴䝜䞊䝗䠄䝝䝤䠅䛜ᑡᩘ䛺䛜䜙Ꮡᅾ䛩䜛䝣䝷䜽䝍䝹ᵓ㐀䛸䝣䝷䜽䝍䝹ḟඖ䠖䠏䠊䝣䝷䜽䝍䝹ᛶࣇࣛࢡࢱࣝ㸻⮬㌟ࡢ୍㒊ศ࡜┦ఝ࡞ᵓ㐀㸦⮬ᕫ┦ఝ㸧㸻≉ᚩⓗ㛗ࡉࡀ↓࠸ᵓ㐀

l1

N v l DCl2䠄䜽䝷䝇䝍䞊ᡂ㛗ἲ䠅䝃䜲䝈 l 䛾⟽䛻ྵ䜎䜜䜛㉁㔞 N 䠄

N B v l B DBᗘ䠅䠄䝪䝑䜽䝇䞉䜹䜴䞁䝔䜱䞁䜾ἲ䠅䝃䜲䝈 lB 䛾⟽䛷⿕そ䛩䜛䛾䛻ᚲせ䛺᭱ᑠ䛾⟽䛾ᩘ NBlB⮬↛⏺䛾⮳䜛䛸䛣䜝䛻㐢ᅾD 䞉䞉䞉 䝣䝷䜽䝍䝹ḟඖ䠄㠀ᩚᩘ䠅㌟䛾ᅇ䜚䛾」㞧䝛䝑䝖䝽䞊䜽䛿䝣䝷䜽䝍䝹䛛䠛

N vlDCWWW䛸ತඃ䛾䝛䝑䝖䝽䞊䜽l v ln N (SW)䝛䝑䝖䝽䞊䜽㊥㞳 lB 䛾䝃䝤䜾䝷䝣䛷⿕そ䛩䜛䛾䛻ᚲせ䛺䝃䝤䜾䝷䝣䛾᭱ᑠᩘ NBN B v l B DN B v l B D䝣䝷䜽䝍䝹lBNB䝣䝷䜽䝍䝹䞉䝛䝑䝖䝽䞊䜽

64"Self-similarity of complex networks"
C. Song, S. Havlin, and H. A. Makse
Nature 433, 392 (2005).

⺮ⓑ㉁┦஫స⏝䝛䝑䝖䝽䞊䜽ከ䛟䛾⌧ᐇ䝛䝑䝖䝽䞊䜽䛻ඹ㏻䛾ᛶ㉁䠍䠊䝇䝰䞊䝹䝽䞊䝹䝗ᛶ䠎䠊䝇䜿䞊䝹䝣䝸䞊ᛶ䠏䠊䝣䝷䜽䝍䝹ᛶ

N B v l B D඲䛶䛾⌧ᐇ䝛䝑䝖䝽䞊䜽䛜䛣䜜䜙䛾ᛶ㉁䜢
ඹ᭷䛧䛶䛔䜛䜟䛡䛷䛿䛺䛔䟿ఱᨾ䚸䛣䛾䜘䛖䛺ඹ㏻ᛶ䛜Ꮡᅾ䛩䜛䛾䛛䠛䝣䝷䜽䝍䝹䞉䝛䝑䝖䝽䞊䜽䝇䝰䞊䝹䝽䞊䝹䝗ᛶ䛾㉳※䜽䝷䝇䝍䞊ಀᩘ

