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ͼоҪߡģһڣѾ˽⣬n G$ \left(\vphantom{n,M}\right.$n, M$ \left.\vphantom{n,M}\right)$ G$ \left(\vphantom{n,p}\right.$n, p$ \left.\vphantom{n,p}\right)$ֱʿռ䡣 N = C2n G$ \left(\vphantom{n,M}\right.$n, M$ \left.\vphantom{n,M}\right)$CMN ܵ磬еȼʣ G$ \left(\vphantom{n,M}\right.$n, M$ \left.\vphantom{n,M}\right)$Ϊʿռ䡣 H $ \in$ G$ \left(\vphantom{n,M}\right.$n, M$ \left.\vphantom{n,M}\right)$ļΪ

P(GM = H) = (CMN)-1. (23)

G$ \left(\vphantom{n,p}\right.$n, p$ \left.\vphantom{n,p}\right)$ʵ2Nпܵ磬ӿͼȫͼ e$ \left(\vphantom{H}\right.$H$ \left.\vphantom{H}\right)$ΪHı H $ \in$ G$ \left(\vphantom{n,p}\right.$n, p$ \left.\vphantom{n,p}\right)$ļΪ

P(Gp = H) = pe$\scriptstyle \left(\vphantom{H}\right.$H$\scriptstyle \left.\vphantom{H}\right)$$\displaystyle \left(\vphantom{1-p}\right.$1 - p$\displaystyle \left.\vphantom{1-p}\right)^{{N-e\left(H\right)}}_{}$. (24)

ͳѧϵ۵ĽǶǿʿռ G$ \left(\vphantom{n,p}\right.$n, p$ \left.\vphantom{n,p}\right)$ǰ e$ \left(\vphantom{H}\right.$H$ \left.\vphantom{H}\right)$ͼHĺ ϵͳ΢״̬ʿռ G$ \left(\vphantom{n,p}\right.$n, p$ \left.\vphantom{n,p}\right)$eķֲΪ

P(e) = CeNpe$\displaystyle \left(\vphantom{1-p}\right.$1 - p$\displaystyle \left.\vphantom{1-p}\right)^{{N-e}}_{}$. (25)
Ϊe΢ͼҲɸʿռ G$ \left(\vphantom{n,e}\right.$n, e$ \left.\vphantom{n,e}\right)$ÿһͼȼʡͼ G$ \left(\vphantom{n,p}\right.$n, p$ \left.\vphantom{n,p}\right)$ϵۣ G$ \left(\vphantom{n,e}\right.$n, e$ \left.\vphantom{n,e}\right)$Ϊ΢ϵۡ

ϵCeNϣͼͳѧвͬĴʽͼ۶ͼ̬ͬϵ--ͼ V$ \left(\vphantom{G_{1}}\right.$G1$ \left.\vphantom{G_{1}}\right)$ V$ \left(\vphantom{G_{2}}\right.$G2$ \left.\vphantom{G_{2}}\right)$ ֮ڿӳ $ \phi$ : V$ \left(\vphantom{G_{1}}\right.$G1$ \left.\vphantom{G_{1}}\right)$ $ \rightarrow$ V$ \left(\vphantom{G_{2}}\right.$G2$ \left.\vphantom{G_{2}}\right)$ʹö $ \forall$x, y $ \in$ V$ \left(\vphantom{G_{1}}\right.$G1$ \left.\vphantom{G_{1}}\right)$ xy $ \in$ E$ \left(\vphantom{G_{1}}\right.$G1$ \left.\vphantom{G_{1}}\right)$ $ \phi$$ \left(\vphantom{x}\right.$x$ \left.\vphantom{x}\right)$$ \phi$$ \left(\vphantom{y}\right.$y$ \left.\vphantom{y}\right)$ $ \in$ E$ \left(\vphantom{G_{2}}\right.$G2$ \left.\vphantom{G_{2}}\right)$ԸͼH൱ڸH ԼH̬ͬͼͳѧ൱ڶ㲻ֵʱṹΪͬΨһ̬ͬͼͬ΢ ״̬ļ򲢶ȡǶڶֵ磬ṹͬ£᷽ͬʽڲͬȻֵֵ塣Ŀǰ оʵϵͳ㶼ǿֵģDz̬ͬ

