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Scale Freeƫģ

ƫģ͵һ㴦Է[18,21]ʷ[20,23]Master[19] ɺ[9]ҲͳѧʹõķһڣΧƫģ͵ʷֲʽԷ ʷMasterڸоеӦá

Scale FreeƫģͣӶĶֵ仯ĽǶȣֵ·̣

$\displaystyle {\frac{{\partial k_{i}}}{{\partial t}}}$ = m$\displaystyle \Pi$$\displaystyle \left(\vphantom{k_{i}}\right.$ki$\displaystyle \left.\vphantom{k_{i}}\right)$ = m$\displaystyle {\frac{{k_{i}}}{{\sum k_{j}}}}$ (28)
ÿһǼmߣ2mֵ $ \sum$kj = 2mt + m0

$\displaystyle {\frac{{\partial k_{i}}}{{\partial t}}}$ = $\displaystyle {\frac{{k_{i}}}{{2t}}}$ (29)
kitiʱ̽ϵͳģʼ ki$ \left(\vphantom{t_{i}}\right.$ti$ \left.\vphantom{t_{i}}\right)$ = m

ki$\displaystyle \left(\vphantom{t}\right.$t$\displaystyle \left.\vphantom{t}\right)$ = m$\displaystyle \left(\vphantom{\frac{t}{t_{i}}}\right.$$\displaystyle {\frac{{t}}{{t_{i}}}}$$\displaystyle \left.\vphantom{\frac{t}{t_{i}}}\right)^{{\frac{1}{2}}}_{}$ (30)
ʽtiķֲͿ֪tʱkiķֲ

P$\displaystyle \left(\vphantom{k_{i}\left(t\right)<k}\right.$ki$\displaystyle \left(\vphantom{t}\right.$t$\displaystyle \left.\vphantom{t}\right)$ < k$\displaystyle \left.\vphantom{k_{i}\left(t\right)<k}\right)$ = P$\displaystyle \left(\vphantom{t_{i}>\frac{m^{2}t}{k^{2}}}\right.$ti > $\displaystyle {\frac{{m^{2}t}}{{k^{2}}}}$$\displaystyle \left.\vphantom{t_{i}>\frac{m^{2}t}{k^{2}}}\right)$. (31)
tʱ̣ ti $ \in$ $ \left[\vphantom{0,t}\right.$0, t$ \left.\vphantom{0,t}\right)$ķֲΪ

P(ti) = $\displaystyle {\frac{{1}}{{m_{0}+t}}}$. (32)
ʽͿԵõkiķֲ

P(k) = $\displaystyle {\frac{{\partial P\left(k_{i}\left(t\right)<k\right)}}{{\partial k}}}$ = $\displaystyle {\frac{{2m^2t}}{{m_{0}+t}}}$$\displaystyle {\frac{{1}}{{k^3}}}$ $\displaystyle \sim$ 2m2k-3(t $\displaystyle \rightarrow$ $\displaystyle \infty$). (33)

(28)һƽϵʵıֻǰ $ \Pi$$ \left(\vphantom{k_{i}}\right.$ki$ \left.\vphantom{k_{i}}\right)$ķֲѡ񣬲ܱ ֤ȷʵÿһĶֵôҲ˳ΪԷԶʵ̷ֲݻҪõֲ p$ \left(\vphantom{k;t_{i},t}\right.$k;ti, t$ \left.\vphantom{k;t_{i},t}\right)$ʾtiʱϵͳĶtʱ̶ֵΪkļʣʼΪ p$ \left(\vphantom{k;t_{i},t_{i}}\right.$k;ti, ti$ \left.\vphantom{k;t_{i},t_{i}}\right)$ = $ \delta$$ \left(\vphantom{k-m}\right.$k - m$ \left.\vphantom{k-m}\right)$Masteṛ

p$\displaystyle \left(\vphantom{k;t_{i},t+1}\right.$k;ti, t + 1$\displaystyle \left.\vphantom{k;t_{i},t+1}\right)$ = p$\displaystyle \left(\vphantom{k;t_{i},t}\right.$k;ti, t$\displaystyle \left.\vphantom{k;t_{i},t}\right)$ + wget$\displaystyle \left(\vphantom{k-1}\right.$k - 1$\displaystyle \left.\vphantom{k-1}\right)$p$\displaystyle \left(\vphantom{k-1;t_{i},t}\right.$k - 1;ti, t$\displaystyle \left.\vphantom{k-1;t_{i},t}\right)$ - wget$\displaystyle \left(\vphantom{k-1}\right.$k - 1$\displaystyle \left.\vphantom{k-1}\right)$p$\displaystyle \left(\vphantom{k;t_{i},t}\right.$k;ti, t$\displaystyle \left.\vphantom{k;t_{i},t}\right)$ (34)
wget$ \left(\vphantom{k}\right.$k$ \left.\vphantom{k}\right)$ = $ {\frac{{k}}{{2t}}}$жĶֲֵвͬʱ̼ϵͳĶƽΪ

P$\displaystyle \left(\vphantom{k}\right.$k$\displaystyle \left.\vphantom{k}\right)$ = $\displaystyle \lim_{{t\rightarrow\infty}}^{}$$\displaystyle {\frac{{\sum_{t_{i}}p\left(k;t_{i},t\right)}}{{t}}}$. (35)
ʽöȷֲ

P$\displaystyle \left(\vphantom{k}\right.$k$\displaystyle \left.\vphantom{k}\right)$ = $\displaystyle {\frac{{2m\left(m+1\right)}}{{k\left(k+1\right)\left(k+2\right)}}}$ $\displaystyle \sim$ k-3. (36)

ʷ̴һֲռtʱ̶ֵΪkĶһţŴСΪ Nk$ \left(\vphantom{t}\right.$t$ \left.\vphantom{t}\right)$¶ϵͳʱ Nkܷı䣬仯·̣

$\displaystyle {\frac{{dN_{k}\left(t\right)}}{{dt}}}$ = mWget$\displaystyle \left(\vphantom{k-1}\right.$k - 1$\displaystyle \left.\vphantom{k-1}\right)$ - mWget$\displaystyle \left(\vphantom{k}\right.$k$\displaystyle \left.\vphantom{k}\right)$ + $\displaystyle \delta_{{k,m}}^{}$. (37)
Wget$ \left(\vphantom{k}\right.$k$ \left.\vphantom{k}\right)$ΪжֵΪkĶµӵļʣ

Wget$\displaystyle \left(\vphantom{k}\right.$k$\displaystyle \left.\vphantom{k}\right)$ = $\displaystyle {\frac{{kN_{k}\left(t\right)}}{{\sum_{d}dN_{d}\left(t\right)}}}$. (38)
Nkͬ㷽 $ \sum_{{k}}^{}$kNk$ \left(\vphantom{t}\right.$t$ \left.\vphantom{t}\right)$ = 2mt + m0

P$\displaystyle \left(\vphantom{k}\right.$k$\displaystyle \left.\vphantom{k}\right)$ = $\displaystyle \lim_{{t\rightarrow\infty}}^{}$$\displaystyle {\frac{{N_{k}\left(t\right)}}{{\sum_{d}N_{d}\left(t\right)}}}$. (39)
һϷ̿Խöȷֲ P$ \left(\vphantom{k}\right.$k$ \left.\vphantom{k}\right)$[20,23]

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wwwwjs 2004-01-04