Watts-Strogatz 䠄1998䠅

p=0䛷䛾䜽䝷䝇䝍䞊ಀᩘつ๎ᛶ䛸䝷䞁䝎䝮ᛶ䛾୧᪉䜢ྲྀ䜚ධ䜜䛯䝛䝑䝖䝽䞊䜽䞉䝰䝕䝹䛻䜘䜚኱䛝䛺䜽䝷䝇䝍䞊ಀᩘ䛸ᑠ䛥䛺䝛䝑䝖䝽䞊䜽ᚄ䜢୧❧䛥䛫䛯䚹(1)ึ䜑䛻Nಶ䛾䝜䞊䝗䛛䜙䛺䜛䝸䞁䜾≧䛾᱁Ꮚ䜢⪃䛘䚸䝜䞊䝗㛫䛿Kಶ䛾᭱㏆᥋(N>>K>>lnN)䜎䛷⤖ྜ䛧䛶䛔䜛䚹䠄つ๎ᛶ䠅
N=83( K  2)
K=4C4( K  1)

C ( 0)

3( K  2)
4( K  1)

䛾䠏ゅᙧ䛜䠏ゅᙧ䛾䜎䜎䛷䛔䜙䜜䜛☜⋡(1  p ) 3

C ( p)
ᖹᆒ䝜䞊䝗㛫㊥㞳(2) ☜⋡p䛷䝷䞁䝎䝮䛻䝸䞁䜽䛾⧅䛞᭰䛘䜢䛩䜛䚹䛯䛰䛧䚸஧㔜䝸䞁䜽䜔⮬ᕫ䝸䞁䜽䜢⚗Ṇ䛩䜛䚹䠄䝷䞁䝎䝮䝛䝇䠅lf (u )42u  4u3( K  2)(1  p)34( K  1)

Nf ( pKN )Ktanh 1u2u  4u

f (u )

䝇䝰䞊䝹䝽䞊䝹䝗ᛶ­ const. u  1®
¯ln(u ) / u u !! 1

K>>1

p=0

p=1

3
4
NKKN
ln N
ln KCl䝇䜿䞊䝹䝣䝸䞊ᛶ䛾㉳※Barabasi-Albert 䝰䝕䝹䠄1999䠅ᡂ㛗䛸ඃඛⓗ㑅ᢥ䛾䝯䜹䝙䝈䝮䛻䜘䜚䝇䜿䞊䝹䞉䝣䝸䞊ᛶ䜢᭷䛩䜛䝛䝑䝖䝽䞊䜽䞉䝰䝕䝹䛾ᵓ⠏䛻ᡂຌ䚹(1) ึ䜑䛻m0 ಶ䠄ᑡᩘ䠅䛾䝜䞊䝗䛜Ꮡᅾ䛧䛶䛔䜛䚹ྛ᫬㛫䝇䝔䝑䝥䛷䠍ಶ䛾䝜䞊䝗䜢ຍ䛘䜛䚹䛣䛾䝜䞊䝗䛿m(ӌm0)ᮏ䛾䝸䞁䜽䜢ᣢ䛳䛶䛚䜚䚸᪤Ꮡ䛾䝜䞊䝗䛾䛹䜜䛛䛸⤖ྜ䛩䜛䚹䠄ᡂ㛗䠅ḟᩘศᕸ㛵ᩘP(k )f (k ,K )¦n 0

K /2Cn (1  p ) n p K / 2 n

( pK / 2) k  K / 2 n  pK / 2e(k  K / 2  n)!

(2) ᪂䛧䛟ຍ䜟䜛䝜䞊䝗䛜᪤Ꮡ䛾䝜䞊䝗䛸䝸䞁䜽䛷⤖䜀䜜䜛☜⋡䛿䚸

3 (ki )

䝷䞁䝎䝮䞉䝛䝑䝖䝽䞊䜽䛸
ᮏ㉁ⓗ䛻ྠ䛨

ki
¦ ki

2/6
3/10

2/10i

䛷୚䛘䜙䜜䜛䚹䠄ඃඛⓗ㑅ᢥ䠅኱䛝䛺 C 䛸ᑠ䛥䛺 l !

