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uuuBAØØØ'''˙˙˙???ØØØ - 上海交通大学数学系math.sjtu.edu.cn/institution/c_s/09-11-15/3-SDh.pdf · 3/16 JJ II J I Back Close ŸŸŸ!!!BAØØØ'''˙˙˙ zzz łcc§Barab´asiÚAlbert3ScienceþuL

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'''uuuBAØØØ©©©���???ØØØ(EEE,,,���äääüüü���êêê¼¼¼óóó���...)

¤¤¤½½½uuu

þþþ°°°���ÆÆÆêêêÆÆÆXXX

>>>eee: [email protected]

2009ccc11���14FFF

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SSSNNNJJJ���

���!!!BAØØØ©©©������zzz

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nnn!!!ÛÛÛ¢¢¢ÃÃÃIIIÝÝÝ���

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���!!!BAØØØ©©©������zzz

�cc§BarabasiÚAlbert3ScienceþuL��mM5�Ø©[1]"Ì�kµ

•JJJÑÑÑJJJ`OOO���������...µµµ /Starting with a small number (m0) of vertices, at

every time step we add a new vertex with m(≤ m0) edges that link the new vertex to m

different vertices already present in the system. To incorporate preferential attachment, we

assume that the probability Π that a new vertex will be connected to a vertex i depends on

the connectivity ki of that vertex, so that Π(ki) = ki/∑

j kj .0¢¢¢SSSþþþm0���:::vvv���ÄÄÄ"

•ÏÏÏ"""ÝÝÝ©©©ÙÙÙÕÕÕáááuuu���mmmµµµ /Because the power law observed for real networksdescribes systems of rather different sizes at different stages of their development, it is expectedthat a correct model should provide a distribution whose main features are independent oftime.0===���(((������...���äääÝÝÝ©©©ÙÙÙAAATTTÕÕÕáááuuu���mmm"

�[Ú²þ|�{(J`²BA�.÷vù��¦"

•ÄÄÄ���ÑÑÑÃÃÃIIIÝÝÝ���äääVVVgggµµµ /This result indicates that large networks self-

organize into a scale-free state.0llldddäääkkk���ÆÆÆÝÝÝ©©©ÙÙÙ������äää¡¡¡���SF���äää"

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���!!!'''uuu���...���½½½

Äk§BollobasÚRiordan[2]'uBA�.Q²kXeµØµ

•BA���...ØØز²²(((µµµ /From a mathematical point of view, however, the de-

scription above, repeated in many papers, does not make sense. The first problem is getting

started. The second problem is with the preferential attachment rule itself, and arises only for

m ≥ 2.0cccöööKKK������äää555���§§§���öööJJJ±±±nnnØØØïïïÄÄÄ"

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HK(HolmeÚKim�p = 1��)�.[6]µ1�^J`ë�§{ö3Ù��Ø­E�

Åë�"Ï�T�.(:iÝ\1�VÇ°(�umΠ(ki)§¤±��CBA�."

•eeeZZZÿÿÿÀÀÀ555���yyy²²²µµµ Ý©Ù­½5µBollobas�<[3]éLCD�.^�Ø�

{¶Cooper�<[4] éWebã�.ÏL�OØ�¶ýƳ�<[7]Úû�Í�<[8]éHK�

.ÏL�OØ�Ú/ÏÄ�Vǧ±9�ä�»Ú­è qyf§��"

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•ÑÑÑWWW���ää䢢¢yyyïïïÄÄĵµµ �lnó��f�Ç�¢yïÄ|±þã*:"

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ooo!!!ÝÝÝ©©©ÙÙÙ���???ØØØ

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�­�5µk�Iݧ6ÄÚuÑ5Æ´�oº�Щ�ä'X��"

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���mmmÕÕÕááá���SF���äää

•ÜÜÜ©©©EEE���OOO������...µµµ KrapivskyÚRedner[12]�éÚ©�JÑ��(Ü©)E�

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J§ã/­Ü�NE��.´²­�SF�ä"

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•CCCm���...µµµ ��Në��O�¯u���O�§DorogovtsevÚMendes[13]J

ÑCm(\�O�)�."b½1i�(:�\�ë�ê�miθ, 0 ≤ θ < 1§ÙÑ{

�BA�.�Ó"n��m:��[(JXeµ

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•���ÚÚÚ���...µµµ Cm�.Ø­½��Ï´ë�êÃ�O�§�¢S�äO�

k�þ�§��m(≥ 2)"·�JÑ���Ú�.§b½1i�(:�\�ë�ê

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¦+ØÓôÚ:`²�m��§�´�ª­½3çÚ:(BA�.)þ"

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•���gggJJJ`OOO������...µµµ Fortunato�<[14]@�#:Ø��U¼��ä�:Ý

ê���&E§¦�JÑ��ÄuÜ©&E��gJ`O��."XJU(:?\

^S5½�g§K#(:Uìc#J`VÇ Πi(t) = (1/i)ν/∑

j(1/j)ν ÀJÎ(:i�

�ë�"(i´�mÏf)

•ccc###JJJ`���...���[[[µµµν = 1�ã/§lþ�e©O�t = 103, 104, 105§ýÿ

Ý©ÙP (k, t) ∼ t−1k−2§�©[14](ØØÓ´�²­SF�ä"§§§­­­½½½���������ÆÆÆ"

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ÊÊÊ!!!(((:::ÝÝÝêêê¼¼¼óóó���...