˵ͼ G$ \left(\vphantom{n,p}\right.$n, p$ \left.\vphantom{n,p}\right)$QָͼĸʿռУƽQ Ķֲֵȶÿһ΢ͼͳ䶥ֲֵɣȻղͬ΢ͼֵļһƽͿԽ⡣ ǣʹ΢ϵ G$ \left(\vphantom{n,e}\right.$n, e$ \left.\vphantom{n,e}\right)$еҲڲͬĶֲֵʽʵʵһ򵥡ǻһƵ ϵ۷ֲ(25)ʽɼֲֵDZƽֵ $ \bar{{e}}$ = NpҶֲ˥ķdz죬ˣ Գڷֲֵ΢ͼƵƽͱ΢ϵͼ G$ \left(\vphantom{n,Np}\right.$n, Np$ \left.\vphantom{n,Np}\right)$Ķֲֵʽ

±£nеȼʵѡȡ $ {\frac{{1}}{{2}}}$n(n - 1)pߣƽ϶㱻ѡȡķֲʽȣij һ㱻ѡȡļ$ \rho$ڵһߵѡȡʱÿһ(n - 1)ԱѡȡıߣѡȡļͬΪÿѡȡ һ߲㣬һ㱻ѡȡļ $ \rho$ = $ {\frac{{2}}{{n}}}$p $ \ll$ 1ѡȡı߲ӰĶٴαѡ ȡļʡ˼Ϊ$ \rho$Ķ㱻ѡȡĽ¼ɸʿռ䣬ֲƽֵΪ $ \lambda$ = $ \rho$ x $ {\frac{{1}}{{2}}}$n(n - 1)p = (n - 1)p $ \doteq$ np IJɷֲ

P(k) = e-np$\displaystyle {\frac{{\left(pn\right)^{k}}}{{k!}}}$. (26)

һѧķ[1]Ǽͼ G$ \left(\vphantom{n,p}\right.$n, p$ \left.\vphantom{n,p}\right)$УһеܴһijĶֵ ĺ壬

P(ki = k) = Ckn-1pk(1 - p)n-1-k. (27)
еkiƵͳƣnܴʱֲ(26)ʽɷֲ档

Ȼȷֱֲ˵ķʽDz㹻ͼ㣬ȻͳơնֲļֻҪͿ õȽ׼ȷʡͼ(5)ͼֻһ΢ͼ㡣ʵõһݽзʱһֱ ͳQǷڣҪͬϵۡͳʵʲһϵͳһġǵ ں״̬΢״̬ĹϵĻϵģǶʵʲҲֻҪһϵͳͿˡ


Fig 5: Ķȷֲͼϸɵһ磬N=20000p=0.0025 ΪϵIJɷֲ

оһҪͳѧϵһҲõоԭ֮һ ΪŲpͻԣںܶļQֻе p > pQcŻ֣չֳڶΪ ŵԴСͼһͨͼΪͼļţжijΪšоpֵ ԴСӰ졣ŴСΪSԴСΪ s = $ {\frac{{S}}{{n}}}$Կ p $ \approx$ 1 С൱ p $ \approx$ 0෴ô0 < p < 1أобΪݻʡ

оЩҪõͼݻйɵĸʿռ $ \tilde{{G}}^{{n}}_{}$Ϊӿͼȫͼͼ $ \tilde{{G}}$ : = G$ \left(\vphantom{n,0}\right.$n, 0$ \left.\vphantom{n,0}\right)$, G$ \left(\vphantom{n,1}\right.$n, 1$ \left.\vphantom{n,1}\right)$,..., G$ \left(\vphantom{n,N}\right.$n, N$ \left.\vphantom{n,N}\right)$ɵĵȼʿռ䣬 G$ \left(\vphantom{n,t}\right.$n, t$ \left.\vphantom{n,t}\right)$ʾtʱ ̵ͼҲʾtߵͼ n $ \rightarrow$ $ \infty$ʱռԿɸ G$ \left(\vphantom{n,M}\right.$n, M$ \left.\vphantom{n,M}\right)$ռ϶ɣ߿ɲ ͬpֵ G$ \left(\vphantom{n,p}\right.$n, p$ \left.\vphantom{n,p}\right)$ռ϶ɵĿռ䡣sͿĸʿռеõоţ p < psc = $ {\frac{{1}}{{n}}}$ʱڴģţ p = psc = $ {\frac{{1}}{{n}}}$ʱŴСΪ S $ \sim$ n$\scriptstyle {\frac{{2}}{{3}}}$ p > psc = $ {\frac{{1}}{{n}}}$ʱŴСS $ \sim$ n

Ϥͳ۵Ļͻ׷۵ϵʵDZͬһ飬ֻӲͬĽǶȽо ֪Уp = pcͨŵĹģչʷֲͬУҲʽĹģֲ

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wwwwjs 2004-01-04