3/10
2/6

2/6
2/10

䖃 䝛䝑䝖䝽䞊䜽ᚄBarabasi-Albert 䝰䝕䝹l|ln N
ln ln N䠄䝇䝰䞊䝹䝽䞊䝹䝗ᛶ䠛䠅䖃 䜽䝷䝇䝍䞊ಀᩘ䝷䞁䝎䝮䞉䝛䝑䝖䝽䞊䜽䛸ྠ⛬ᗘ⌧ᐇ䛾䝛䝑䝖䝽䞊䜽䛸୙୍⮴䝇䜿䞊䝹䞉䝣䝸䞊ᛶP ( k ) v k J : J䝝䝤ᵓ㐀3䛥䜙䛺䜛䝛䝑䝖䝽䞊䜽䞉䝰䝕䝹䛾ᚲせᛶ䝣䝷䜽䝍䝹ᛶ䛾㉳※䝣䝷䜽䝍䝹ᵓ㐀

Song el at. (Nature, 2005)」㞧䝛䝑䝖䝽䞊䜽䛻䝣䝷䜽䝍䝹ᛶ䛜䛒䜛䛣䛸䜢ⓎぢN v l DC䠄䝣䝷䜽䝍䝹ᛶ䠅l v log N䠄䝇䝰䞊䝹䝽䞊䝹䝗ᛶ䠅⮫⏺⌧㇟≉ᚩⓗ㛗䛥䛾↓䛔ᨭ㓄᪉⛬ᘧ

ᖹ⾮⣔⮫⏺⌧㇟㠀ᖹ⾮⣔ ⮬ᕫ⤌⧊໬⮫⏺ᛶ (SOC)'I 0N B v l B DB 䠄䝪䝑䜽䝇䞉䜹䜴䞁䝔䜱䞁䜾䠅

Yook el at. (PRL, 2005)
䝣䝷䜽䝍䝹䝛䝑䝖䝽䞊䜽䛿disassortative

Goh el at. (PRL, 2006)
䝣䝷䜽䝍䝹䞉䝛䝑䝖䝽䞊䜽䛾䝇䜿䝹䝖䞁ᵓ㐀䛿⮫⏺ⓗSong el at. (Nature Phys., 2006)
䝰䞊䝆䝳䞊䝹䛾ධ䜜Ꮚᵓ㐀䝣䝷䜽䝍䝹ᛶ䛾㉳※䛿?

