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'''uuuBAØØØ©©©���???ØØØ(EEE,,,���äääüüü���êêê¼¼¼óóó���...)
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SSSNNNJJJ���
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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"
4/16
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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"
•m ≥ 2XXXÛÛÛJJJ`ëëë���LCD(linearized chord diagram)�.[3]µæ^Ü¿(:§�#Ngë�ÚEë�"Webã�.[4](�©áÚ�.[5])µkÀ½m�(:,�2ë�§�#NEë�"
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§��"
5/16
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nnn!!!ÛÛÛ¢¢¢ÃÃÃIIIÝÝÝ���
Ùg§\³nóÆ�Li�<[9]@�=±Ý©Ù´�Æ��ÃIÝ�½ÂئÜ
n§Ï�Ý©ÙØU���L�äÿÀ"
•ÃÃÃIIIÝÝÝ���IIIÝÝÝ´LLLµµµ du�Æ´p�C©Ù§ék�Ó�ÆÝ©Ù��
䧦�½Â���ïþA½�äIݧÝ�þ"-D = {d1, d2, · · · , dN}L«(:ê�N��äÝS�§G(D)L«duë�ØÓ äk�ÓÝS���ä�N"é?
¿A½�äg ∈ G(D)§O�s(g) =∑
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•���ÝÝÝëëëeee���KKK���µµµ ?Ø>I�§oâz§ÌK(Motif)�ÃIݧÝ
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•ÑÑÑWWW���ää䢢¢yyyïïïÄÄĵµµ �lnó��f�Ç�¢yïÄ|±þã*:"
6/16
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ooo!!!ÝÝÝ©©©ÙÙÙ���???ØØØ
·�@�BA�.�äÝ©ÙÕáu�m�´n�z�¹§ù��¦éNõ¢S
�äÚ�.�äÑÃ{÷v§7L�Ä�ä5�"
•AAA������...���äää���[[[µµµ éuL3Phys. Rev.þA��.�ä?1õ��m:
�[§·�uy���õõõêêê���äääÝÝÝ©©©ÙÙÙ´���mmm������������ÆÆƧ=P (k, t) = C(t)k−γ(t)§¢S
�äÚ�.�äÑ�A^§��Ý©Ùx3Véê�Iþ¤�^��"©aXeµ���mmmÕÕÕááá���SF���äääµµµP (k, t) = Ck−㧧§°°°(((���555ÝÝÝJJJ`§§§XXXEEE������..."""���²²²���SF���äääµµµP (k, t) = Ctαk−㧧§α > 0ÃÃÃ444���(CCCm)¶¶¶α < 0kkk444���(ccc###)"""
ªªªuuu½½½���SF���äääµµµP (k, t) → Ck−㧧§ìììCCC���555ÝÝÝJJJ`§§§XXX���ÚÚÚ���..."""
•üüü���ããã���êêê¼¼¼LLL§§§µµµ ·�Ú\µ(:Ý�Ùgê¼ó[10]µ{Ki(t), i =
1, 2, · · · ; t = i, i + 1, · · · }¶(:ê�þê¼ó[11]µN(t) = {Nk(t), k = 1, 2, · · · }"•nnn������nnnØØدKKKµµµ §�´Ý©ÙP (k, t)�½5µP (k)�3íº´Ä
kP (k) = Ck−γº(:êN(t)�4�½nµNk(t)%CtP (k)�1�XÛº��ÝKm(t)
��5µk�Iݧ6ÄÚuÑ5Æ´�oº�Щ�ä'X��"
7/16
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���mmmÕÕÕááá���SF���äää
•ÜÜÜ©©©EEE���OOO������...µµµ KrapivskyÚRedner[12]�éÚ©�JÑ��(Ü©)E�
�.µ(1)�ÅÀJ��Î(:E��#(:¶(2)#(:��Î(:ë�^�¶(3)#
(:�E�Î(:���P(:�ë�"¦�í�\Ý©ÙPi(k, t) ∼ k−2§�.�
ä�²þë�êUéêO�"555¿¿¿µµµEEE���ÅÅÅ������ddduuuÝÝÝJJJ`"""
•ÚÚÚ©©©������¢¢¢yyyïïïÄÄĵµµ�>ã/´�110cPhysical ReviewØ©§ÚOz�©
Ù�کꧧ�NÚ©ê�éêO�A5"m>ã/´t = 104Út = 105��[(
J§ã/Ü�NE��.´²�SF�ä"
8/16
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���²²²���SF���äää(1)
•CCCm���...µµµ ��Në��O�¯u���O�§DorogovtsevÚMendes[13]J
ÑCm(\�O�)�."b½1i�(:�\�ë�ê�miθ, 0 ≤ θ < 1§ÙÑ{
�BA�.�Ó"n��m:��[(JXeµ
•���²²²���SF���äääµµµCm�.�Ý©Ù�P (k, t) ∼ tαk−γ§Ù¥α = 2θ/(1 −θ)Úγ = (3− θ)/(1− θ)§¡��²�SF�ä"ã/�N§´Ø½�SF�ä"
9/16
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ªªªuuu½½½SF���äää
•���ÚÚÚ���...µµµ Cm�.ؽ��Ï´ë�êÃ�O�§�¢S�äO�
k�þ�§��m(≥ 2)"·�JÑ���Ú�.§b½1i�(:�\�ë�ê
�[m(1− e−ri)] + 1§Ù{�BA�.�Ó"n��m:��[(JXeµ
•���mmm���������SF���äääµµµ�Ú�.Ý©Ù�P (k, t) ∼ C(t)k−γ(t) −→ 2m2k−3§
¦+ØÓôÚ:`²�m��§�´�ª½3çÚ:(BA�.)þ"
10/16
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���²²²���SF���äää(2)
•���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"
11/16
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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§§§���...· · · · · ·"
12/16
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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"""
13/16
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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"
14/16
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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!§ ЧE���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
⇒ N(0, σ2k)§Ù¥ti ´?¿k��½¢ê§⇒L�©ÙÂñ"
•���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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