•���...���½½½Âµµµ �{z§·���ÄOOO������äää"e?Û�.üz5K��

^u�c��ä§Kti��\\�(:i3t��(:ÝKi(t)��àgê¼ó[10]"�Ð

©�äkm0�(:§Tê¼óde¡�Щ©ÙÚ=£VÇ��(½µ1 ÐÐЩ©©©©©ÙÙÙµµµÏØ­Eë�§b½1i�(:�\�Ýê©Ù�αi(h)§Ù¥ 0 ≤

h ≤ i− 1 + m0"2�1i�(:Ý\1�VÇ�fi(k, t)§BA�.fi(k, t) ' mΠ(ki)"

2 ===£££VVVÇÇǵµµlt�t + 1�Ýki(t)�Cz5Æpk,l(i, t) = P{Ki(t + 1) = l|Ki(t) = k}"w,Ý\1 �pk,k+1(i, t) = fi(k, t)¶ÝØC�pk,k(t) = 1− fi(k, t)¶Ù§pk,l(t) = 0"

ê¼óx{Ki(t), i = 1, 2, · · · ; t = i, i + 1, · · · }Ò´O��ä�êÆ�."•äääNNN���~~~fffµµµ ­��±�ÑЩ�ä§]�7L�ÄЩ�ä"

1 ���ÚÚÚ���...µµµαi(h) = δh([m(1−e−ci)]+1)§fi(k, t) = {[m(1−e−ct)]+1}k2

∫ t

0{[m(1−e−cx)]+1}dx

�iÃ'"

2 ccc###JJJ`���...µµµαi(h) = δhm§fi(k, t) = m(1/i)ν∑j(1/j)ν�ik'§�kÃ'"

3 ������555ÝÝÝJJJ`���...µµµαi(h) = δhm§fi(k, t) = mkr∑j kr

j, 0 ≤ r < ∞�iÃ'"�r =

0�§éA�Åë�¶�0 < r ≤ 1�§∑

j krj = µt¶�r > 1�§

∑j kr

j ∝ tr"

4 EEE������...µµµ�Ä\Ýkµαi(h) = δh0§fi(k, t) = (k + 1)/t"ÙÙÙ§§§���...· · · · · ·"

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OOO������äää���­­­½½½555

•ÌÌÌ���§§§ÚÚÚ���©©©���§§§µµµ dê¼ó�ÑP (k, i, t)�Ì�§§2�b½ fi(k, t) ≡f(k, t)§¦Ú��P (k, t)��©�§µ at+1 − at + bt

at

ct= dt §Ù¥at = tP (k, t)§bt =

tf(k, t)§ct = t§dt = tf(k − 1, t)P (k − 1, t) + αt+1(k)"

•]]]���ÝÝÝ©©©ÙÙÙ���OOO���µµµ�6Щ©Ù§¦)�©�§§4íúªXeµ

P (k, t) = 1+m0

t+m0

t−1∏i=1

[1− f(k, i)]

P (k, 1) +t−1∑l=1

(l+m0)f(k−1,l)P (k−1,l)+αl+1(k)

(1+m0)l∏

j=1

[1−f(k,j)]

.

•���©©©���§§§444���½½½nnn[12]µµµ elimt→∞ dt = l, ct+1− ct = 1Úlimt→∞ bt = b ≥ 0§

Klimt→∞at

t = limt→∞(at+1 − at) = l1+b"·�^§y²eã

•üüü���������555���½½½nnnµµµ �äÝ©Ù�3^�§¤�SF�ä�^�"½½½nnn1 elimi→∞ αi(h) = α(h)Úf(k, t) = g(k)O(t−r), r ≥ 1§K�äÝ©Ù�3¶½½½nnn2 e�3�êM¦�α(M) = 0Úlimt→∞ tf(k, t) = Ak + B�A > 0§K�SF�

ä"A = β´ÄåÆ�ê§?�ÚB 6= 0¡£ SF�¶A = 0���Å�"

NNN555µµµûûû���ÍÍÍ���ÇÇÇ^êêê¼¼¼óóóÄÄÄ���VVVÇÇÇ���������aaaqqq(((JJJ"""

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888!!!(((:::êêêêêê¼¼¼óóó���...

