BA模型的数学基础

1316

t+1 Pi (t + 1)
Ô Ç
i
Vol.30A
£¨ ½Ê i
½ º Ç HK ¸ Í ¡ Å BA Û ¦ ¡ ¤½ Ê i ¶ à t º k (t), ¶ £ ½ ÊÄÃ Í ÅÒ Ç ½ mΠ(k (t)). ½ ½ Ê i t + 1 ¶ È t + 1 × ½ ÊÄÃ Ì £ Í ÅÒ £
0
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 depends on the connectivity ki of that vertex, so that Π(ki ) = ki / j kj . After t steps the model leads to a random network with t + m0 vertices and mt edges”. [3] ¡ BA Å ¡ γ = 2.9 ± 0.1. [4] ¡ · ¡ ÅÁ¾ Å 2 −3 P (k ) ∼ 2m k , γ = 3. Á ¼
Þ Ï
Î¥ Û × Û¦ Þ Ï Â Ó ¼¶ Ç ¨ » ¿ Ú Ì
∂ki ki = mΠ(ki ) = , ∂t 2t
i
º Ô ¨ » Û¦
É ¿Õ¿
ki (i) = m ,
i
Þ [4] ³ ° k Å ½ Ê i ÄÃ Ì £ Í ÅÒ © ¦ Ç ½ mΠ(k ) (° Ú µ º mΠ - ³ °), ϶ ½ Ê i Å k (t) ° £ ½ Ê À m » à £ BA Û ¦ Å ­ Ý Ï ¡ mΠ - ³ °§ · ¶ £Ê ¡ Krapivsky Ç , Ë ½ Ê i Å k (t), t¶ k Å ½ Ê N (t), Äà ¼
BA
¦Ð ¾ ÆÓ ¨ ¬ ¢Ú¿ BA Ó Ü§ Å «Æ Ö ÅØ¢ Æ ¤ Â Æ Ö ¼ ¤ Á · Æ » ÏÆ £ «¢ ·¾Ü ­Æ Ó § ܧ¤ Þ Ö¦BA ܧ©Ó © © ¤ MR(2000) © Ü ¦05C80; 60J10 ªÜ ߦO231.5 ¬¯Ò¥¢¦A ¬ Ñߦ1003-3998(2010)05-1313-09
½
Ï