䝣䝷䜽䝍䝹ᛶ䛸䝇䜿䞊䝹䝣䝸䞊ᛶ䜢᭷䛩䜛䝰䝕䝹䝣䝷䜽䝍䝹䞉䝛䝑䝖䝽䞊䜽䛾ᙧᡂ䖃 (u,v)-flower 䝰䝕䝹䠄2007䠅

䖃 Song-Havlin-Makse 䝰䝕䝹䠄2006䠅

u+vᮏ䛾䜶䝑䝆䛛䜙ᡂ䜛䝃䜲䜽䝹䞉䜾䝷䝣ᫍᆺ䜾䝷䝣ḟᩘk䛾䝜䞊䝗䛻mkಶ䛾䝜䞊䝗䜢⧅䛢䜛1ୡ௦๓䛾䜶䝑䝆䜢䛿䛪䛧ᪧ䝜䞊䝗䛻㞄᥋䛩䜛䝜䞊䝗ྠኈ䜢⧅䛢䜛ྛ䜶䝑䝆䜢u+vᮏ䛾䜶䝑䝆䛛䜙ᡂ䜛䝃䜲䜽䝹䞉䜾䝷䝣䛻⨨䛝᥮䛘䜛௙௙䠄䛯䛰䛧䚸ʹ ൑ ‫ ݑ‬൑ ‫ݒ‬䠅」㞧䝛䝑䝖䝽䞊䜽䛾⛉Ꮫ䛾ᒎ㛤䛭䛾௚䛾ၥ㢟䖃 SF䚸SW䚸䝣䝷䜽䝍䝹ᛶ䛾┦஫㛵ಀ䖃 ⮬ᕫ⤌⧊໬⮫⏺䛻䜘䜛」㞧䝛䝑䝖䝽䞊䜽ᙧᡂఱᨾ⌧ᐇ䛾䝛䝑䝖䝽䞊䜽䛜䝣䝷䜽䝍䝹ᵓ㐀䜢ྲྀ䜛䛛䠛䝇䜿䞊䝹䝣䝸䞊ᛶ䖃 ᦆയ䛻䜘䜛䝛䝑䝖䝽䞊䜽䛾㡹ᙉᛶ䞉⬤ᙅᛶ䖃 㔜䜏௜䛡䜙䜜䛯䝛䝑䝖䝽䞊䜽䛾ᛶ㉁ྠ᫬ᡂ❧䛩䜛䝰䝕䝹䛿䠛⯟✵㊰⥺䝛䝑䝖䝽䞊䜽ᰴ౯䛾┦㛵䝛䝑䝖䝽䞊䜽⾑⟶⥙䞉䞉䞉䞉䃒䛸D䛻㛵ಀ䛜䛒䜛䛛䝇䝰䞊䝹䝽䞊䝹䝗ᛶ䝣䝷䜽䝍䝹ᛶ䜽䝻䝇䜸䞊䝞䞊䖃 䝛䝑䝖䝽䞊䜽ୖ䛾≀⌮შ䛾⶝ᘏ䛚㔠䛾ὶ㏻

ᣑᩓၥ㢟ୡㄽᙧᡂ䝇䝢䞁ၥ㢟ཧ⪃᭩୍⯡᭩䛂᪂䝛䝑䝖䝽䞊䜽ᛮ⪃ ୡ⏺䛾䛧䛟䜏䜢ㄞ䜏ゎ䛟䛃䝞䝷䝞䝅 䠄᪥ᮏᨺ㏦ฟ∧༠఍䠅䛂」㞧䛺ୡ⏺䚸༢⣧䛺ἲ๎ 䝛䝑䝖䝽䞊䜽⛉Ꮫ䛾᭱๓⥺䛃䝬䞊䜽䞉䝤䜻䝱䝘䞁 䠄ⲡᛮ♫䠅䛂⚾䛯䛱䛿䛹䛖䛴䛺䛜䛳䛶䛔䜛䛾䛛 䝛䝑䝖䝽䞊䜽䛾⛉Ꮫ䜢ᛂ⏝䛩䜛䛃ቑ⏣┤⣖䠄୰බ᪂᭩䠅䛂䛺䛬3ே䛔䜛䛸შ䛜ᗈ䜎䜛䛾䛛䛃ቑ⏣┤⣖䠄᪥⤒䝥䝺䝭䜰䝅䝸䞊䝈䠅䝺䝫䞊䝖ㄢ㢟䠖( , )-flower䝰䝕䝹䛾䝇䜿䞊䝹䝣䝸䞊ᣦᩘ 䛸䝣䝷䜽䝍䝹ḟඖ ௙ 䛜䛭䜜䛮䜜䚸䜔䜔ᑓ㛛ⓗ䛂」㞧䝛䝑䝖䝽䞊䜽䛸䛭䛾ᵓ㐀䛃▮ஂಖ⪃௓ 䠄ඹ❧ฟ∧, 2013䠅䛂」㞧䝛䝑䝖䝽䞊䜽䠖ᇶ♏䛛䜙ᛂ⏝䜎䛷䛃ቑ⏣┤⣖䚸௒㔝⣖㞝 䠄㏆௦⛉Ꮫ♫, 2010䠅

ᑓ㛛ⓗ“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).

௙䛷୚䛘䜙䜜䜛䛣䛸䜢♧䛫䚹䝠䞁䝖䠖 䛂」㞧䝛䝑䝖䝽䞊䜽䛸䛭䛾ᵓ㐀䛃 ▮ஂಖ⪃௓ 䠄ඹ❧ฟ∧䠅