•���...���½½½Âµµµ ·�Ó���ÄOOO������äää§e?Û�.üz5K��^u�

c��ä§K3t�Ý�k�(:êN(t) = {Nk(t), k = 1, 2, · · · }��þê¼ó[11]"1 OOOþþþLLL§§§µµµ½ÂOþYk(t) = Nk(t + 1)−Nk(t)§¡Y(t)�N(t)�OþL§"

2 ===£££VVVÇÇǵµµO�^�VÇP{Y(t)|N(t)}§(½lt�t + 1�Nk(t)�Cz5Æ"

•äääNNN���~~~fffµµµ�{B§·���Ä�Åä§=zg\�^ë�"1 BAäääµµµ~XkµP{Y1(t) = 0|N1(t)} = N1(t)/2t¶P{Y1(t) = 1|N1(t)} = 1 −

(N1(t)/2t)"e-ejL«1j�©þ�1Ù{�0§K���

P{Y(t) = ej+1 − ej + e1|N(t)} = j2tNj(t), 1 ≤ j ≤ t.

2 LCDäääµµµ½Â�[3]§[11]O�=£Vǧ���

P{Y(t) = ej+1 − ej + e1|N(t)} = j2t+1Nj(t) + δj1

2t+1 , 1 ≤ j ≤ t.

•ÝÝÝ©©©ÙÙÙ­­­½½½555µµµ y²limt→∞E[Nk(t)]

t = P (k)¿¦ÑP (k)"

1 LCDäääµµµBollobas�<[3]^|Ü�{y²­½5"P (k) = 4k(k+1)(k+2)"

2 BAäääÚÚÚLCDäääµµµ·�[11]ÏLO�^�Ï"E[E[Y(t)|N(t)]]��E[Nk(t)]��©

�§§2|^4�½ny²­½5"ù«�{'|Ü�{{B"

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���ÅÅÅäää���444���½½½nnn

•Polya---���...µµµ ²;�Polya-�.1923cJѧy®í2�2ÂPolya-

½Ã¡Polya-"-f¥Cka�x¥Úb�ç¥"zg�ÅÄ�¥§XÄ�x¥§K

\\1�x¥Úc�祶XÄ�祧K\\d�ç¥ Ø\\x¥"ØÓÄ¥5K

��þ!§ Ð§E���Åä"þãê¼ó�.�ïÄ�Åä�4�½n"•���ÅÅÅäää������êêê½½½ÆÆƵµµ éBA�Åä§Bollobas�<^��Ø�ªy²

Nk(t)�f�ê½Æ¶ Mori[15]K^�Øy²Nk(t)ÚKm(t)�r�ê½Æ"=

limt→∞Nk(t)

t = P (k), a.s.; limt→∞ t−1/2Km(t) = ξ, a.s.§�ÅCþξýéëY"

•���ÅÅÅäää���¥¥¥%%%444���½½½nnnµµµ Mori[15]�^�Øy²eã¥%4�½n∑ki=1

ti[Ni(t)−tP (i)]√t

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•���ÅÅÅäää������   ������nnnµµµ2009cBryc�<[16]|^þãê¼óÚ�©�§éBA�Åäy²eã� ��n

lim sup(inf)t→∞log P{|N1(t)/t−P (1)|≤x}

t = −I(x).

ù�(J`²Ø��êÂñ"é���kXÛy²ºE�)û"

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ÌÌÌ���ëëë���©©©zzz[1] Barabasi A.-L. and Albert R., Science 286, 1999, 509-512[2] Bollobas B and Riordan O M. Mathematical results on scale-free random graphs,

Bornholdt S. and Schuster H. G. (eds), Wiley-VCH, 2002, 1-34[3] Bollobas B. et al., Random Structures and Algorithms 18, 2001, 279-290[4] Cooper C and Frieze A., Random Structures and Algorithms 22, 2003, 311-335[5] Dorogovtsev S. N. et al., Phys. Rev. Lett. 85, 2000, 4633-4636[6] Holme P. and Kim B. J., Phys. Rev. E 65, 2002, 026107[7] Du C. F. and Gong F. Z., Stability of random networks, preprint[8] Hou Z. T. et al., Degree-distribution stability of growing networks, //www.paper.edu.cn[9] Li L., Towards a theory of scale-free graphs, Internet Math. 2, 2005, 431-523[10] Shi D. H., Chen Q. H. and Liu L. M., Phys. Rev. E 71, 2005, 036140[11] Xu H. and Shi D. H., Chinese Phys. Lett. 71, 2009, 038901[12] Krapivsky P. L. and Redner S., Phys. Rev. E 71, 2005, 036118[13] Dorogovtsev S. N. and Mendes J. F. F., Phys. Rev. E 63, 2001, 056125[14] Fortunato S., Flammini A. and Menczer1 F., Phys. Rev. Lett. 96, 2006, 218701[15] Mori T. F., Studia Scientiarum Mathematicarum Hungarica 39 2002, 143-155

[16] Bryc W., Minda D. and Sethuraman S., Appl. Prob. Trust 14, 2009, 1-31

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