½

am,d = Bollob´ as
2m(m + 1) . (d + m)(d + m + 1)(d + m + 2)
Ç [9] Õ Á¾ Ú Õ Å ¾¤
n→∞ n m
lim E [#n m (d)]/n = am,d .
m,d
½Ð Ç ¡ # (d)/n Ò Î ½ a . ¹½ BA Û ¦ Ò Ï Å ª § ª ¡Ù à ºË Å£ Õ Õ Â Æ ¡º Ú ℄ ÐÉ Æ £ 1 À m ≥ 2 ¶ ¡ Æ ¹ Ä × ¾ ÊÄÃ Í ÅÒ Ç ½ mΠ? 2 BA Û ¦ Å ¤ ¡ Å »¿· Ð ª À ¿ ·¡ ¼¡ ¼ § » À × ÐÉ © ¶ £
2
[ ª Ï Â©Å ·£
¿
¸
ÛÝÓØ ­ Ú Û¡ÑÓ È ×ÐÉ£Holme § Kim Ⱦ Í ¡ (º HK ¸Í ¡ ): À £ ½ Ê ² ¶ ¡È Ì °¸ Ò Π × Å½Ê Ç Í£ ¥¿ m − 1 ÌÍ °Ò p È ÌÍ ÏÅ½Ê È Í ¡ °Ò 1 − p Å ½ Ê Ï¸ Í £ ¡Ñ Ó p = 1 Å Ç ¯ Ä ¡Í ¡ ¥È Ì °¸ Ò Π × Å ½ Ê Ç Í £¥ ¿ m − 1 Ì Í È Ì Í Ï ½ Ê È Ï¶ Í £ ½À Ç Å Í ¡ ¡Â Ä Ú ¾¤ £
=
(kl (t) − 1)!/[(m − 2)!(kl (t) − m + 1)!] m−1 = , kl (t)!/[(m − 1)!(kl (t) − m + 1)!] kl (t)
½ ½ Ê l Å O (t) Ï m − 1 × ½ Ê Ï¶ ͼ ¶ ¡ Ï ½ Ê i ÅÒ £ ÕÅÚÉ Ñ Ó È ¾ÅÈ 1 × ÐÉ ¡ ­ × Ç Å HK ¸ Í ¡ ¿¸ Ý mΠ - ³ ° Å Í » Ç £ Ù¾ ¬ BA Û¦ Î ª Å Ä × £ Û ¡Ù¾ × Ç ÎÅ BA Û ¦ £½ º × º m × ½ ÊÅ ¡¥ Å N = m (m − 1), À¢ ½ Ê −m , · · · , −1, ­Õ Å Æ Ï Ñ Ó ¿ ³ ° mΠ - ³ ° » Ë £ ½ Û ¬ BA Û¦¡ » ­ ± Æ £ Dorogovtsev Ç Ï¡ k (t), P (k, i, t) := P {k (t) = k } ½ ½ Ê i ¶ à t k ÅÒ £ Šβ ¶Ã t ÏØ½Ê Å ¡
i [5] i k
dNk (t) (k − 1)Nk−1 (t) − kNk (t) =m + δkm , dt k kNk (t)
¥Ï Á³¶ mΠ - ³°£  ° ÄÃ × ½ P (k) Å´ » ÂÓ Õ
«¾Ê
Î ¼¡
P (k ) =
Å Ì ¸ Ú ¡ É ¶ Î Ç N t(t) → P (k), ÄÃ m = 1 ¶ ¡ BA Û ¦ Î Å¿Õ
i
Pi (t + 1) = =
ki (t) + j kj (t) ki (t) + j kj (t)
l∈Oi (t)
m− 2 kl (t) Ckl (t)−1 m− 1 j kj (t) Ckl (t)
l∈Oi (t)
m−1 =m j kj (t)
ki (t) , j kj (t)
¥Ï
m− 2 Ck l (t)−1 m− 1 Ck l (t) l
¿´
P (k ) =
2m(m + 1) . k (k + 1)(k + 2)
­ É ¾Å ¿ ¡ ¹
¯ Í ¡ mΠ - ³ °
­
£
No.5 Dorogovtsev Ç
[7]
¬ Á Í Ç¥ BA Û ¦ Å ºÒ Ǫ¡ Ò° ŠΪ ² ¶´ t » ÊŲ á £
¹ £À
1315
2010,30A(5):1313–1321

¤ צ Õ∗ Ç (È ½Ä Ä ÄÅ ÄÆ ¹ 410075) Âà (È ½Ä Ä ÄÅ ÄÆ ¹ 410075) ¿ ( ½Ä ÄÀ 200444) º (Á » ½Ä¾É ¼ÄÀ Á ) ± ²·³µ¶ ´ 100 ¸°
[1] [2] [1] [3]
1314

−γ
Ô Ç
Vol.30A
Ò Ô ±» Ê ¡ ¥ k , ¥ Ï γ ¿ ¡Û Ò Å¸ £Â Ó Î ² k Å ½ Ê Ê P (k) Ò Å £ ¿ À × Ý ¡Â Ó È ¾ ų» Í¡ ℄ Ù Å BA Û ¦ £ BA Û ¦ Ú ¥ “· · · starting with a small number (m ) of vertices,
Bollob´ as[8] ¥ “From a mathematical point of view, however, the description 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. In order to prove results about the BA model, one must first decide on the details of the model itself. It turns out to be convenient to allow multiple edges and loops”. ¡ ¾ LCD ¡ “We start with the case m = 1. Consider a fixed sequence of vertices v1 , v2 , · · · . We t shall inductively define a random graph process {Gt 1 }t≥0 so that G1 is a directed graph on 1 {vi : 1 ≤ i ≤ t}, as follows. Start with G0 1 the “graph” with no vertices, or with G1 the graph t−1 t with one vertex and one loop. Given G1 , form G1 by adding the vertex vt together with a single edge directed from vt to vi , where i is chosen randomly with dGt−1 (vs ) , 1 ≤ s ≤ t − 1, 1 2t − 1 Π(i = s) = 1 , s = t. 2t − 1
k
4 . k (k + 1)(k + 2)